Complex numbers
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[{"Name":"Complex numbers","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"7m 51s","ChapterTopicVideoID":4925,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4925.jpeg","UploadDate":"2020-09-29T10:46:07.4170000","DurationForVideoObject":"PT7M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"We\u0027re going to start a new subject called complex numbers."},{"Start":"00:03.960 ","End":"00:09.360","Text":"We\u0027re going to extend the whole concept of number as we know it up till now."},{"Start":"00:09.360 ","End":"00:13.710","Text":"Up to now, the only kind of numbers we came across were"},{"Start":"00:13.710 ","End":"00:18.360","Text":"called real numbers and they are numbers on the number line,"},{"Start":"00:18.360 ","End":"00:23.785","Text":"which include whole numbers, negative numbers, fractions,"},{"Start":"00:23.785 ","End":"00:31.010","Text":"and even irrational numbers like Pi and the square root of 2 and e and so on."},{"Start":"00:31.010 ","End":"00:33.365","Text":"But they\u0027re all on the number line."},{"Start":"00:33.365 ","End":"00:36.400","Text":"Now we\u0027re going to go outside the box,"},{"Start":"00:36.400 ","End":"00:38.990","Text":"so beyond the number line."},{"Start":"00:38.990 ","End":"00:42.680","Text":"The problem begins that we\u0027re limited when we"},{"Start":"00:42.680 ","End":"00:46.265","Text":"don\u0027t know what the square root of a negative number is."},{"Start":"00:46.265 ","End":"00:49.415","Text":"For example, what is the square root of minus 1?"},{"Start":"00:49.415 ","End":"00:52.430","Text":"Up till now, we said,"},{"Start":"00:52.430 ","End":"00:55.415","Text":"\u0027\u0027Well, we can\u0027t have a square root of a negative number.\u0027\u0027"},{"Start":"00:55.415 ","End":"01:04.085","Text":"But supposing that we invent a new entity and we give it a name,"},{"Start":"01:04.085 ","End":"01:06.800","Text":"call it the letter i, and say,"},{"Start":"01:06.800 ","End":"01:10.820","Text":"the square root of minus 1 is i. i is a new number."},{"Start":"01:10.820 ","End":"01:12.545","Text":"It\u0027s not on the number line."},{"Start":"01:12.545 ","End":"01:13.759","Text":"It\u0027s not a variable,"},{"Start":"01:13.759 ","End":"01:15.920","Text":"even though it is a letter,"},{"Start":"01:15.920 ","End":"01:17.501","Text":"it\u0027s a number,"},{"Start":"01:17.501 ","End":"01:19.820","Text":"and we say it\u0027s the square root of minus 1."},{"Start":"01:19.820 ","End":"01:22.475","Text":"If we continue with this,"},{"Start":"01:22.475 ","End":"01:28.610","Text":"we can develop a whole mathematics which is consistent and which is actually useful,"},{"Start":"01:28.610 ","End":"01:30.815","Text":"but that the use comes later on."},{"Start":"01:30.815 ","End":"01:33.890","Text":"Meanwhile, let\u0027s just take this as an axiom of"},{"Start":"01:33.890 ","End":"01:37.415","Text":"faith and say that there is a square root of minus 1."},{"Start":"01:37.415 ","End":"01:42.695","Text":"It\u0027s a new number and it\u0027s called i. i stands by the way, for imaginary."},{"Start":"01:42.695 ","End":"01:45.170","Text":"We\u0027re actually going to develop, first of all,"},{"Start":"01:45.170 ","End":"01:50.660","Text":"imaginary numbers, and then we\u0027ll go onto complex numbers."},{"Start":"01:50.660 ","End":"01:57.320","Text":"Imaginary numbers are something times i."},{"Start":"01:57.320 ","End":"02:03.815","Text":"3i would be an imaginary number,"},{"Start":"02:03.815 ","End":"02:09.755","Text":"minus 8i, 1.74i."},{"Start":"02:09.755 ","End":"02:13.670","Text":"Even, say square root of 2 times i,"},{"Start":"02:13.670 ","End":"02:18.110","Text":"where the number here is a real number."},{"Start":"02:18.110 ","End":"02:23.970","Text":"The numbers that we know up to now and all these are called imaginary numbers."},{"Start":"02:24.220 ","End":"02:26.795","Text":"They can be drawn."},{"Start":"02:26.795 ","End":"02:28.130","Text":"There is a diagram,"},{"Start":"02:28.130 ","End":"02:32.720","Text":"but I don\u0027t feel it\u0027s necessary to do that."},{"Start":"02:32.720 ","End":"02:36.650","Text":"They actually can go from the number line to the number plane,"},{"Start":"02:36.650 ","End":"02:40.096","Text":"and we might do that later on but meanwhile not."},{"Start":"02:40.096 ","End":"02:45.590","Text":"One of the things that this gives us is that we can now solve"},{"Start":"02:45.590 ","End":"02:52.845","Text":"the equations like x squared equals minus 1."},{"Start":"02:52.845 ","End":"02:59.510","Text":"If x squared is minus 1 then just like we would do with real numbers, we say,"},{"Start":"02:59.510 ","End":"03:05.737","Text":"the next is plus or minus the square root of minus 1,"},{"Start":"03:05.737 ","End":"03:11.855","Text":"and then we can say that x is plus or minus I. Yeah."},{"Start":"03:11.855 ","End":"03:16.200","Text":"Minus i is also an imaginary number, it\u0027s minus 1i."},{"Start":"03:17.410 ","End":"03:23.825","Text":"For example, if I wanted to know x squared is minus 9,"},{"Start":"03:23.825 ","End":"03:31.160","Text":"then I could say that x is plus or minus the square root of minus 9."},{"Start":"03:31.160 ","End":"03:34.040","Text":"If the usual rules are still going to hold,"},{"Start":"03:34.040 ","End":"03:36.709","Text":"this is going to be plus or minus."},{"Start":"03:36.709 ","End":"03:41.330","Text":"Since minus 9 is 9 times i will"},{"Start":"03:41.330 ","End":"03:46.280","Text":"get plus or minus the square root of 9 times the square root of minus 1,"},{"Start":"03:46.280 ","End":"03:49.750","Text":"which is plus or minus 3i."},{"Start":"03:49.750 ","End":"03:52.520","Text":"So much for imaginary numbers."},{"Start":"03:52.520 ","End":"03:56.645","Text":"Now, let\u0027s introduce complex numbers."},{"Start":"03:56.645 ","End":"04:06.395","Text":"Now, complex numbers are just a combination of a real number plus an imaginary number."},{"Start":"04:06.395 ","End":"04:12.740","Text":"For example, 3 plus 4i is a complex number,"},{"Start":"04:12.740 ","End":"04:14.435","Text":"a typical complex number."},{"Start":"04:14.435 ","End":"04:17.464","Text":"In general, a complex number,"},{"Start":"04:17.464 ","End":"04:22.850","Text":"we could define as a plus bi is"},{"Start":"04:22.850 ","End":"04:29.180","Text":"the general complex number where a and b are real numbers,"},{"Start":"04:29.180 ","End":"04:32.160","Text":"the numbers that we know up to now,"},{"Start":"04:32.450 ","End":"04:42.200","Text":"and i has the property that i equals the square root of minus 1,"},{"Start":"04:42.200 ","End":"04:44.150","Text":"and that\u0027s complex numbers."},{"Start":"04:44.150 ","End":"04:47.360","Text":"Now, let\u0027s learn how to work with them."},{"Start":"04:47.360 ","End":"04:49.695","Text":"How to add, multiply, subtract,"},{"Start":"04:49.695 ","End":"04:53.195","Text":"divide, take exponents of, and so on."},{"Start":"04:53.195 ","End":"04:58.020","Text":"Let\u0027s do that one operation at a time."},{"Start":"04:58.090 ","End":"05:03.830","Text":"Before that, this thing is so important that I feel I\u0027ve got a highlight it."},{"Start":"05:03.830 ","End":"05:07.190","Text":"I\u0027m also going to highlight the fact that,"},{"Start":"05:07.190 ","End":"05:12.965","Text":"in other words, this says that i squared is minus 1."},{"Start":"05:12.965 ","End":"05:16.044","Text":"Let\u0027s also highlight that."},{"Start":"05:16.044 ","End":"05:17.660","Text":"That\u0027s the basis for everything."},{"Start":"05:17.660 ","End":"05:21.275","Text":"A complex number is a plus bi,"},{"Start":"05:21.275 ","End":"05:28.400","Text":"where i is this new mathematical entity and let\u0027s just go along with it."},{"Start":"05:28.400 ","End":"05:32.480","Text":"People used to say that negative numbers don\u0027t exist."},{"Start":"05:32.480 ","End":"05:36.170","Text":"They were almost imaginary but we got used to"},{"Start":"05:36.170 ","End":"05:40.790","Text":"them and negative numbers seem to you just as valid as any other."},{"Start":"05:40.790 ","End":"05:45.590","Text":"After a while, we\u0027ll get used to the fact that there\u0027s a new kind of number called i."},{"Start":"05:45.590 ","End":"05:49.969","Text":"The name imaginary stuck,"},{"Start":"05:49.969 ","End":"05:54.785","Text":"but it doesn\u0027t mean that they are really imaginary."},{"Start":"05:54.785 ","End":"05:56.330","Text":"It depends on your perspective."},{"Start":"05:56.330 ","End":"06:00.620","Text":"Anyway, let\u0027s start with addition and subtraction."},{"Start":"06:00.620 ","End":"06:05.840","Text":"Addition and subtraction work in the obvious way that you might expect."},{"Start":"06:05.840 ","End":"06:07.895","Text":"Let\u0027s just make up an example."},{"Start":"06:07.895 ","End":"06:17.565","Text":"Suppose I have one complex number is 3 plus 4i just like I took it from here."},{"Start":"06:17.565 ","End":"06:20.265","Text":"I\u0027ll just put it in brackets for emphasis."},{"Start":"06:20.265 ","End":"06:22.550","Text":"I want to add another complex number."},{"Start":"06:22.550 ","End":"06:27.655","Text":"Let\u0027s say 7 minus 2i."},{"Start":"06:27.655 ","End":"06:31.695","Text":"You do what you would expect to do where you would just"},{"Start":"06:31.695 ","End":"06:36.870","Text":"add the 3 to the 7 and say that\u0027s 10."},{"Start":"06:36.870 ","End":"06:40.005","Text":"Then 4i minus 2i,"},{"Start":"06:40.005 ","End":"06:41.895","Text":"4 minus 2 is 2,"},{"Start":"06:41.895 ","End":"06:45.645","Text":"so it\u0027s plus 2i."},{"Start":"06:45.645 ","End":"06:49.240","Text":"Same thing works for subtraction."},{"Start":"06:49.240 ","End":"06:52.205","Text":"Just make an example up."},{"Start":"06:52.205 ","End":"06:56.225","Text":"Minus 5, I don\u0027t know plus 2i."},{"Start":"06:56.225 ","End":"07:01.685","Text":"Let say less minus"},{"Start":"07:01.685 ","End":"07:11.340","Text":"8 minus 7i."},{"Start":"07:11.340 ","End":"07:12.720","Text":"Again, just the obvious."},{"Start":"07:12.720 ","End":"07:19.170","Text":"Minus 5 take away minus 8 is minus 5 plus 8 is 3,"},{"Start":"07:19.170 ","End":"07:24.915","Text":"and 2i minus minus 7i is 2i plus 7i."},{"Start":"07:24.915 ","End":"07:28.595","Text":"This is the answer, just the obvious."},{"Start":"07:28.595 ","End":"07:33.515","Text":"I\u0027m not going to write a general rule for a plus bi plus c plus di."},{"Start":"07:33.515 ","End":"07:38.120","Text":"I could, but it\u0027s just the obvious thing to do."},{"Start":"07:38.120 ","End":"07:44.060","Text":"Let\u0027s take a break now and after the break,"},{"Start":"07:44.060 ","End":"07:51.120","Text":"we\u0027ll continue with the next topic which is multiplication of complex numbers."}],"ID":4918},{"Watched":false,"Name":"Multiplication","Duration":"7m 22s","ChapterTopicVideoID":4926,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4926.jpeg","UploadDate":"2020-09-29T10:50:03.7000000","DurationForVideoObject":"PT7M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.620","Text":"Here we are after the break."},{"Start":"00:01.620 ","End":"00:03.990","Text":"We are continuing with complex numbers."},{"Start":"00:03.990 ","End":"00:07.590","Text":"We learned what they are and we learned about addition and"},{"Start":"00:07.590 ","End":"00:11.520","Text":"subtraction and we\u0027re about to do multiplication."},{"Start":"00:11.520 ","End":"00:17.080","Text":"Is just something I forgot to mention earlier that when we write a complex number,"},{"Start":"00:17.080 ","End":"00:19.240","Text":"a plus bi,"},{"Start":"00:19.240 ","End":"00:21.455","Text":"then there\u0027s a name."},{"Start":"00:21.455 ","End":"00:31.530","Text":"This part here is called the real part and the bi part is called the imaginary part."},{"Start":"00:31.930 ","End":"00:37.115","Text":"Another thing is that when we work with real numbers,"},{"Start":"00:37.115 ","End":"00:40.910","Text":"the typical variable is the letter x."},{"Start":"00:40.910 ","End":"00:43.279","Text":"But when we work with complex numbers,"},{"Start":"00:43.279 ","End":"00:46.100","Text":"then we would use the letter z."},{"Start":"00:46.100 ","End":"00:52.615","Text":"You might say that z is equal to a plus bi."},{"Start":"00:52.615 ","End":"00:57.215","Text":"Z posed to x gives us a hint that we\u0027re talking about complex numbers."},{"Start":"00:57.215 ","End":"01:00.580","Text":"Now about that multiplication. Let\u0027s see."},{"Start":"01:00.580 ","End":"01:05.125","Text":"Let\u0027s start off with something easy."},{"Start":"01:05.125 ","End":"01:15.130","Text":"Let\u0027s say we have 4 times 3 plus 2i."},{"Start":"01:15.130 ","End":"01:17.530","Text":"This is a real number,"},{"Start":"01:17.530 ","End":"01:19.315","Text":"which is also a complex number,"},{"Start":"01:19.315 ","End":"01:21.085","Text":"times another complex number."},{"Start":"01:21.085 ","End":"01:23.430","Text":"In this case, it\u0027s just straightforward,"},{"Start":"01:23.430 ","End":"01:32.160","Text":"4 times 3 is 12 and 4 times 2i is 8i so the answer is 12 plus 8i."},{"Start":"01:32.160 ","End":"01:33.550","Text":"Let\u0027s make it a little bit more difficult."},{"Start":"01:33.550 ","End":"01:36.985","Text":"Let\u0027s take an imaginary times a general complex number."},{"Start":"01:36.985 ","End":"01:38.910","Text":"I\u0027ll take the same 3 plus 2i,"},{"Start":"01:38.910 ","End":"01:45.980","Text":"but suppose I had 4i here times 3 plus 2i,"},{"Start":"01:45.980 ","End":"01:47.785","Text":"what would it be?"},{"Start":"01:47.785 ","End":"01:53.290","Text":"As usual, the distributive law applies and we bracket so meaning, this times this,"},{"Start":"01:53.290 ","End":"02:00.255","Text":"this times this, so 4i times 3 is 4 times 3 times i is 12i."},{"Start":"02:00.255 ","End":"02:04.035","Text":"Then 4i times 2i,"},{"Start":"02:04.035 ","End":"02:05.940","Text":"4 times 2 is 8,"},{"Start":"02:05.940 ","End":"02:11.620","Text":"so it\u0027s plus 8i^2."},{"Start":"02:11.620 ","End":"02:16.675","Text":"But it doesn\u0027t end here because remember the most important thing about"},{"Start":"02:16.675 ","End":"02:23.465","Text":"complex numbers is that i^2 is minus 1 or that i is the square root of minus 1."},{"Start":"02:23.465 ","End":"02:27.155","Text":"I replace i^2 by minus 1,"},{"Start":"02:27.155 ","End":"02:29.910","Text":"not 1 minus 1."},{"Start":"02:29.910 ","End":"02:36.700","Text":"This is equal to 12i minus 8."},{"Start":"02:36.700 ","End":"02:39.290","Text":"But when we write complex numbers,"},{"Start":"02:39.290 ","End":"02:42.140","Text":"it\u0027s customary to write the real part first,"},{"Start":"02:42.140 ","End":"02:47.700","Text":"so I write the answer as minus 8 plus 12i."},{"Start":"02:48.440 ","End":"02:51.680","Text":"Now let\u0027s go to the most general case,"},{"Start":"02:51.680 ","End":"02:53.615","Text":"a complex times a complex."},{"Start":"02:53.615 ","End":"02:56.975","Text":"Let\u0027s take, I don\u0027t know,"},{"Start":"02:56.975 ","End":"03:07.570","Text":"minus 5 plus 4i and let\u0027s keep the same 3 plus 2i and see what we get this time."},{"Start":"03:07.570 ","End":"03:11.360","Text":"Remember when we have a sum of two things times the sum of two things,"},{"Start":"03:11.360 ","End":"03:13.460","Text":"we just multiply everything in"},{"Start":"03:13.460 ","End":"03:18.260","Text":"this bracket with everything in this bracket and sometimes we put little arcs."},{"Start":"03:18.260 ","End":"03:21.260","Text":"This with this, this with this, this with this,"},{"Start":"03:21.260 ","End":"03:25.630","Text":"this with this, the order doesn\u0027t matter. But let\u0027s see."},{"Start":"03:25.630 ","End":"03:31.590","Text":"Minus 5 times 3 is minus 15,"},{"Start":"03:31.590 ","End":"03:38.430","Text":"minus 5 with 2i is minus 10i."},{"Start":"03:38.430 ","End":"03:43.605","Text":"Then 4i times 3 is 12i."},{"Start":"03:43.605 ","End":"03:50.800","Text":"Finally 4i times 2i is 8i^2."},{"Start":"03:50.800 ","End":"03:53.110","Text":"Now once you get into the hang of it,"},{"Start":"03:53.110 ","End":"03:56.665","Text":"and you\u0027ll remember that i^2 is minus 1,"},{"Start":"03:56.665 ","End":"03:59.410","Text":"which I\u0027ll highlight again,"},{"Start":"03:59.410 ","End":"04:04.250","Text":"then you would have written this straightaway as minus 8."},{"Start":"04:04.250 ","End":"04:10.475","Text":"In fact, I\u0027ll just erase this and say this is minus 8 because i^2 is minus 1."},{"Start":"04:10.475 ","End":"04:12.610","Text":"Then we just collect stuff together."},{"Start":"04:12.610 ","End":"04:17.910","Text":"Minus 15 and minus 8 gives me minus 23,"},{"Start":"04:17.910 ","End":"04:26.360","Text":"and minus 10i plus 12i is plus 2i and that\u0027s the answer to the multiplication."},{"Start":"04:26.540 ","End":"04:30.910","Text":"I\u0027d like to do one more example with multiplication."},{"Start":"04:30.910 ","End":"04:32.880","Text":"Let\u0027s take the following."},{"Start":"04:32.880 ","End":"04:43.060","Text":"Let\u0027s take 7 plus 5i times 7 minus 5i."},{"Start":"04:44.440 ","End":"04:46.925","Text":"Let\u0027s see what we get."},{"Start":"04:46.925 ","End":"04:50.360","Text":"Of course we could do it the same as before or we could"},{"Start":"04:50.360 ","End":"04:54.800","Text":"notice that this is the same thing just with a plus and with a minus."},{"Start":"04:54.800 ","End":"05:00.185","Text":"Then you might recall a certain formula called the difference of squares formula,"},{"Start":"05:00.185 ","End":"05:04.010","Text":"that a plus b times a minus b."},{"Start":"05:04.010 ","End":"05:05.660","Text":"If you have two things ones with a plus,"},{"Start":"05:05.660 ","End":"05:08.929","Text":"and ones with a minus, the answer is the difference of the squares."},{"Start":"05:08.929 ","End":"05:12.575","Text":"It\u0027s the first one squared less the second one squared."},{"Start":"05:12.575 ","End":"05:15.095","Text":"If I apply that rule here,"},{"Start":"05:15.095 ","End":"05:18.535","Text":"then I get that this is 7^2,"},{"Start":"05:18.535 ","End":"05:23.890","Text":"I\u0027m putting the brackets just for emphasis, minus 5i^2."},{"Start":"05:23.890 ","End":"05:26.120","Text":"Here I do need the brackets,"},{"Start":"05:26.120 ","End":"05:28.925","Text":"otherwise it would just be the i^2."},{"Start":"05:28.925 ","End":"05:31.295","Text":"This is equal to,"},{"Start":"05:31.295 ","End":"05:35.840","Text":"7^2 squared is 49 and initially we would"},{"Start":"05:35.840 ","End":"05:41.940","Text":"write minus 25i^2 because 5i times 5i,"},{"Start":"05:41.940 ","End":"05:49.049","Text":"but remember that i^2 is minus 1 so really this is equal to"},{"Start":"05:49.049 ","End":"05:58.065","Text":"49 plus 25 and"},{"Start":"05:58.065 ","End":"06:00.180","Text":"that is equal to,"},{"Start":"06:00.180 ","End":"06:07.090","Text":"let\u0027s see, 74, I make it."},{"Start":"06:07.090 ","End":"06:11.615","Text":"Turns out the answer is a real number without the imaginary part."},{"Start":"06:11.615 ","End":"06:18.470","Text":"Now, I actually knew this was going to happen because whenever you have a pair like this,"},{"Start":"06:18.470 ","End":"06:20.630","Text":"ones with a plus and ones with a minus,"},{"Start":"06:20.630 ","End":"06:23.855","Text":"it comes out real and there is actually a name."},{"Start":"06:23.855 ","End":"06:26.120","Text":"If a complex number."},{"Start":"06:26.120 ","End":"06:32.925","Text":"Let\u0027s call it z is 7 plus 5i in this case,"},{"Start":"06:32.925 ","End":"06:39.830","Text":"then the same thing with a negative on the imaginary part,"},{"Start":"06:39.830 ","End":"06:44.090","Text":"the opposite, is called the complex conjugate."},{"Start":"06:44.090 ","End":"06:47.930","Text":"Sometimes we call that Z bar."},{"Start":"06:47.930 ","End":"06:51.355","Text":"This is the, I\u0027ll write the word,"},{"Start":"06:51.355 ","End":"06:53.750","Text":"two words, complex conjugate."},{"Start":"06:53.750 ","End":"06:56.480","Text":"Sometimes in the context of complex numbers,"},{"Start":"06:56.480 ","End":"06:59.720","Text":"I just say this is the conjugate of this and the other way round,"},{"Start":"06:59.720 ","End":"07:01.310","Text":"and this is the conjugate of this."},{"Start":"07:01.310 ","End":"07:05.190","Text":"In general, when we have a plus bi,"},{"Start":"07:05.190 ","End":"07:10.380","Text":"then the conjugate is a minus bi."},{"Start":"07:10.380 ","End":"07:16.430","Text":"These two are conjugates and this will be very useful later on for example,"},{"Start":"07:16.430 ","End":"07:19.385","Text":"when we do the section on division."},{"Start":"07:19.385 ","End":"07:22.170","Text":"Let\u0027s take a break now."}],"ID":4919},{"Watched":false,"Name":"Exponents","Duration":"10m 29s","ChapterTopicVideoID":4927,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4927.jpeg","UploadDate":"2020-09-29T10:53:50.8600000","DurationForVideoObject":"PT10M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"Here we are back with complex numbers."},{"Start":"00:02.640 ","End":"00:09.270","Text":"We\u0027ve just done addition and subtraction and multiplication."},{"Start":"00:09.270 ","End":"00:14.340","Text":"Let\u0027s continue now with exponents or if you like,"},{"Start":"00:14.340 ","End":"00:19.170","Text":"powers, and after that we\u0027ll get to division."},{"Start":"00:19.170 ","End":"00:22.785","Text":"Let\u0027s start with powers of i."},{"Start":"00:22.785 ","End":"00:28.365","Text":"Now, the main starting point is that i^2 is minus 1."},{"Start":"00:28.365 ","End":"00:32.625","Text":"We\u0027ll keep highlighting it again and again, it\u0027s so important."},{"Start":"00:32.625 ","End":"00:34.890","Text":"But I\u0027d like to look at some other powers."},{"Start":"00:34.890 ","End":"00:39.780","Text":"In fact, let\u0027s look at natural numbers as 0, 1, 2, 3,"},{"Start":"00:39.780 ","End":"00:47.010","Text":"4. i^0, and anything to the 0 is 1 and i^1 is just i,"},{"Start":"00:47.010 ","End":"00:49.625","Text":"I\u0027m just writing that for completeness."},{"Start":"00:49.625 ","End":"00:51.455","Text":"We\u0027ve got 0, 1, 2."},{"Start":"00:51.455 ","End":"00:55.910","Text":"What about i^3?"},{"Start":"00:55.910 ","End":"01:04.935","Text":"Well, I can write i^3 as i^2 times i^1 and i^2 is minus 1,"},{"Start":"01:04.935 ","End":"01:07.620","Text":"so it\u0027s minus 1 times i,"},{"Start":"01:07.620 ","End":"01:11.535","Text":"so the answer is minus i."},{"Start":"01:11.535 ","End":"01:14.655","Text":"Now let\u0027s go to the next one, i^4."},{"Start":"01:14.655 ","End":"01:21.000","Text":"i^4, I could write it as"},{"Start":"01:21.000 ","End":"01:28.170","Text":"( i^2)^2 using the laws of exponents."},{"Start":"01:28.170 ","End":"01:30.510","Text":"Since i^2 is minus 1,"},{"Start":"01:30.510 ","End":"01:32.715","Text":"it\u0027s minus 1^2,"},{"Start":"01:32.715 ","End":"01:36.570","Text":"which is 1, so the answer is 1."},{"Start":"01:36.570 ","End":"01:39.120","Text":"Let\u0027s do one more."},{"Start":"01:39.120 ","End":"01:42.390","Text":"i^5, let\u0027s see what that equals."},{"Start":"01:42.390 ","End":"01:51.585","Text":"i^5= I can either start from the beginning or I could use i^4 times i."},{"Start":"01:51.585 ","End":"01:55.140","Text":"Let\u0027s say I use it\u0027s i^4 times i,"},{"Start":"01:55.140 ","End":"01:57.180","Text":"i^4 is 1,"},{"Start":"01:57.180 ","End":"01:59.265","Text":"1 times i,"},{"Start":"01:59.265 ","End":"02:02.170","Text":"so it\u0027s equal to i. i^6."},{"Start":"02:04.160 ","End":"02:08.790","Text":"I can either say i^5 times i or we can start a fresh and say"},{"Start":"02:08.790 ","End":"02:13.970","Text":"it\u0027s (i^2)^3 when it\u0027s an even number, I can do that."},{"Start":"02:13.970 ","End":"02:17.210","Text":"Then i^2 is (minus 1)^3,"},{"Start":"02:17.210 ","End":"02:19.650","Text":"which is minus 1."},{"Start":"02:20.890 ","End":"02:24.750","Text":"Just do as many as I have room for."},{"Start":"02:26.120 ","End":"02:33.290","Text":"i^7, I\u0027ll do it as if from scratch."},{"Start":"02:33.290 ","End":"02:35.030","Text":"What we can do is we always say,"},{"Start":"02:35.030 ","End":"02:42.343","Text":"if it\u0027s an odd number I can say that it\u0027s i^6 times i"},{"Start":"02:42.343 ","End":"02:50.570","Text":"and then (i^2 )^3s in laws of exponents times i,"},{"Start":"02:50.570 ","End":"02:55.920","Text":"i^2 is (minus 1)^3 times i."},{"Start":"02:55.920 ","End":"02:58.020","Text":"(Minus 1)^3 is minus 1,"},{"Start":"02:58.020 ","End":"03:01.695","Text":"so it\u0027s minus i. i^8,"},{"Start":"03:01.695 ","End":"03:05.340","Text":"it\u0027s the last one we\u0027ll do here, i^8,"},{"Start":"03:05.340 ","End":"03:10.125","Text":"you can say it\u0027s (i^2)^4,"},{"Start":"03:10.125 ","End":"03:13.710","Text":"which is (minus 10)^4,"},{"Start":"03:13.710 ","End":"03:16.555","Text":"which is plus 1."},{"Start":"03:16.555 ","End":"03:19.250","Text":"Notice that it\u0027s actually periodic."},{"Start":"03:19.250 ","End":"03:26.115","Text":"It goes,1i minus i. I said that all wrong. I\u0027ll start again."},{"Start":"03:26.115 ","End":"03:29.190","Text":"1i minus 1 minus i,"},{"Start":"03:29.190 ","End":"03:32.765","Text":"then 1i minus 1 minus i, and so on."},{"Start":"03:32.765 ","End":"03:34.190","Text":"It has a period of 4;"},{"Start":"03:34.190 ","End":"03:36.875","Text":"every 4 it repeats itself."},{"Start":"03:36.875 ","End":"03:39.755","Text":"The reason for that is that i^4 is 1,"},{"Start":"03:39.755 ","End":"03:42.080","Text":"so i^8 is 1,"},{"Start":"03:42.080 ","End":"03:43.260","Text":"i^12 is 1,"},{"Start":"03:43.260 ","End":"03:45.690","Text":"every 4 we start over again."},{"Start":"03:45.690 ","End":"03:48.705","Text":"That\u0027s as far as exponents of just i."},{"Start":"03:48.705 ","End":"03:51.980","Text":"Now let\u0027s make it a bit more complicated."},{"Start":"03:51.980 ","End":"03:54.830","Text":"Let me first clean the board."},{"Start":"03:54.830 ","End":"03:59.580","Text":"Let\u0027s try something like minus i^11."},{"Start":"04:01.870 ","End":"04:06.530","Text":"What I would do here is, first of all,"},{"Start":"04:06.530 ","End":"04:17.460","Text":"I see minus i is minus 1 times i. I can say it\u0027s minus 1^11 times i^11."},{"Start":"04:18.400 ","End":"04:20.870","Text":"Let\u0027s do each piece separately."},{"Start":"04:20.870 ","End":"04:24.740","Text":"This piece here is minus"},{"Start":"04:24.740 ","End":"04:31.175","Text":"1^11 because it\u0027s an odd power of minus 1, it\u0027s minus 1."},{"Start":"04:31.175 ","End":"04:37.940","Text":"But i^11, we use a usual trick of saying that"},{"Start":"04:37.940 ","End":"04:45.240","Text":"(i^2)^5 is i^10 times i. I skipped a step,"},{"Start":"04:45.240 ","End":"04:51.390","Text":"it\u0027s i^10 times i and i^10 is (i^2)^5 so on."},{"Start":"04:51.390 ","End":"04:56.325","Text":"i^2 is minus 1^5 times i,"},{"Start":"04:56.325 ","End":"04:58.320","Text":"which is minus i."},{"Start":"04:58.320 ","End":"05:04.470","Text":"Finally, I combine this minus 1 with this minus i,"},{"Start":"05:04.470 ","End":"05:10.770","Text":"so actually, the answer comes out to be just 1 plus 1."},{"Start":"05:10.900 ","End":"05:18.445","Text":"Another example, 3 minus (4i)^2."},{"Start":"05:18.445 ","End":"05:21.440","Text":"Power of 2 we can use the formula."},{"Start":"05:21.440 ","End":"05:29.565","Text":"I\u0027m talking about a minus b squared is a^2 minus 2ab plus b^2,"},{"Start":"05:29.565 ","End":"05:31.220","Text":"same if it was a plus here,"},{"Start":"05:31.220 ","End":"05:33.245","Text":"it would be a plus here."},{"Start":"05:33.245 ","End":"05:39.195","Text":"We get 3^2 is 9 minus 2ab,"},{"Start":"05:39.195 ","End":"05:42.525","Text":"2 times 3 times 4 is 24,"},{"Start":"05:42.525 ","End":"05:44.535","Text":"so minus 24i,"},{"Start":"05:44.535 ","End":"05:51.940","Text":"and b^2 is (4i)^2."},{"Start":"05:51.940 ","End":"05:59.175","Text":"Now, (4i)^2 is 4^2 times i^2,"},{"Start":"05:59.175 ","End":"06:01.275","Text":"which is (16i)^2,"},{"Start":"06:01.275 ","End":"06:06.540","Text":"actually minus 16 because i^2 is minus 1,"},{"Start":"06:06.540 ","End":"06:15.400","Text":"and so the answer is 9 minus 16 is minus 7 minus 24i."},{"Start":"06:16.820 ","End":"06:20.894","Text":"Now, let\u0027s take a higher power."},{"Start":"06:20.894 ","End":"06:23.785","Text":"Now let\u0027s stick to the power of 2."},{"Start":"06:23.785 ","End":"06:26.060","Text":"Let\u0027s make it with a plus in the middle."},{"Start":"06:26.060 ","End":"06:37.080","Text":"But let\u0027s say I have here something like i^9 plus i^15 squared."},{"Start":"06:37.310 ","End":"06:40.560","Text":"Using the corresponding formula,"},{"Start":"06:40.560 ","End":"06:42.520","Text":"well, I\u0027ll just make it 2 in 1."},{"Start":"06:42.520 ","End":"06:44.345","Text":"If it\u0027s plus here,"},{"Start":"06:44.345 ","End":"06:46.715","Text":"then it\u0027s also a plus here."},{"Start":"06:46.715 ","End":"06:49.800","Text":"We use a plus b^2."},{"Start":"06:50.020 ","End":"06:52.940","Text":"This by the way, is one way of solving it"},{"Start":"06:52.940 ","End":"06:55.055","Text":"because at the end I\u0027m going to show you an easier way."},{"Start":"06:55.055 ","End":"06:57.935","Text":"But let\u0027s do it this way with the formula."},{"Start":"06:57.935 ","End":"07:01.510","Text":"We get (i^9)^2."},{"Start":"07:01.510 ","End":"07:06.065","Text":"We can straight away see that that\u0027s i^18 with the rules of exponents."},{"Start":"07:06.065 ","End":"07:12.610","Text":"Then plus twice i^9 times i^15."},{"Start":"07:12.610 ","End":"07:14.370","Text":"Again, the rules of exponents,"},{"Start":"07:14.370 ","End":"07:18.735","Text":"I add 9 plus 15 is 24,"},{"Start":"07:18.735 ","End":"07:26.820","Text":"and then i^15 all squared is i^30."},{"Start":"07:26.820 ","End":"07:28.275","Text":"This=."},{"Start":"07:28.275 ","End":"07:32.740","Text":"Now, at this point you should already be able to"},{"Start":"07:32.740 ","End":"07:37.825","Text":"without doing all these expansions with i^2,"},{"Start":"07:37.825 ","End":"07:41.635","Text":"you could straight away see that i^16,"},{"Start":"07:41.635 ","End":"07:43.475","Text":"anything that divides by 4,"},{"Start":"07:43.475 ","End":"07:45.525","Text":"i to its power is 1."},{"Start":"07:45.525 ","End":"07:48.930","Text":"This is i^16 times i^2,"},{"Start":"07:48.930 ","End":"07:51.670","Text":"so that\u0027s minus 1."},{"Start":"07:51.890 ","End":"07:54.660","Text":"Then here, i^24,"},{"Start":"07:54.660 ","End":"07:56.850","Text":"if it\u0027s divisible by 4,"},{"Start":"07:56.850 ","End":"07:58.710","Text":"it\u0027s going to be 1."},{"Start":"07:58.710 ","End":"08:00.888","Text":"See, divisible by 4 is 1,"},{"Start":"08:00.888 ","End":"08:03.156","Text":"divisible by 4 is 1,"},{"Start":"08:03.156 ","End":"08:04.800","Text":"divisible by 4 is 1."},{"Start":"08:04.800 ","End":"08:06.030","Text":"All the 0,"},{"Start":"08:06.030 ","End":"08:08.190","Text":"4, 8, 12, 16,"},{"Start":"08:08.190 ","End":"08:10.200","Text":"all those powers of 1,"},{"Start":"08:10.200 ","End":"08:11.875","Text":"that\u0027s a shortcut way of doing it,"},{"Start":"08:11.875 ","End":"08:13.324","Text":"and you see I have this leftover,"},{"Start":"08:13.324 ","End":"08:15.485","Text":"leftover is 2, so it\u0027s i^2."},{"Start":"08:15.485 ","End":"08:18.215","Text":"Here, it\u0027s an even, so it\u0027s 1."},{"Start":"08:18.215 ","End":"08:22.535","Text":"This is just 2 times 1. i^30,"},{"Start":"08:22.535 ","End":"08:27.620","Text":"well, 28 divides by 4."},{"Start":"08:27.620 ","End":"08:31.275","Text":"1^28 is 1. Then so it\u0027s again just i^2,"},{"Start":"08:31.275 ","End":"08:33.480","Text":"which is minus 1,"},{"Start":"08:33.480 ","End":"08:34.530","Text":"plus minus 1,"},{"Start":"08:34.530 ","End":"08:36.375","Text":"which is minus 1."},{"Start":"08:36.375 ","End":"08:41.160","Text":"What we get is minus 1 plus 2 minus 1, 0."},{"Start":"08:41.160 ","End":"08:42.810","Text":"Okay, that\u0027s the answer."},{"Start":"08:42.810 ","End":"08:45.200","Text":"I said I\u0027d show you another way of doing it."},{"Start":"08:45.200 ","End":"08:51.440","Text":"The other way of doing it is right from the beginning to say i^9 is i^8 times"},{"Start":"08:51.440 ","End":"08:58.680","Text":"i. i^8 is 1 times i. I could say this is=i."},{"Start":"08:58.680 ","End":"09:03.945","Text":"Then again, i^15 is,"},{"Start":"09:03.945 ","End":"09:06.460","Text":"I\u0027m going to show you this one on the side,"},{"Start":"09:06.460 ","End":"09:12.190","Text":"i^15 is i^12 times i^3."},{"Start":"09:12.190 ","End":"09:15.710","Text":"You take out the multiple of 4. This is 1."},{"Start":"09:15.710 ","End":"09:19.910","Text":"Once again, it\u0027s because (i^4)^3 is 1,"},{"Start":"09:19.910 ","End":"09:23.480","Text":"you can either say it\u0027s (i^4)^3,"},{"Start":"09:23.480 ","End":"09:26.990","Text":"which is 1^3, which is 1."},{"Start":"09:26.990 ","End":"09:28.790","Text":"If you didn\u0027t remember that,"},{"Start":"09:28.790 ","End":"09:32.765","Text":"you could always go back to (i^2)^6,"},{"Start":"09:32.765 ","End":"09:37.380","Text":"which is (minus 1)^6,"},{"Start":"09:37.380 ","End":"09:39.460","Text":"minus 1 to an even power is 1."},{"Start":"09:39.460 ","End":"09:42.010","Text":"I\u0027m showing you alternative ways of doing all things."},{"Start":"09:42.010 ","End":"09:44.420","Text":"Sure, this is not just one way of doing it."},{"Start":"09:44.420 ","End":"09:48.218","Text":"Either way, the i^12 is 1 and i^3,"},{"Start":"09:48.218 ","End":"09:49.800","Text":"well, let\u0027s not repeat our work,"},{"Start":"09:49.800 ","End":"09:52.155","Text":"let\u0027s just look it up, it\u0027s minus i."},{"Start":"09:52.155 ","End":"09:54.552","Text":"What we get is here it\u0027s 1,"},{"Start":"09:54.552 ","End":"09:56.798","Text":"here it\u0027s minus i,"},{"Start":"09:56.798 ","End":"10:02.670","Text":"so altogether, i^15 is minus i."},{"Start":"10:02.670 ","End":"10:05.100","Text":"If i^15 is minus i,"},{"Start":"10:05.100 ","End":"10:08.835","Text":"we have i plus minus i squared,"},{"Start":"10:08.835 ","End":"10:11.400","Text":"but I can do i plus minus i, it\u0027s just 0."},{"Start":"10:11.400 ","End":"10:12.726","Text":"It\u0027s i minus i,"},{"Start":"10:12.726 ","End":"10:15.030","Text":"0^2, which is 0,"},{"Start":"10:15.030 ","End":"10:19.115","Text":"and it\u0027s a good job we got the same answer if we did it with 2 different ways,"},{"Start":"10:19.115 ","End":"10:23.160","Text":"0, 0, so that obviously is the answer."},{"Start":"10:23.160 ","End":"10:26.355","Text":"That\u0027s enough for exponents."},{"Start":"10:26.355 ","End":"10:28.690","Text":"Let\u0027s take a break here."}],"ID":4920},{"Watched":false,"Name":"Division","Duration":"6m 51s","ChapterTopicVideoID":4918,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4918.jpeg","UploadDate":"2020-09-29T10:42:50.8570000","DurationForVideoObject":"PT6M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.519","Text":"Continuing with complex numbers,"},{"Start":"00:02.519 ","End":"00:07.244","Text":"this time we are going talk about division of complex numbers."},{"Start":"00:07.244 ","End":"00:10.635","Text":"But to start with, I would like to remind you"},{"Start":"00:10.635 ","End":"00:14.040","Text":"of something we did in the section on multiplication."},{"Start":"00:14.040 ","End":"00:17.520","Text":"In fact, I\u0027m just going to copy and paste the example."},{"Start":"00:17.520 ","End":"00:22.320","Text":"This is the example we had in the chapter on multiplication I just"},{"Start":"00:22.320 ","End":"00:27.300","Text":"copied it and we had the product of 2 numbers,"},{"Start":"00:27.300 ","End":"00:31.730","Text":"which were not just any 2 complex numbers that were conjugates of each other,"},{"Start":"00:31.730 ","End":"00:36.269","Text":"remember if we have the imaginary part once with a plus and once with a minus,"},{"Start":"00:36.269 ","End":"00:39.443","Text":"then these 22 complex numbers are conjugates."},{"Start":"00:39.443 ","End":"00:41.325","Text":"When we multiply them together,"},{"Start":"00:41.325 ","End":"00:44.180","Text":"we got a positive real number."},{"Start":"00:44.180 ","End":"00:45.995","Text":"In fact, if you look at it,"},{"Start":"00:45.995 ","End":"00:48.155","Text":"what this 74 is,"},{"Start":"00:48.155 ","End":"00:54.225","Text":"it\u0027s actually 7^2 plus 5^2 it came out to be 49 plus 25,"},{"Start":"00:54.225 ","End":"00:59.030","Text":"and this is actually true in general as a rule that if we take a plus"},{"Start":"00:59.030 ","End":"01:04.625","Text":"bi as one complex number and then take its conjugate a minus bi,"},{"Start":"01:04.625 ","End":"01:07.700","Text":"the answer is always going to be positive real,"},{"Start":"01:07.700 ","End":"01:11.585","Text":"and it\u0027s equal to a^2 plus b^2."},{"Start":"01:11.585 ","End":"01:15.949","Text":"This is going to be very useful to us when we talk about division."},{"Start":"01:15.949 ","End":"01:18.319","Text":"Let\u0027s start straight away with an example."},{"Start":"01:18.319 ","End":"01:26.400","Text":"I want to compute 11 plus 2i divided by 2 minus i."},{"Start":"01:26.400 ","End":"01:29.585","Text":"The trick is this."},{"Start":"01:29.585 ","End":"01:34.160","Text":"We multiply apply this fraction top and bottom by the same thing,"},{"Start":"01:34.160 ","End":"01:38.765","Text":"and we multiply it by the conjugate of the denominator."},{"Start":"01:38.765 ","End":"01:41.330","Text":"To remind you again, the word conjugate,"},{"Start":"01:41.330 ","End":"01:44.060","Text":"these 2 numbers are conjugates of each other."},{"Start":"01:44.060 ","End":"01:47.660","Text":"Complex conjugates and these 2 are complex conjugates."},{"Start":"01:47.660 ","End":"01:49.700","Text":"It\u0027s the same real part,"},{"Start":"01:49.700 ","End":"01:52.460","Text":"but the imaginary part with the opposite sign."},{"Start":"01:52.460 ","End":"01:58.960","Text":"In this case, the conjugate of 2 minus i is going to be 2 plus i,"},{"Start":"01:58.960 ","End":"02:02.465","Text":"and if I multiply both top and bottom,"},{"Start":"02:02.465 ","End":"02:06.185","Text":"really I\u0027m just multiplying by 1 because this fraction is 1."},{"Start":"02:06.185 ","End":"02:09.635","Text":"This is going to help us, as you will see."},{"Start":"02:09.635 ","End":"02:17.875","Text":"What we get, is let\u0027s do the denominator first."},{"Start":"02:17.875 ","End":"02:20.430","Text":"That\u0027s going to be the easier part,"},{"Start":"02:20.430 ","End":"02:25.130","Text":"2 minus i times 1 plus i we already discussed from"},{"Start":"02:25.130 ","End":"02:30.430","Text":"this formula that this is 2^2 plus 1^2."},{"Start":"02:30.430 ","End":"02:36.540","Text":"Because this is 1i, this is 2 plus 1i that will be 5,"},{"Start":"02:36.540 ","End":"02:37.890","Text":"of course in the moment."},{"Start":"02:37.890 ","End":"02:41.250","Text":"Then on the numerator we just multiply,"},{"Start":"02:41.250 ","End":"02:49.175","Text":"just like we would multiply 2 complex numbers or just write it."},{"Start":"02:49.175 ","End":"02:50.960","Text":"We\u0027re doing the product here,"},{"Start":"02:50.960 ","End":"02:53.610","Text":"and then we get,"},{"Start":"02:53.720 ","End":"02:57.940","Text":"let\u0027s see, 11 times 2."},{"Start":"02:57.940 ","End":"02:59.675","Text":"Let\u0027s try and do it all in 1 go,"},{"Start":"02:59.675 ","End":"03:04.565","Text":"11 times 2 is 22,"},{"Start":"03:04.565 ","End":"03:07.095","Text":"I\u0027m just going to make a note."},{"Start":"03:07.095 ","End":"03:14.595","Text":"That\u0027s 22 and 2i times I is 2i^2,"},{"Start":"03:14.595 ","End":"03:16.290","Text":"so that\u0027s minus 2,"},{"Start":"03:16.290 ","End":"03:22.520","Text":"so I get 20 for the real part and then on the imaginary parts I\u0027ve got"},{"Start":"03:22.520 ","End":"03:28.780","Text":"2 times 2 is 4i and 11 times 1 is 11i,"},{"Start":"03:28.780 ","End":"03:33.520","Text":"so I have got 4 plus 11, it\u0027s plus 15i."},{"Start":"03:35.300 ","End":"03:39.010","Text":"I\u0027m going to erase the scratch work."},{"Start":"03:39.230 ","End":"03:44.865","Text":"Over the dividing line and 2^2 plus 1^2 is 5,"},{"Start":"03:44.865 ","End":"03:49.545","Text":"and finally, we can do the division,"},{"Start":"03:49.545 ","End":"03:51.380","Text":"when we divide by a number,"},{"Start":"03:51.380 ","End":"04:01.320","Text":"we just divide each bit separately 20 divided by 5 is 4 and 15 over 5 is 3,"},{"Start":"04:01.320 ","End":"04:04.995","Text":"so we get the answer as 4 plus 3i,"},{"Start":"04:04.995 ","End":"04:11.040","Text":"and that\u0027s all there is to division of complex numbers."},{"Start":"04:14.480 ","End":"04:17.630","Text":"When we divide by a complex number,"},{"Start":"04:17.630 ","End":"04:22.560","Text":"what we do is we then multiply top and bottom by the conjugate."},{"Start":"04:23.390 ","End":"04:26.865","Text":"Let\u0027s just do 1 more for practice,"},{"Start":"04:26.865 ","End":"04:34.480","Text":"3 plus 7i divided by 2 minus 5i,"},{"Start":"04:34.760 ","End":"04:41.270","Text":"so what we do is we multiply top and bottom by a fraction,"},{"Start":"04:41.270 ","End":"04:44.600","Text":"which is the conjugate of the denominator."},{"Start":"04:44.600 ","End":"04:48.350","Text":"The conjugate of 2 minus 5i is 2 plus 5i,"},{"Start":"04:48.350 ","End":"04:51.440","Text":"but we have to do the same thing to the numerator so as not"},{"Start":"04:51.440 ","End":"04:55.155","Text":"to change the sum the answer,"},{"Start":"04:55.155 ","End":"05:00.655","Text":"and then what we get is on the denominator,"},{"Start":"05:00.655 ","End":"05:04.300","Text":"we always get from this formula,"},{"Start":"05:04.300 ","End":"05:07.370","Text":"2^2 plus 5^2,"},{"Start":"05:08.030 ","End":"05:10.750","Text":"and on the numerator,"},{"Start":"05:10.750 ","End":"05:12.100","Text":"we just multiply it out."},{"Start":"05:12.100 ","End":"05:15.240","Text":"Let\u0027s see, I will do it slightly differently than I did before,"},{"Start":"05:15.240 ","End":"05:25.270","Text":"3 times 2 is 6 plus 7i times 2i is 14i."},{"Start":"05:27.290 ","End":"05:31.630","Text":"Then these 2 give me plus"},{"Start":"05:32.150 ","End":"05:39.800","Text":"15i and 7i times 5i is 35i^2 minus 35."},{"Start":"05:39.800 ","End":"05:42.670","Text":"Remember, i^2 is minus 1."},{"Start":"05:42.670 ","End":"05:46.845","Text":"Again, i^2 is minus 1."},{"Start":"05:46.845 ","End":"05:51.510","Text":"Then what we get is, let\u0027s see,"},{"Start":"05:51.510 ","End":"06:00.435","Text":"6 minus 35 is minus 29,"},{"Start":"06:00.435 ","End":"06:09.270","Text":"14 plus 15 is 29i and on the denominator,"},{"Start":"06:09.270 ","End":"06:13.350","Text":"4 plus 25 is 29."},{"Start":"06:13.350 ","End":"06:16.175","Text":"That works out nicely."},{"Start":"06:16.175 ","End":"06:22.155","Text":"That comes out to be minus 1 plus 1i,"},{"Start":"06:22.155 ","End":"06:25.025","Text":"which is just i, and that\u0027s the answer."},{"Start":"06:25.025 ","End":"06:29.435","Text":"Of course often you will get fractions in the answer just,"},{"Start":"06:29.435 ","End":"06:33.654","Text":"again examples where it came out neatly, where everything divided."},{"Start":"06:33.654 ","End":"06:37.095","Text":"That\u0027s enough for division,"},{"Start":"06:37.095 ","End":"06:42.650","Text":"and of course there are more solved example exercises besides the tutorial,"},{"Start":"06:42.650 ","End":"06:45.420","Text":"go and look at them and do them."},{"Start":"06:46.300 ","End":"06:48.530","Text":"Let\u0027s take a break now,"},{"Start":"06:48.530 ","End":"06:51.390","Text":"and after that the next topic."}],"ID":4921},{"Watched":false,"Name":"Quadratic Equations","Duration":"9m 59s","ChapterTopicVideoID":4928,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4928.jpeg","UploadDate":"2020-09-29T10:57:04.5170000","DurationForVideoObject":"PT9M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"Continuing with complex numbers is"},{"Start":"00:02.700 ","End":"00:08.220","Text":"one more topic which is related to quadratic equations."},{"Start":"00:08.220 ","End":"00:09.615","Text":"You might say, \"Oh yeah,"},{"Start":"00:09.615 ","End":"00:12.210","Text":"we\u0027ve already learned quadratic equations.\""},{"Start":"00:12.210 ","End":"00:19.065","Text":"But I want to relate to those quadratic equations which we used to say have no solution,"},{"Start":"00:19.065 ","End":"00:22.090","Text":"maybe now they do somehow."},{"Start":"00:22.340 ","End":"00:24.930","Text":"A note on notation,"},{"Start":"00:24.930 ","End":"00:30.060","Text":"we used to usually use x as the variable,"},{"Start":"00:30.060 ","End":"00:33.495","Text":"but when we talk about complex numbers,"},{"Start":"00:33.495 ","End":"00:38.835","Text":"we usually use the letter z and I put a line through it to distinguish it from a 2."},{"Start":"00:38.835 ","End":"00:48.740","Text":"So our first example is z^2 plus 9=0."},{"Start":"00:48.740 ","End":"00:50.000","Text":"We don\u0027t need to crank out"},{"Start":"00:50.000 ","End":"00:53.900","Text":"the whole quadratic formula because there\u0027s a missing middle term,"},{"Start":"00:53.900 ","End":"01:00.470","Text":"so what I can do is say z^2 is minus 9,"},{"Start":"01:00.470 ","End":"01:03.870","Text":"and then I take the square root and then z is going to"},{"Start":"01:03.870 ","End":"01:09.125","Text":"be plus or minus the square root of minus 9,"},{"Start":"01:09.125 ","End":"01:13.550","Text":"and we can simplify this because we did this actually in"},{"Start":"01:13.550 ","End":"01:18.410","Text":"the very first chapter on complex numbers, we\u0027ll do it again."},{"Start":"01:18.410 ","End":"01:25.570","Text":"This is equal to plus or minus the square root of 9 times the square root of minus 1,"},{"Start":"01:25.570 ","End":"01:27.560","Text":"because the properties of the square roots,"},{"Start":"01:27.560 ","End":"01:30.365","Text":"square root of the product is the product of the square roots."},{"Start":"01:30.365 ","End":"01:34.640","Text":"Then we finally get the answer plus or minus 3,"},{"Start":"01:34.640 ","End":"01:37.860","Text":"and the square root of minus 1 is i."},{"Start":"01:38.750 ","End":"01:42.240","Text":"This used to have no solution,"},{"Start":"01:42.240 ","End":"01:44.930","Text":"when we were talking about the numbers,"},{"Start":"01:44.930 ","End":"01:47.225","Text":"we used to be familiar with the real numbers,"},{"Start":"01:47.225 ","End":"01:49.385","Text":"but in terms of complex numbers,"},{"Start":"01:49.385 ","End":"01:52.190","Text":"it now has two solutions."},{"Start":"01:52.190 ","End":"01:56.570","Text":"Let\u0027s try another one that used to be impossible."},{"Start":"01:56.570 ","End":"02:01.610","Text":"Let\u0027s take now z^2"},{"Start":"02:01.610 ","End":"02:10.255","Text":"plus 4z plus 5=0."},{"Start":"02:10.255 ","End":"02:17.120","Text":"This time we will use the quadratic formula and say that z is"},{"Start":"02:17.120 ","End":"02:24.480","Text":"equal to minus b plus or minus the square root of b^2,"},{"Start":"02:24.480 ","End":"02:27.120","Text":"which is 16 minus 4ac,"},{"Start":"02:27.120 ","End":"02:30.780","Text":"4 times 1 times 5 is 20,"},{"Start":"02:30.780 ","End":"02:36.045","Text":"and all this over 2a which is 2 times 1 is 2."},{"Start":"02:36.045 ","End":"02:42.990","Text":"So what we get is equal to minus 4 plus or minus."},{"Start":"02:42.990 ","End":"02:46.503","Text":"Now this is the square root of minus 4."},{"Start":"02:46.503 ","End":"02:49.045","Text":"Let\u0027s just do that at the side."},{"Start":"02:49.045 ","End":"02:58.010","Text":"The square root of minus 4 is the square root of 4 times the square root of minus 1,"},{"Start":"02:58.010 ","End":"03:03.060","Text":"because minus 4 is 4 times minus 1 and this is equal to 2,"},{"Start":"03:03.060 ","End":"03:04.755","Text":"and this is equal to i."},{"Start":"03:04.755 ","End":"03:12.675","Text":"So we get minus 4 plus or minus 2i over 2,"},{"Start":"03:12.675 ","End":"03:15.865","Text":"and that gives us two answers."},{"Start":"03:15.865 ","End":"03:17.815","Text":"If I take the plus,"},{"Start":"03:17.815 ","End":"03:21.290","Text":"I\u0027ll get minus 4 plus 2i over 2,"},{"Start":"03:21.290 ","End":"03:23.010","Text":"and then we just cancel by 2,"},{"Start":"03:23.010 ","End":"03:25.725","Text":"so we get minus 2 plus i,"},{"Start":"03:25.725 ","End":"03:27.380","Text":"and if we take the minus,"},{"Start":"03:27.380 ","End":"03:30.365","Text":"we get minus 2 minus i."},{"Start":"03:30.365 ","End":"03:35.250","Text":"So these are the two solutions to this quadratic equation."},{"Start":"03:35.480 ","End":"03:39.240","Text":"Let\u0027s do one more,"},{"Start":"03:39.240 ","End":"03:42.045","Text":"but to point something out,"},{"Start":"03:42.045 ","End":"03:46.610","Text":"notice that the two solutions here are complex conjugates of each other,"},{"Start":"03:46.610 ","End":"03:49.655","Text":"minus 2 and it\u0027s plus or minus i."},{"Start":"03:49.655 ","End":"03:51.430","Text":"The same thing here,"},{"Start":"03:51.430 ","End":"03:52.900","Text":"the real part is zero."},{"Start":"03:52.900 ","End":"03:57.230","Text":"I could have written 0 plus or minus 3i."},{"Start":"03:57.230 ","End":"04:03.065","Text":"0 plus 3i is the conjugate of 0 minus 3i,"},{"Start":"04:03.065 ","End":"04:06.350","Text":"the opposite imaginary part, same as here."},{"Start":"04:06.350 ","End":"04:16.340","Text":"It happens when the coefficients of the quadratic equation, the real numbers."},{"Start":"04:16.340 ","End":"04:19.970","Text":"But actually even the coefficients could be complex numbers."},{"Start":"04:19.970 ","End":"04:23.197","Text":"Let\u0027s take another last example."},{"Start":"04:23.197 ","End":"04:26.325","Text":"Here it is I wrote it out when you weren\u0027t looking,"},{"Start":"04:26.325 ","End":"04:29.180","Text":"and now even the coefficients are complex,"},{"Start":"04:29.180 ","End":"04:30.650","Text":"but I still have an a, b,"},{"Start":"04:30.650 ","End":"04:33.400","Text":"and a c. This bit is a,"},{"Start":"04:33.400 ","End":"04:37.230","Text":"this bit is b and this bit is c,"},{"Start":"04:37.230 ","End":"04:40.299","Text":"and I can still use the quadratic equation."},{"Start":"04:40.299 ","End":"04:45.620","Text":"So what I get is that z is equal to minus"},{"Start":"04:45.620 ","End":"04:51.290","Text":"b plus or minus the square root of b^2,"},{"Start":"04:51.290 ","End":"04:54.680","Text":"which is 4 minus 4ac,"},{"Start":"04:54.680 ","End":"04:56.975","Text":"so it\u0027s 4 times a,"},{"Start":"04:56.975 ","End":"05:00.655","Text":"which is 1 plus i times c,"},{"Start":"05:00.655 ","End":"05:03.970","Text":"which is 1 minus i,"},{"Start":"05:03.970 ","End":"05:08.990","Text":"and all this is over 2a,"},{"Start":"05:08.990 ","End":"05:13.350","Text":"which is twice 1 plus i."},{"Start":"05:13.350 ","End":"05:15.190","Text":"There\u0027s a bit more work in"},{"Start":"05:15.190 ","End":"05:17.950","Text":"this exercise than the previous ones because we\u0027re going to have to do"},{"Start":"05:17.950 ","End":"05:22.000","Text":"some complex number multiplications and then we\u0027re going to have a division,"},{"Start":"05:22.000 ","End":"05:24.400","Text":"but we\u0027ll get there."},{"Start":"05:24.400 ","End":"05:27.720","Text":"Okay, so let\u0027s continue,"},{"Start":"05:27.720 ","End":"05:32.785","Text":"and what I suggest is to compute what\u0027s under the square root sign."},{"Start":"05:32.785 ","End":"05:35.555","Text":"Maybe we can do that at the side."},{"Start":"05:35.555 ","End":"05:39.810","Text":"Let\u0027s see, first of all, I\u0027ll do the 1 plus i times"},{"Start":"05:39.810 ","End":"05:43.975","Text":"1 minus i this bit and see what we get here."},{"Start":"05:43.975 ","End":"05:47.920","Text":"Well, this is a number times its conjugate and remember"},{"Start":"05:47.920 ","End":"05:52.990","Text":"the formula that when you multiply a complex number by its conjugate,"},{"Start":"05:52.990 ","End":"05:56.645","Text":"we get a^2 plus b^2."},{"Start":"05:56.645 ","End":"05:59.550","Text":"I\u0027m going to use that formula here,"},{"Start":"05:59.550 ","End":"06:03.810","Text":"and it\u0027s going to be 1^2 plus 1^2,"},{"Start":"06:03.810 ","End":"06:05.745","Text":"other words, 2."},{"Start":"06:05.745 ","End":"06:07.380","Text":"This part is 2,"},{"Start":"06:07.380 ","End":"06:09.180","Text":"and then under the square root sign,"},{"Start":"06:09.180 ","End":"06:15.660","Text":"I get the square root now 4 minus 4 times 2 because this part was 2 from here,"},{"Start":"06:15.660 ","End":"06:20.340","Text":"and that\u0027s 4 minus 8 is minus 4 so the square root of minus 4."},{"Start":"06:20.340 ","End":"06:27.800","Text":"Back here, we get minus 2 plus or minus"},{"Start":"06:27.800 ","End":"06:37.560","Text":"the square root of minus 4 over twice 1 plus i."},{"Start":"06:38.660 ","End":"06:44.445","Text":"Now back here, the square root of minus 4 is the square root of 4 times minus 1,"},{"Start":"06:44.445 ","End":"06:49.330","Text":"so its square root of 4 is 2 times square root of minus 1 is i, that\u0027s 2i."},{"Start":"06:50.030 ","End":"06:56.355","Text":"What I get, two solutions,"},{"Start":"06:56.355 ","End":"07:01.200","Text":"minus 2 plus 2i"},{"Start":"07:01.200 ","End":"07:09.030","Text":"over twice 1 plus i and minus 2,"},{"Start":"07:09.030 ","End":"07:17.340","Text":"this time, minus 2i over twice 1 plus i."},{"Start":"07:17.340 ","End":"07:23.165","Text":"Certainly, I can divide top and bottom by 2 in each case."},{"Start":"07:23.165 ","End":"07:25.460","Text":"In this case, if I cancel the 2,"},{"Start":"07:25.460 ","End":"07:32.490","Text":"I\u0027ve got minus 1 plus i over 1 plus i,"},{"Start":"07:32.490 ","End":"07:35.190","Text":"and in the other case, if I cancel by 2,"},{"Start":"07:35.190 ","End":"07:37.755","Text":"I have minus 1,"},{"Start":"07:37.755 ","End":"07:42.675","Text":"minus i over 1 plus i,"},{"Start":"07:42.675 ","End":"07:49.865","Text":"and you can see here that if I take the minus outside the brackets,"},{"Start":"07:49.865 ","End":"07:56.395","Text":"this is equal to minus 1 plus i over 1 plus i,"},{"Start":"07:56.395 ","End":"07:59.730","Text":"and that\u0027s going to equal minus 1."},{"Start":"07:59.730 ","End":"08:01.965","Text":"That\u0027s one of the solutions."},{"Start":"08:01.965 ","End":"08:03.810","Text":"As to this,"},{"Start":"08:03.810 ","End":"08:09.675","Text":"let\u0027s just continue this over here,"},{"Start":"08:09.675 ","End":"08:11.880","Text":"I\u0027m going to rewrite it. What was it?"},{"Start":"08:11.880 ","End":"08:16.590","Text":"Minus 1 plus i over 1 plus i,"},{"Start":"08:16.590 ","End":"08:20.165","Text":"and remember when we have a division,"},{"Start":"08:20.165 ","End":"08:25.110","Text":"we multiply top and bottom by the conjugate of the denominator,"},{"Start":"08:25.110 ","End":"08:31.020","Text":"so it\u0027s 1 minus i and 1 minus i."},{"Start":"08:31.020 ","End":"08:33.885","Text":"Let\u0027s see what we get."},{"Start":"08:33.885 ","End":"08:35.670","Text":"On the denominator,"},{"Start":"08:35.670 ","End":"08:37.070","Text":"it\u0027s always a positive number,"},{"Start":"08:37.070 ","End":"08:40.105","Text":"it\u0027s 1^2 plus 1^2,"},{"Start":"08:40.105 ","End":"08:42.420","Text":"pretty much like we did here."},{"Start":"08:42.420 ","End":"08:45.045","Text":"So that comes out to be 2."},{"Start":"08:45.045 ","End":"08:46.770","Text":"On the numerator, let\u0027s see,"},{"Start":"08:46.770 ","End":"08:54.150","Text":"we have minus 1 times 1 is minus 1,1 times i is"},{"Start":"08:54.150 ","End":"09:04.050","Text":"i minus 1 times minus i is also plus i and i times minus i is minus i^2,"},{"Start":"09:04.050 ","End":"09:09.360","Text":"but minus i^2 is plus 1,"},{"Start":"09:09.360 ","End":"09:11.830","Text":"because i^2 is minus 1."},{"Start":"09:11.830 ","End":"09:16.960","Text":"What we get is 1 is 1 with 1 cancels to get 2i over 2,"},{"Start":"09:16.960 ","End":"09:20.050","Text":"which is just equal to i."},{"Start":"09:20.050 ","End":"09:22.429","Text":"So collecting the two together,"},{"Start":"09:22.429 ","End":"09:32.585","Text":"we\u0027ve got that the solution is the Z is either equal to minus 1 or i two solutions."},{"Start":"09:32.585 ","End":"09:35.150","Text":"So from now on,"},{"Start":"09:35.150 ","End":"09:36.770","Text":"one word in complex numbers,"},{"Start":"09:36.770 ","End":"09:39.785","Text":"all quadratic equations will have two solutions."},{"Start":"09:39.785 ","End":"09:42.260","Text":"Well, except the case that there\u0027s only one solution,"},{"Start":"09:42.260 ","End":"09:45.410","Text":"but that\u0027s really two solutions that just happened to be the same."},{"Start":"09:45.410 ","End":"09:48.880","Text":"In any events, we don\u0027t get the situation where there\u0027s no solution,"},{"Start":"09:48.880 ","End":"09:51.495","Text":"because the square root can be of anything,"},{"Start":"09:51.495 ","End":"09:53.755","Text":"and that\u0027s what was impeding us."},{"Start":"09:53.755 ","End":"09:55.910","Text":"That\u0027s it. That\u0027s the last chapter."},{"Start":"09:55.910 ","End":"09:58.890","Text":"Done with complex numbers."}],"ID":4922},{"Watched":false,"Name":"Exercise 1","Duration":"4m 20s","ChapterTopicVideoID":4919,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4919.jpeg","UploadDate":"2016-02-02T09:20:08.1970000","DurationForVideoObject":"PT4M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"In this exercise,"},{"Start":"00:02.070 ","End":"00:05.700","Text":"a little bit of practice in computations"},{"Start":"00:05.700 ","End":"00:09.720","Text":"with complex numbers and addition or subtraction,"},{"Start":"00:09.720 ","End":"00:17.955","Text":"and then multiplied by a real number or an imaginary number or a mixture."},{"Start":"00:17.955 ","End":"00:19.725","Text":"Anyway, let\u0027s get started with"},{"Start":"00:19.725 ","End":"00:26.070","Text":"the first one which is a simple addition of two complex numbers."},{"Start":"00:26.070 ","End":"00:30.495","Text":"What we do is we add the real part;"},{"Start":"00:30.495 ","End":"00:32.055","Text":"4 plus 3,"},{"Start":"00:32.055 ","End":"00:34.960","Text":"and that would be 7."},{"Start":"00:34.960 ","End":"00:37.415","Text":"Then the imaginary part,"},{"Start":"00:37.415 ","End":"00:39.545","Text":"i and minus 2i,"},{"Start":"00:39.545 ","End":"00:44.545","Text":"so that\u0027s minus i and that\u0027s all there is to it."},{"Start":"00:44.545 ","End":"00:49.980","Text":"Here we have a subtraction almost the same as addition."},{"Start":"00:49.980 ","End":"00:52.965","Text":"We do the real part separately;"},{"Start":"00:52.965 ","End":"00:58.710","Text":"minus 2, minus 4 would be minus 6,"},{"Start":"00:58.710 ","End":"01:05.565","Text":"and then 3i takeaway minus 5i is 3i plus 5i,"},{"Start":"01:05.565 ","End":"01:09.180","Text":"so the answer is minus 6 plus 8i."},{"Start":"01:09.180 ","End":"01:13.935","Text":"Very straightforward. The next one,"},{"Start":"01:13.935 ","End":"01:21.785","Text":"we have to multiply first each bit by a real number."},{"Start":"01:21.785 ","End":"01:24.920","Text":"Multiplying a complex number by a real number is very"},{"Start":"01:24.920 ","End":"01:28.700","Text":"straightforward also because you just multiply each part."},{"Start":"01:28.700 ","End":"01:34.830","Text":"In the first one 3 times 2 is 6 and 3 times 1 is 3,"},{"Start":"01:34.830 ","End":"01:37.200","Text":"so we get 6 plus 3i."},{"Start":"01:37.200 ","End":"01:38.670","Text":"That\u0027s the first bit."},{"Start":"01:38.670 ","End":"01:42.315","Text":"The second bit multiplied by the 5,"},{"Start":"01:42.315 ","End":"01:44.010","Text":"but it\u0027s a minus 5,"},{"Start":"01:44.010 ","End":"01:48.300","Text":"and we\u0027ll need to put it in brackets because the whole thing is being subtracted."},{"Start":"01:48.300 ","End":"01:50.730","Text":"So 5 times 6 is 30,"},{"Start":"01:50.730 ","End":"01:54.600","Text":"5 times 2i is 10i."},{"Start":"01:54.600 ","End":"01:56.805","Text":"Now we have a subtraction problem,"},{"Start":"01:56.805 ","End":"02:02.205","Text":"so 6 minus 30 for the real part is minus 24,"},{"Start":"02:02.205 ","End":"02:07.335","Text":"and 3i, minus 10i is minus 7i,"},{"Start":"02:07.335 ","End":"02:09.790","Text":"and that\u0027s our answer."},{"Start":"02:10.310 ","End":"02:14.820","Text":"In this one, we multiply not by a real number,"},{"Start":"02:14.820 ","End":"02:17.070","Text":"but by a pure imaginary number,"},{"Start":"02:17.070 ","End":"02:19.690","Text":"and that changes things a bit."},{"Start":"02:19.690 ","End":"02:23.330","Text":"The way to do this is to multiply each bit,"},{"Start":"02:23.330 ","End":"02:24.800","Text":"but also reverse the order,"},{"Start":"02:24.800 ","End":"02:31.875","Text":"because notice that if I multiply i by i that gives me i^2 which is minus 1,"},{"Start":"02:31.875 ","End":"02:33.840","Text":"and that\u0027s the real part,"},{"Start":"02:33.840 ","End":"02:38.770","Text":"and i times 4 is 4i."},{"Start":"02:38.930 ","End":"02:42.930","Text":"When you have an imaginary in front just be prepared to change the order"},{"Start":"02:42.930 ","End":"02:46.890","Text":"and then that will be the standard representation."},{"Start":"02:46.890 ","End":"02:48.660","Text":"We still have the other part."},{"Start":"02:48.660 ","End":"02:52.920","Text":"So again, do the 3i by the 2i first,"},{"Start":"02:52.920 ","End":"02:55.170","Text":"and that gives us 6i ^2,"},{"Start":"02:55.170 ","End":"03:00.330","Text":"and 6i ^2 is minus 6."},{"Start":"03:00.330 ","End":"03:04.170","Text":"Let\u0027s just make a reminder,"},{"Start":"03:04.170 ","End":"03:07.935","Text":"i^2 is minus 1."},{"Start":"03:07.935 ","End":"03:14.500","Text":"Then the 3i times the minus 5 is minus 15i."},{"Start":"03:14.900 ","End":"03:18.135","Text":"Because it\u0027s a plus, I don\u0027t need brackets here,"},{"Start":"03:18.135 ","End":"03:21.870","Text":"and we now just collect together the real part is minus 1,"},{"Start":"03:21.870 ","End":"03:24.855","Text":"minus 6 is minus 7."},{"Start":"03:24.855 ","End":"03:31.540","Text":"4i minus 15i is minus 11i."},{"Start":"03:32.540 ","End":"03:35.150","Text":"Now the last one."},{"Start":"03:35.150 ","End":"03:38.010","Text":"This part has an imaginary multiplier,"},{"Start":"03:38.010 ","End":"03:40.365","Text":"this is one is a real multiplier."},{"Start":"03:40.365 ","End":"03:42.450","Text":"For the imaginary one, once again,"},{"Start":"03:42.450 ","End":"03:47.610","Text":"I do it the 2i times the i first, that\u0027s 2i^2,"},{"Start":"03:47.610 ","End":"03:52.110","Text":"and therefore, we have minus 2,"},{"Start":"03:52.110 ","End":"03:57.090","Text":"and then the 2i times the 3 is 6i."},{"Start":"03:57.090 ","End":"04:00.035","Text":"The other one, because the 4 is real,"},{"Start":"04:00.035 ","End":"04:02.045","Text":"you don\u0027t have to change the order or anything,"},{"Start":"04:02.045 ","End":"04:07.775","Text":"so we get plus 4 times 2 is 8 and minus 12i and then collect together."},{"Start":"04:07.775 ","End":"04:10.310","Text":"Minus 2 and 8 is 6,"},{"Start":"04:10.310 ","End":"04:14.810","Text":"6 minus 12i is minus 6i."},{"Start":"04:14.810 ","End":"04:20.410","Text":"This is just warm-up exercises and we\u0027re done."}],"ID":4923},{"Watched":false,"Name":"Exercise 2","Duration":"10m 27s","ChapterTopicVideoID":4920,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4920.jpeg","UploadDate":"2016-02-02T09:21:50.9370000","DurationForVideoObject":"PT10M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.074","Text":"In this exercise, we\u0027ll be practicing multiplication of complex numbers."},{"Start":"00:06.074 ","End":"00:12.940","Text":"There\u0027s a long way and a short way and I\u0027ll start with the long way of doing things."},{"Start":"00:13.820 ","End":"00:22.530","Text":"I guess we\u0027ll start with number 1 and we just expand using the general distributive law,"},{"Start":"00:22.530 ","End":"00:25.215","Text":"meaning each 1 of these multiplied with each 1 of these."},{"Start":"00:25.215 ","End":"00:34.305","Text":"Let\u0027s say 4 times 3 is 12 and then 4 times minus 2i is minus 8i,"},{"Start":"00:34.305 ","End":"00:38.115","Text":"i times 3 is 3i,"},{"Start":"00:38.115 ","End":"00:41.025","Text":"you don\u0027t write i3, you write 3i."},{"Start":"00:41.025 ","End":"00:46.480","Text":"Then i times minus 2i is minus 2i^2."},{"Start":"00:46.850 ","End":"00:49.385","Text":"Then in the next step,"},{"Start":"00:49.385 ","End":"00:53.580","Text":"you collect the real part."},{"Start":"00:53.620 ","End":"00:57.205","Text":"I just want to remind you"},{"Start":"00:57.205 ","End":"01:05.150","Text":"that i^2 is minus 1, that\u0027s very basic."},{"Start":"01:05.150 ","End":"01:08.990","Text":"It\u0027s the most essential thing there is in complex numbers."},{"Start":"01:08.990 ","End":"01:13.050","Text":"This i^2 is minus 1,"},{"Start":"01:13.050 ","End":"01:15.030","Text":"so what I have here is plus 2."},{"Start":"01:15.030 ","End":"01:23.680","Text":"So 12 plus 2 is 14 and minus 8i plus 3i is minus 5i."},{"Start":"01:23.990 ","End":"01:27.930","Text":"Next one. Just expand,"},{"Start":"01:27.930 ","End":"01:31.275","Text":"minus 2 times 4 is minus 8,"},{"Start":"01:31.275 ","End":"01:36.345","Text":"minus 2 times minus 5i is plus 10i,"},{"Start":"01:36.345 ","End":"01:43.330","Text":"and then plus 12i and minus 15i^2."},{"Start":"01:43.330 ","End":"01:45.740","Text":"Then I collect this with this."},{"Start":"01:45.740 ","End":"01:51.035","Text":"This is minus 8 plus 15 and"},{"Start":"01:51.035 ","End":"01:56.385","Text":"that gives us 7 and 10 and 12,"},{"Start":"01:56.385 ","End":"02:02.220","Text":"that makes it 22i."},{"Start":"02:02.220 ","End":"02:08.520","Text":"Next one. Let\u0027s see if we can get all 3 of them in, that\u0027s it."},{"Start":"02:08.520 ","End":"02:15.090","Text":"Minus 4 times minus 2 is 8 and then this with this minus"},{"Start":"02:15.090 ","End":"02:20.805","Text":"12i plus 4i"},{"Start":"02:20.805 ","End":"02:28.485","Text":"and minus 6i^2."},{"Start":"02:28.485 ","End":"02:31.680","Text":"So 8 plus 6, I\u0027ll write it again,"},{"Start":"02:31.680 ","End":"02:35.670","Text":"it\u0027s so important, i^2 is minus 1,"},{"Start":"02:35.670 ","End":"02:44.610","Text":"is 14 and minus 12 plus 4 is minus 8i."},{"Start":"02:44.610 ","End":"02:47.580","Text":"That\u0027s that one and the next,"},{"Start":"02:47.580 ","End":"02:54.750","Text":"2 minus 6i minus"},{"Start":"02:54.750 ","End":"03:00.960","Text":"4i plus 12i^2,"},{"Start":"03:00.960 ","End":"03:09.720","Text":"2 minus 12 is minus 10 and minus 10i."},{"Start":"03:09.720 ","End":"03:13.215","Text":"Then 4 times 4, 16,"},{"Start":"03:13.215 ","End":"03:16.270","Text":"4 times 2 plus 8i"},{"Start":"03:16.730 ","End":"03:25.510","Text":"minus 8i and then minus 4i^2."},{"Start":"03:26.600 ","End":"03:30.540","Text":"This cancels with this,"},{"Start":"03:30.540 ","End":"03:36.360","Text":"so we get just 16 plus 4,"},{"Start":"03:36.360 ","End":"03:38.520","Text":"that makes it 20."},{"Start":"03:38.520 ","End":"03:41.979","Text":"These 2 cancel, so 0i, so just 20."},{"Start":"03:41.979 ","End":"03:49.630","Text":"Of course, we could have done this also with the formula for a minus b,"},{"Start":"03:49.630 ","End":"03:58.610","Text":"a plus b and said straight away that this is a^2 minus b^2,"},{"Start":"03:58.610 ","End":"04:08.875","Text":"which in our case came out to be 4^2 minus 2i^2."},{"Start":"04:08.875 ","End":"04:14.470","Text":"But 2i^2 squared is 4i^2,"},{"Start":"04:14.470 ","End":"04:19.470","Text":"which is minus 4,"},{"Start":"04:19.470 ","End":"04:21.633","Text":"so minus minus 4 is plus 4."},{"Start":"04:21.633 ","End":"04:26.910","Text":"Anyway, you get 16 plus 4 and that\u0027s the same as the 20."},{"Start":"04:26.910 ","End":"04:30.190","Text":"I\u0027m just observing that we could have used difference of squares here."},{"Start":"04:30.190 ","End":"04:34.550","Text":"We\u0027re not saying it\u0027s easier, just noting it."},{"Start":"04:34.550 ","End":"04:38.290","Text":"What I\u0027d like to do now is go back to the beginning and do it with"},{"Start":"04:38.290 ","End":"04:41.650","Text":"the short way once you practice with this."},{"Start":"04:41.650 ","End":"04:45.740","Text":"I\u0027m going to go back up to number 1."},{"Start":"04:46.320 ","End":"04:52.270","Text":"Just clean up a bit."},{"Start":"04:52.270 ","End":"04:54.160","Text":"I\u0027ll use a different color."},{"Start":"04:54.160 ","End":"05:00.185","Text":"In this technique, we just can do everything in our heads without any intermediate steps."},{"Start":"05:00.185 ","End":"05:05.960","Text":"You say we get a real part from the first times the first and the last times the last."},{"Start":"05:05.960 ","End":"05:12.415","Text":"4 times 3 is 12 and then minus 2i^2,"},{"Start":"05:12.415 ","End":"05:15.660","Text":"so you just change the sign, it\u0027s plus 2."},{"Start":"05:15.660 ","End":"05:20.490","Text":"Basically you say this times this minus the 1 times minus 2,"},{"Start":"05:20.490 ","End":"05:23.860","Text":"so it\u0027s 12 plus 2 is 14."},{"Start":"05:23.860 ","End":"05:26.990","Text":"I\u0027m just reminding you this is 12."},{"Start":"05:26.990 ","End":"05:34.850","Text":"In our heads, we say 12 minus the product of these 2 is minus 2, for the first bit."},{"Start":"05:34.850 ","End":"05:41.670","Text":"For the i bit we just say 1 times 3 is 3,"},{"Start":"05:41.670 ","End":"05:45.225","Text":"4 times minus 2 is minus 8,"},{"Start":"05:45.225 ","End":"05:47.670","Text":"so we say minus 5i,"},{"Start":"05:47.670 ","End":"05:51.515","Text":"but mentally we did the exercise of"},{"Start":"05:51.515 ","End":"05:59.200","Text":"3 minus 8 to get to the minus 5."},{"Start":"05:59.200 ","End":"06:02.010","Text":"I\u0027m just going to erase that."},{"Start":"06:02.010 ","End":"06:06.705","Text":"That was just what takes place mentally."},{"Start":"06:06.705 ","End":"06:09.680","Text":"Let me be a bit more precise in this one,"},{"Start":"06:09.680 ","End":"06:11.495","Text":"I was a bit vague here."},{"Start":"06:11.495 ","End":"06:13.190","Text":"You took the real part,"},{"Start":"06:13.190 ","End":"06:15.170","Text":"you multiply this times this,"},{"Start":"06:15.170 ","End":"06:17.735","Text":"minus this times this, ignore the i,"},{"Start":"06:17.735 ","End":"06:23.180","Text":"say minus 2 times 4 is 8 and then subtract 3 times minus 5,"},{"Start":"06:23.180 ","End":"06:24.815","Text":"which is minus 15."},{"Start":"06:24.815 ","End":"06:35.010","Text":"We get 8 minus minus 15 is 7."},{"Start":"06:35.840 ","End":"06:41.520","Text":"Then for the imaginary bit you just take this times this plus this times this,"},{"Start":"06:41.520 ","End":"06:45.285","Text":"12 plus 10 is 22i."},{"Start":"06:45.285 ","End":"06:52.160","Text":"In fact, I think it will help if I just write a little formula in general, when we have,"},{"Start":"06:52.160 ","End":"06:59.950","Text":"let\u0027s say 1 of them is a plus bi and the other 1 is c plus di."},{"Start":"06:59.980 ","End":"07:10.040","Text":"What we get the real part is this times this minus this times this."},{"Start":"07:10.040 ","End":"07:15.385","Text":"In other words, it\u0027s ac minus bd."},{"Start":"07:15.385 ","End":"07:18.890","Text":"The reason for the minus is because that\u0027s what takes"},{"Start":"07:18.890 ","End":"07:22.370","Text":"the place of the i^2 if we were multiplying it out."},{"Start":"07:22.370 ","End":"07:26.050","Text":"That\u0027s the real part and the imaginary part,"},{"Start":"07:26.050 ","End":"07:28.245","Text":"we do this with this,"},{"Start":"07:28.245 ","End":"07:31.000","Text":"the inners and the outers."},{"Start":"07:31.520 ","End":"07:36.930","Text":"That would give us ad plus bc or bc plus ad."},{"Start":"07:36.930 ","End":"07:39.945","Text":"Let\u0027s say ad plus bc,"},{"Start":"07:39.945 ","End":"07:43.290","Text":"that\u0027s the i part."},{"Start":"07:43.290 ","End":"07:45.110","Text":"If I wanted to phrase that,"},{"Start":"07:45.110 ","End":"07:49.100","Text":"I say that for the real part"},{"Start":"07:49.100 ","End":"07:53.540","Text":"we get the first and the first minus the second and the second."},{"Start":"07:53.540 ","End":"07:55.775","Text":"Then for the i part,"},{"Start":"07:55.775 ","End":"08:00.765","Text":"the inner pair plus the outer pair."},{"Start":"08:00.765 ","End":"08:05.925","Text":"Let\u0027s practice that on this one, on the third one."},{"Start":"08:05.925 ","End":"08:09.255","Text":"Minus 4 and minus 2,"},{"Start":"08:09.255 ","End":"08:12.750","Text":"we multiply it by, is 8,"},{"Start":"08:12.750 ","End":"08:16.245","Text":"minus 2 times 3 is minus 6,"},{"Start":"08:16.245 ","End":"08:22.095","Text":"so I do 8 subtract minus 6 and that gives me 14."},{"Start":"08:22.095 ","End":"08:24.620","Text":"I can think of that 14 as being from here,"},{"Start":"08:24.620 ","End":"08:30.930","Text":"this times this minus this times this and for the i part,"},{"Start":"08:30.930 ","End":"08:34.540","Text":"this times this plus this times this."},{"Start":"08:34.690 ","End":"08:39.575","Text":"Sorry, plus 4 minus 12,"},{"Start":"08:39.575 ","End":"08:44.040","Text":"that\u0027s minus 8 and that gets the i stuck onto it."},{"Start":"08:45.100 ","End":"08:49.980","Text":"I\u0027ll continue. I\u0027ll just copy this down there."},{"Start":"08:52.090 ","End":"08:54.680","Text":"The formula, let\u0027s see,"},{"Start":"08:54.680 ","End":"08:58.530","Text":"I copied it, I\u0027ll paste it, there we are."},{"Start":"08:58.870 ","End":"09:01.310","Text":"Applying it to this one,"},{"Start":"09:01.310 ","End":"09:04.670","Text":"it doesn\u0027t hurt actually to draw the arcs in, I see."},{"Start":"09:04.670 ","End":"09:06.950","Text":"The first with the first, the second with the second,"},{"Start":"09:06.950 ","End":"09:08.330","Text":"then we do a subtraction."},{"Start":"09:08.330 ","End":"09:10.280","Text":"2 times 1 is 2,"},{"Start":"09:10.280 ","End":"09:13.400","Text":"minus 4 times minus 3 is 12,"},{"Start":"09:13.400 ","End":"09:17.685","Text":"2 minus 12 minus 10."},{"Start":"09:17.685 ","End":"09:22.450","Text":"Then this and this and this time we add."},{"Start":"09:22.450 ","End":"09:26.120","Text":"2 times minus 3 is minus 6,"},{"Start":"09:26.120 ","End":"09:27.590","Text":"this is minus 4,"},{"Start":"09:27.590 ","End":"09:34.790","Text":"minus 6 add minus 4 gives us minus 10."},{"Start":"09:34.790 ","End":"09:39.380","Text":"Same as this, but this time it\u0027s with an i on the second bit."},{"Start":"09:41.200 ","End":"09:44.970","Text":"Don\u0027t need this now."},{"Start":"09:45.100 ","End":"09:49.795","Text":"Once again, this and this and this and this and subtract."},{"Start":"09:49.795 ","End":"09:51.790","Text":"4 times 4 is 16,"},{"Start":"09:51.790 ","End":"09:54.690","Text":"minus 2 times 2 is minus 4,"},{"Start":"09:54.690 ","End":"09:58.695","Text":"16 less minus 4 is 20."},{"Start":"09:58.695 ","End":"10:02.190","Text":"Then this with this, this with this."},{"Start":"10:02.190 ","End":"10:05.370","Text":"Minus 2 times 4 is minus 8,"},{"Start":"10:05.370 ","End":"10:06.990","Text":"4 times 2 is plus 8,"},{"Start":"10:06.990 ","End":"10:09.450","Text":"minus 8 and 8 is 0."},{"Start":"10:09.450 ","End":"10:12.830","Text":"I could put plus 0 and then the i and of course,"},{"Start":"10:12.830 ","End":"10:15.740","Text":"we don\u0027t actually need this part,"},{"Start":"10:15.740 ","End":"10:19.320","Text":"just drop it, get the same as before."},{"Start":"10:19.460 ","End":"10:22.595","Text":"We did this exercise twice,"},{"Start":"10:22.595 ","End":"10:27.420","Text":"long and short, and you can choose whichever one you like. We\u0027re done."}],"ID":4924},{"Watched":false,"Name":"Exercise 3","Duration":"9m 41s","ChapterTopicVideoID":4921,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4921.jpeg","UploadDate":"2016-02-02T09:23:25.0230000","DurationForVideoObject":"PT9M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.390","Text":"In this exercise, there are six separate computations."},{"Start":"00:05.390 ","End":"00:09.050","Text":"They all have exponents or powers,"},{"Start":"00:09.050 ","End":"00:13.125","Text":"and mostly they\u0027re based on powers of i."},{"Start":"00:13.125 ","End":"00:16.575","Text":"Now the most fundamental thing that we could"},{"Start":"00:16.575 ","End":"00:20.145","Text":"use in all of these exercises is the following."},{"Start":"00:20.145 ","End":"00:23.850","Text":"That i to the power of 2,"},{"Start":"00:23.850 ","End":"00:32.250","Text":"i^2 rather is = minus 1."},{"Start":"00:32.250 ","End":"00:36.810","Text":"Also useful is i^3 is = minus i,"},{"Start":"00:36.810 ","End":"00:41.815","Text":"and very useful also is i^4 is =1."},{"Start":"00:41.815 ","End":"00:48.110","Text":"Just for completeness, I\u0027ll throw also i^1 =i in here."},{"Start":"00:48.110 ","End":"00:51.445","Text":"So we have i^1, 2, 3, and 4."},{"Start":"00:51.445 ","End":"00:55.730","Text":"It turns out that it goes periodically that i^5 is i,"},{"Start":"00:55.730 ","End":"00:59.747","Text":"i^6, i^7,"},{"Start":"00:59.747 ","End":"01:01.200","Text":"i^8, i^9, 10, 11, 12, and so on."},{"Start":"01:01.200 ","End":"01:05.413","Text":"There are only four combinations of i to the power of,"},{"Start":"01:05.413 ","End":"01:07.100","Text":"and this is the basis of it."},{"Start":"01:07.100 ","End":"01:10.670","Text":"Let\u0027s just assume that we remember these by heart and we\u0027ll use those in"},{"Start":"01:10.670 ","End":"01:15.270","Text":"all the six exercises that follow."},{"Start":"01:15.270 ","End":"01:19.570","Text":"We\u0027ll start with the first one."},{"Start":"01:20.660 ","End":"01:24.195","Text":"i^8, I can write,"},{"Start":"01:24.195 ","End":"01:26.200","Text":"this is one possibility,"},{"Start":"01:26.200 ","End":"01:30.765","Text":"as i^4 times i^4."},{"Start":"01:30.765 ","End":"01:33.570","Text":"Since i^4 is 1,"},{"Start":"01:33.570 ","End":"01:34.755","Text":"that is very important."},{"Start":"01:34.755 ","End":"01:36.525","Text":"This is 1 times 1,"},{"Start":"01:36.525 ","End":"01:38.295","Text":"which is just 1."},{"Start":"01:38.295 ","End":"01:42.480","Text":"Or I could have written (i^4)^2."},{"Start":"01:42.480 ","End":"01:46.840","Text":"Anyway, the answer is just 1."},{"Start":"01:47.300 ","End":"01:50.995","Text":"The other way what I meant was (i^4)^2."},{"Start":"01:50.995 ","End":"01:53.915","Text":"The reason I might do it this way is if it was a larger number,"},{"Start":"01:53.915 ","End":"01:57.770","Text":"say i^80, I wouldn\u0027t want to write i^4,"},{"Start":"01:57.770 ","End":"02:00.720","Text":"i^4, i^4 20 times."},{"Start":"02:00.720 ","End":"02:03.325","Text":"I would rather say (i^4)^20."},{"Start":"02:03.325 ","End":"02:06.110","Text":"In this case, I\u0027ll do it this way."},{"Start":"02:06.110 ","End":"02:07.610","Text":"But if it was a very large number,"},{"Start":"02:07.610 ","End":"02:09.350","Text":"I would maybe like,"},{"Start":"02:09.350 ","End":"02:11.625","Text":"I don\u0027t know i^400 here."},{"Start":"02:11.625 ","End":"02:13.775","Text":"Then I would say (i^4)^100,"},{"Start":"02:13.775 ","End":"02:16.880","Text":"and then it\u0027s 1^2 and it\u0027s 1."},{"Start":"02:16.880 ","End":"02:20.070","Text":"So this is just an alternative."},{"Start":"02:21.790 ","End":"02:25.415","Text":"I\u0027ll change the color. It\u0027s an alternative solution."},{"Start":"02:25.415 ","End":"02:28.700","Text":"Now here, i^15,"},{"Start":"02:28.700 ","End":"02:30.935","Text":"it doesn\u0027t divide by 4,"},{"Start":"02:30.935 ","End":"02:36.170","Text":"but I could write it as i to the power of the nearest thing"},{"Start":"02:36.170 ","End":"02:41.540","Text":"that divides by 4 that\u0027s below 15, is 12."},{"Start":"02:41.540 ","End":"02:48.120","Text":"I could write it then as i^12 times i^3."},{"Start":"02:48.120 ","End":"02:50.330","Text":"Then I could use this."},{"Start":"02:50.330 ","End":"02:53.150","Text":"I actually prefer to go with the exponent."},{"Start":"02:53.150 ","End":"02:54.875","Text":"This one would be the better way,"},{"Start":"02:54.875 ","End":"02:59.890","Text":"because in this case I can use it as (i^4)^3."},{"Start":"02:59.890 ","End":"03:08.255","Text":"I hope you remember your rules of exponents, so times i^3."},{"Start":"03:08.255 ","End":"03:10.340","Text":"Then this one,"},{"Start":"03:10.340 ","End":"03:15.750","Text":"i^4 is 1, 1^3 is 1."},{"Start":"03:16.070 ","End":"03:18.330","Text":"Write it as 1^3."},{"Start":"03:18.330 ","End":"03:22.810","Text":"An i^3 from here is minus i."},{"Start":"03:22.810 ","End":"03:28.380","Text":"Altogether the answer is minus i."},{"Start":"03:28.720 ","End":"03:34.175","Text":"We use this quite heavily that i^4 is 1."},{"Start":"03:34.175 ","End":"03:43.160","Text":"I will say you should remember that i^3 is minus i. P is fairly often i^2= minus 1,"},{"Start":"03:43.160 ","End":"03:44.900","Text":"of course is a very basic,"},{"Start":"03:44.900 ","End":"03:48.245","Text":"that\u0027s almost the definition of i."},{"Start":"03:48.245 ","End":"03:50.225","Text":"Of course, you know this."},{"Start":"03:50.225 ","End":"03:54.670","Text":"Continuing with the next one."},{"Start":"03:54.670 ","End":"04:01.120","Text":"The formulas, the stuff in red is gone but that\u0027s okay."},{"Start":"04:01.120 ","End":"04:05.710","Text":"Change my mind, I think I just copied them over."},{"Start":"04:05.710 ","End":"04:08.360","Text":"We\u0027ll have them right in front of us."},{"Start":"04:08.360 ","End":"04:11.930","Text":"Number 3 is not just i to the power of,"},{"Start":"04:11.930 ","End":"04:14.110","Text":"its minus i to the power of."},{"Start":"04:14.110 ","End":"04:20.720","Text":"Now minus i is minus 1 times i."},{"Start":"04:20.720 ","End":"04:23.630","Text":"When we take a product to the power of,"},{"Start":"04:23.630 ","End":"04:25.805","Text":"we take each bit to the power."},{"Start":"04:25.805 ","End":"04:29.700","Text":"So this thing will equal"},{"Start":"04:30.490 ","End":"04:37.870","Text":"minus 1^6 times i^6."},{"Start":"04:37.870 ","End":"04:43.880","Text":"Now minus 1^6 is because the 6 is an even number, this is =1."},{"Start":"04:43.880 ","End":"04:52.950","Text":"I^6, I can write as i^4 times i^2 because I want to use these."},{"Start":"04:52.960 ","End":"04:58.290","Text":"The i^4 is 1, i^2 is minus 1."},{"Start":"04:58.290 ","End":"05:04.177","Text":"So altogether I get 1 times 1 times minus 1,"},{"Start":"05:04.177 ","End":"05:07.920","Text":"and so the answer is minus 1."},{"Start":"05:08.600 ","End":"05:11.810","Text":"We have another one with minus i."},{"Start":"05:11.810 ","End":"05:13.555","Text":"Same technique."},{"Start":"05:13.555 ","End":"05:21.860","Text":"Write this as minus 1^9, i^9."},{"Start":"05:21.860 ","End":"05:26.015","Text":"This gives us, because 9 is an odd number,"},{"Start":"05:26.015 ","End":"05:29.345","Text":"minus 1^9 is minus 1."},{"Start":"05:29.345 ","End":"05:33.155","Text":"I^9, I can write as"},{"Start":"05:33.155 ","End":"05:41.105","Text":"i^8 times i. I take the multiple of 4 closest but below 9."},{"Start":"05:41.105 ","End":"05:44.795","Text":"Then this is equal to,"},{"Start":"05:44.795 ","End":"05:48.595","Text":"i^8 is also 1 because it\u0027s (i^4)^2,"},{"Start":"05:48.595 ","End":"05:51.735","Text":"or i^4 times i^4."},{"Start":"05:51.735 ","End":"05:58.469","Text":"This gives us minus 1 times 1 times i."},{"Start":"05:58.469 ","End":"06:02.635","Text":"This is equal to minus i."},{"Start":"06:02.635 ","End":"06:08.165","Text":"In general, i to the power of any number that\u0027s divisible by 4 is 1."},{"Start":"06:08.165 ","End":"06:10.805","Text":"So it\u0027s not just that i^4 is 1,"},{"Start":"06:10.805 ","End":"06:12.965","Text":"i^8 is 1,"},{"Start":"06:12.965 ","End":"06:18.010","Text":"i^12 is 1, i^16 is 1 and so on."},{"Start":"06:18.010 ","End":"06:25.975","Text":"In the next one, it\u0027s more involved."},{"Start":"06:25.975 ","End":"06:32.155","Text":"We have this whole thing to the power of 2 or squared."},{"Start":"06:32.155 ","End":"06:37.220","Text":"We just use the algebraic formula,"},{"Start":"06:38.060 ","End":"06:42.054","Text":"special binomial products or expansions."},{"Start":"06:42.054 ","End":"06:44.380","Text":"In general in algebra,"},{"Start":"06:44.380 ","End":"06:47.730","Text":"(a+b)^2"},{"Start":"06:47.730 ","End":"06:55.470","Text":"is a^2 plus 2ab plus b^2."},{"Start":"06:55.470 ","End":"06:58.205","Text":"Looking ahead at number 6,"},{"Start":"06:58.205 ","End":"07:00.400","Text":"we might need the same thing with a minus."},{"Start":"07:00.400 ","End":"07:03.345","Text":"Actually, I don\u0027t have to write another formula."},{"Start":"07:03.345 ","End":"07:05.690","Text":"Usually, I combine the 2 formulas in 1."},{"Start":"07:05.690 ","End":"07:07.175","Text":"If there\u0027s a minus here,"},{"Start":"07:07.175 ","End":"07:09.514","Text":"then the middle term gets the minus."},{"Start":"07:09.514 ","End":"07:11.600","Text":"So it\u0027s either plus here and plus here,"},{"Start":"07:11.600 ","End":"07:13.310","Text":"or minus here and minus here."},{"Start":"07:13.310 ","End":"07:15.930","Text":"That will be good for these two."},{"Start":"07:16.240 ","End":"07:22.385","Text":"In this case, using the formula with plus a is 2i, b is 3."},{"Start":"07:22.385 ","End":"07:28.170","Text":"I get (2i)^2 plus"},{"Start":"07:28.300 ","End":"07:35.000","Text":"twice 2i times 3 plus 3^2."},{"Start":"07:35.000 ","End":"07:41.510","Text":"This gives us 4i^2"},{"Start":"07:41.510 ","End":"07:46.226","Text":"plus 2 times 2 times 3 is 12i,"},{"Start":"07:46.226 ","End":"07:48.510","Text":"plus 3^2 is 9."},{"Start":"07:48.510 ","End":"07:52.170","Text":"Now i^2 is minus 1."},{"Start":"07:52.170 ","End":"07:58.490","Text":"So this is minus 4 plus 9 is 5."},{"Start":"07:58.490 ","End":"08:01.450","Text":"I\u0027m not going to write it again. i^2 is minus 1."},{"Start":"08:01.450 ","End":"08:05.215","Text":"That\u0027s the most basic thing that there is to know about complex numbers."},{"Start":"08:05.215 ","End":"08:08.185","Text":"Then plus 12i."},{"Start":"08:08.185 ","End":"08:11.900","Text":"This is our answer to this."},{"Start":"08:12.050 ","End":"08:17.770","Text":"Here, we\u0027re going to use the formula with a minus."},{"Start":"08:17.770 ","End":"08:20.559","Text":"But the thing is, I think we should first"},{"Start":"08:20.559 ","End":"08:23.920","Text":"simplify the powers of I and then use the formula,"},{"Start":"08:23.920 ","End":"08:25.810","Text":"but you could do it the other way round."},{"Start":"08:25.810 ","End":"08:28.335","Text":"So I suggest saying as follows."},{"Start":"08:28.335 ","End":"08:30.680","Text":"That for I expand it out the side,"},{"Start":"08:30.680 ","End":"08:36.275","Text":"I\u0027ll say i^5 is i^4 times i^1,"},{"Start":"08:36.275 ","End":"08:40.305","Text":"break up the 5 into power of 4,"},{"Start":"08:40.305 ","End":"08:43.605","Text":"multiple of 4 closest to 5."},{"Start":"08:43.605 ","End":"08:47.990","Text":"Then i^4 is 1, i^1 is i."},{"Start":"08:47.990 ","End":"08:50.045","Text":"So it is just i."},{"Start":"08:50.045 ","End":"08:52.985","Text":"In the case of i^13,"},{"Start":"08:52.985 ","End":"08:59.990","Text":"I\u0027ll write this as i to the multiple of 4 before 13 is 12."},{"Start":"08:59.990 ","End":"09:03.470","Text":"This is i^12 and also i^1 is 13,"},{"Start":"09:03.470 ","End":"09:05.030","Text":"is 12 plus 1."},{"Start":"09:05.030 ","End":"09:07.340","Text":"This is also equal to 1,"},{"Start":"09:07.340 ","End":"09:10.460","Text":"and this is also equal to i."},{"Start":"09:10.460 ","End":"09:13.040","Text":"This is interesting."},{"Start":"09:13.040 ","End":"09:15.080","Text":"These are both equal to i."},{"Start":"09:15.080 ","End":"09:16.280","Text":"Why is that interesting?"},{"Start":"09:16.280 ","End":"09:23.600","Text":"Because then I get i minus i^2 and I don\u0027t even care what power it is."},{"Start":"09:23.600 ","End":"09:30.174","Text":"0 to any power 0 happens to be 2, which is just 0."},{"Start":"09:30.174 ","End":"09:33.575","Text":"We don\u0027t even, I guess we don\u0027t even need this formula."},{"Start":"09:33.575 ","End":"09:39.410","Text":"I thought we would. Part 6 was the last."},{"Start":"09:39.410 ","End":"09:42.030","Text":"So we are done."}],"ID":4925},{"Watched":false,"Name":"Exercise 4","Duration":"14m 6s","ChapterTopicVideoID":4922,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4922.jpeg","UploadDate":"2016-02-02T09:25:41.1500000","DurationForVideoObject":"PT14M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.440","Text":"In this exercise,"},{"Start":"00:01.440 ","End":"00:04.050","Text":"there are 5 separate problems,"},{"Start":"00:04.050 ","End":"00:06.195","Text":"each one a division exercise,"},{"Start":"00:06.195 ","End":"00:07.380","Text":"except the last one,"},{"Start":"00:07.380 ","End":"00:08.970","Text":"whether there\u0027s something squared in,"},{"Start":"00:08.970 ","End":"00:11.775","Text":"but generally this is practice for division."},{"Start":"00:11.775 ","End":"00:16.275","Text":"I\u0027d like to remind you how we do division of complex numbers."},{"Start":"00:16.275 ","End":"00:20.625","Text":"We multiply top and bottom by what is called the conjugate."},{"Start":"00:20.625 ","End":"00:25.080","Text":"I\u0027ll just get ready for number 1 here."},{"Start":"00:25.080 ","End":"00:27.990","Text":"And I\u0027ll remind you what a conjugate is."},{"Start":"00:27.990 ","End":"00:33.945","Text":"If we have a complex number a plus bi,"},{"Start":"00:33.945 ","End":"00:38.490","Text":"then its conjugate is a minus bi,"},{"Start":"00:38.490 ","End":"00:40.260","Text":"or if this was a minus bi,"},{"Start":"00:40.260 ","End":"00:44.625","Text":"the conjugate would be with the plus it\u0027s the reverse sign."},{"Start":"00:44.625 ","End":"00:50.550","Text":"Let\u0027s write the word conjugate and as I say,"},{"Start":"00:50.550 ","End":"00:54.585","Text":"the trick is to multiply top and bottom"},{"Start":"00:54.585 ","End":"01:00.224","Text":"of the division by the conjugate of the denominator."},{"Start":"01:00.224 ","End":"01:02.280","Text":"How does that help us?"},{"Start":"01:02.280 ","End":"01:05.610","Text":"Because in general, if you multiply a number by"},{"Start":"01:05.610 ","End":"01:10.930","Text":"its conjugate a plus bi times a minus bi."},{"Start":"01:10.940 ","End":"01:16.605","Text":"What we get actually is a^2 plus b^2,"},{"Start":"01:16.605 ","End":"01:20.535","Text":"which is a real number without any imaginary part."},{"Start":"01:20.535 ","End":"01:24.750","Text":"If you use the difference of squares formula on this,"},{"Start":"01:24.750 ","End":"01:27.915","Text":"you actually get a^2 minus b^2 i^2,"},{"Start":"01:27.915 ","End":"01:31.035","Text":"but the i^2 is minus 1 and so this is what we get."},{"Start":"01:31.035 ","End":"01:41.310","Text":"I\u0027ll just make a note that we multiply top and bottom of"},{"Start":"01:41.310 ","End":"01:46.060","Text":"the fraction by conjugate"},{"Start":"01:46.850 ","End":"01:56.350","Text":"of the bottom denominator of course when I say top and bottom numerator and denominator."},{"Start":"01:57.320 ","End":"02:00.975","Text":"Let\u0027s apply that to the first one."},{"Start":"02:00.975 ","End":"02:03.645","Text":"We multiply top and bottom."},{"Start":"02:03.645 ","End":"02:06.405","Text":"What I\u0027m really saying is I\u0027m multiplying by 1,"},{"Start":"02:06.405 ","End":"02:12.525","Text":"but the conjugate of the bottom is 2 minus i."},{"Start":"02:12.525 ","End":"02:19.890","Text":"That\u0027s how we do it and then we have a multiplication on the top."},{"Start":"02:19.890 ","End":"02:21.870","Text":"So the top is easy enough,"},{"Start":"02:21.870 ","End":"02:27.300","Text":"5 times 2 minus i. I\u0027ll just write it in full the first time."},{"Start":"02:27.300 ","End":"02:34.080","Text":"5 times 2 minus i over 2 plus i,"},{"Start":"02:34.080 ","End":"02:44.560","Text":"2 minus i and this is equal to 5 times 2 minus i is 10 minus 5i."},{"Start":"02:45.050 ","End":"02:56.010","Text":"The bottom, using this formula is 2^2 plus 1^2 and 2^2 plus 1^2 is 5."},{"Start":"02:56.010 ","End":"02:57.450","Text":"Let\u0027s make a note of that."},{"Start":"02:57.450 ","End":"03:00.855","Text":"That is 2^2 plus 1^2,"},{"Start":"03:00.855 ","End":"03:03.580","Text":"1 being the 1i."},{"Start":"03:04.280 ","End":"03:12.180","Text":"I should have not multiplied it out if I would have left it like this,"},{"Start":"03:12.180 ","End":"03:13.440","Text":"it would have been easier to see."},{"Start":"03:13.440 ","End":"03:16.860","Text":"We get back to 2 minus i."},{"Start":"03:16.860 ","End":"03:19.995","Text":"Because this is 5 times 2 minus i,"},{"Start":"03:19.995 ","End":"03:25.360","Text":"I\u0027m canceling this with this and that\u0027s the answer."},{"Start":"03:26.270 ","End":"03:30.010","Text":"Onto the next one."},{"Start":"03:30.410 ","End":"03:37.260","Text":"As before, we multiply top and bottom by the conjugate of the denominator."},{"Start":"03:37.260 ","End":"03:39.075","Text":"Instead of 5 minus 2i,"},{"Start":"03:39.075 ","End":"03:41.100","Text":"it\u0027ll be 5 plus 2i."},{"Start":"03:41.100 ","End":"03:44.820","Text":"The same thing here, 5 plus 2i."},{"Start":"03:44.820 ","End":"03:50.955","Text":"This will equal we know how to multiply complex numbers."},{"Start":"03:50.955 ","End":"03:56.085","Text":"We do this times this minus this times this for the real part,"},{"Start":"03:56.085 ","End":"03:58.380","Text":"2 times 5 is 10,"},{"Start":"03:58.380 ","End":"04:00.390","Text":"3 times 2 is 6,"},{"Start":"04:00.390 ","End":"04:08.280","Text":"10 minus 6 is 4 and then the imaginary part is this times this plus this"},{"Start":"04:08.280 ","End":"04:16.875","Text":"times this so it\u0027s 15 plus 4 is 19i over,"},{"Start":"04:16.875 ","End":"04:23.670","Text":"and using the formula for multiplying by a conjugate it\u0027s 5^2 plus 2^2,"},{"Start":"04:23.670 ","End":"04:29.290","Text":"which we can mentally do is 25 plus 4 is 29."},{"Start":"04:29.480 ","End":"04:34.035","Text":"We would leave it like this except that it says in standard form."},{"Start":"04:34.035 ","End":"04:35.340","Text":"In standard form,"},{"Start":"04:35.340 ","End":"04:41.565","Text":"it\u0027s 4 over 29 plus 19 over 29i."},{"Start":"04:41.565 ","End":"04:48.310","Text":"It\u0027s always a plus bi in standard form and that\u0027s the answer to number 2."},{"Start":"04:48.410 ","End":"04:52.140","Text":"Next is number 3."},{"Start":"04:52.140 ","End":"04:55.095","Text":"And in number 3,"},{"Start":"04:55.095 ","End":"05:01.965","Text":"the usual multiply top and bottom by the conjugate of the bottom,"},{"Start":"05:01.965 ","End":"05:12.490","Text":"as we can think of it as 0 minus 4i and the conjugate of 0 minus 4i is 0 plus 4i."},{"Start":"05:13.810 ","End":"05:18.620","Text":"If it helps out a minus bi a plus bi."},{"Start":"05:18.620 ","End":"05:20.510","Text":"But we don\u0027t really need the 0s,"},{"Start":"05:20.510 ","End":"05:22.650","Text":"I guess I\u0027ll erase that."},{"Start":"05:27.440 ","End":"05:30.840","Text":"Of course, same thing on the numerator because we can\u0027t"},{"Start":"05:30.840 ","End":"05:34.740","Text":"change the exercise and multiply it by 1 essentially."},{"Start":"05:34.740 ","End":"05:42.315","Text":"On the numerator we get the real part comes from minus i times 4i,"},{"Start":"05:42.315 ","End":"05:44.760","Text":"which is minus 4i^2,"},{"Start":"05:44.760 ","End":"05:47.835","Text":"which is plus 4,"},{"Start":"05:47.835 ","End":"05:52.455","Text":"and 6 times 4 is 24i."},{"Start":"05:52.455 ","End":"05:54.330","Text":"That\u0027s the numerator,"},{"Start":"05:54.330 ","End":"05:59.745","Text":"on the denominator we could use the formula of a^2 plus b^2,"},{"Start":"05:59.745 ","End":"06:03.945","Text":"where a is 0 and it\u0027s just 4^2 is 16."},{"Start":"06:03.945 ","End":"06:09.075","Text":"Of course, you could also see it by just saying this times this is 4 times 4 is 16."},{"Start":"06:09.075 ","End":"06:16.590","Text":"It\u0027s minus 16i^2 minus 16i^2 squared is 16 because minus i^2 squared is 1."},{"Start":"06:16.590 ","End":"06:20.415","Text":"Let\u0027s write it in standard form."},{"Start":"06:20.415 ","End":"06:23.715","Text":"Standard form is 4 over 16,"},{"Start":"06:23.715 ","End":"06:32.460","Text":"4 over 16 mentally we can do that\u0027s a 1/4 and 24 over 16,"},{"Start":"06:32.460 ","End":"06:33.900","Text":"if you cancel by 8,"},{"Start":"06:33.900 ","End":"06:36.480","Text":"you get 3 over 2, which is 1/2."},{"Start":"06:36.480 ","End":"06:39.045","Text":"But I\u0027ll leave it as 3 over 2."},{"Start":"06:39.045 ","End":"06:44.010","Text":"Somehow looks better in this context as an improper fraction,"},{"Start":"06:44.010 ","End":"06:53.295","Text":"dividing by 8i and this is the answer to 3 onto number 4."},{"Start":"06:53.295 ","End":"06:57.165","Text":"As before, we multiply by one,"},{"Start":"06:57.165 ","End":"07:02.383","Text":"but just something over itself and that something is the conjugate of the denominator."},{"Start":"07:02.383 ","End":"07:07.390","Text":"To 4+2i here and 4+2i here."},{"Start":"07:07.640 ","End":"07:10.750","Text":"Let\u0027s see what we get."},{"Start":"07:11.780 ","End":"07:14.610","Text":"Trivial, I\u0027ll begin with the denominator."},{"Start":"07:14.610 ","End":"07:17.175","Text":"Using the a^2+b^2 formula,"},{"Start":"07:17.175 ","End":"07:21.330","Text":"4^2+2^2, 16+4, 20."},{"Start":"07:21.330 ","End":"07:24.550","Text":"Let\u0027s see, on the numerator."},{"Start":"07:24.830 ","End":"07:32.070","Text":"In fact, why don\u0027t we use the a+b^2 formula,"},{"Start":"07:32.070 ","End":"07:35.940","Text":"the special binomial, because this is 4+2i^2,"},{"Start":"07:35.940 ","End":"07:44.780","Text":"so it\u0027s 4^2+2*4*2i+2i^2,"},{"Start":"07:46.190 ","End":"07:52.470","Text":"then that will equal the real part is 16"},{"Start":"07:52.470 ","End":"08:01.545","Text":"and 4i^2 is -4, 16-4 is 12."},{"Start":"08:01.545 ","End":"08:11.925","Text":"The middle one, 2*4*2 is 16i/20."},{"Start":"08:11.925 ","End":"08:14.850","Text":"But we want to put it in standard form,"},{"Start":"08:14.850 ","End":"08:20.280","Text":"so it\u0027s 12/20 that can be canceled by 4 to get"},{"Start":"08:20.280 ","End":"08:27.270","Text":"3/5 and 16/20i is"},{"Start":"08:27.270 ","End":"08:31.510","Text":"also can be canceled by 4 and we get 4/5."},{"Start":"08:33.120 ","End":"08:36.305","Text":"This is the answer to Number 4."},{"Start":"08:36.305 ","End":"08:38.475","Text":"On to Number 5."},{"Start":"08:38.475 ","End":"08:40.895","Text":"I\u0027ll just scroll a bit."},{"Start":"08:40.895 ","End":"08:45.190","Text":"Here, it\u0027s not just a complex number over a complex number,"},{"Start":"08:45.190 ","End":"08:46.910","Text":"this one is squared."},{"Start":"08:46.910 ","End":"08:55.505","Text":"We can just modify the general method of multiplying by the conjugate of the denominator."},{"Start":"08:55.505 ","End":"09:03.490","Text":"What we can do is multiply by 2i-3,"},{"Start":"09:06.000 ","End":"09:09.730","Text":"take 2 on the last bit."},{"Start":"09:10.970 ","End":"09:13.725","Text":"Number 5 is a bit different."},{"Start":"09:13.725 ","End":"09:17.190","Text":"Well, for one thing, this is not written in the right order."},{"Start":"09:17.190 ","End":"09:20.659","Text":"Let me just do that first before I do anything."},{"Start":"09:20.659 ","End":"09:24.770","Text":"Let me write it as 3+2i^2."},{"Start":"09:24.770 ","End":"09:29.060","Text":"It\u0027s the same thing, but we usually write the real and then the imaginary."},{"Start":"09:29.060 ","End":"09:31.315","Text":"Now, there\u0027s two ways of doing this."},{"Start":"09:31.315 ","End":"09:33.130","Text":"One way is to square this,"},{"Start":"09:33.130 ","End":"09:35.570","Text":"and then we have a single complex number and we"},{"Start":"09:35.570 ","End":"09:38.645","Text":"multiply top and bottom by the conjugate of the denominator."},{"Start":"09:38.645 ","End":"09:41.570","Text":"But there\u0027s actually a slightly easier way to go,"},{"Start":"09:41.570 ","End":"09:44.765","Text":"and that is to multiply top and bottom."},{"Start":"09:44.765 ","End":"09:47.675","Text":"Actually, the conjugate turns out"},{"Start":"09:47.675 ","End":"09:50.420","Text":"that if you take the conjugate of a square of something,"},{"Start":"09:50.420 ","End":"09:54.910","Text":"it\u0027s the same as taking the conjugate and then squaring it."},{"Start":"09:54.910 ","End":"09:57.435","Text":"But even if you didn\u0027t know that,"},{"Start":"09:57.435 ","End":"09:59.365","Text":"if you see something to the power,"},{"Start":"09:59.365 ","End":"10:02.230","Text":"you multiply it by the conjugate to that same power,"},{"Start":"10:02.230 ","End":"10:04.170","Text":"and it will do the trick."},{"Start":"10:04.170 ","End":"10:09.410","Text":"Let\u0027s do it here and here; 3-2i^2."},{"Start":"10:10.940 ","End":"10:13.570","Text":"Now, what should I start with;"},{"Start":"10:13.570 ","End":"10:15.570","Text":"the numerator or the denominator?"},{"Start":"10:15.570 ","End":"10:18.420","Text":"I\u0027ll say let\u0027s start with the denominator."},{"Start":"10:18.420 ","End":"10:28.165","Text":"The denominator, I can rewrite it as (3+2i)(3-2i),"},{"Start":"10:28.165 ","End":"10:35.680","Text":"this whole thing squared because the square of a product is the product of the squares."},{"Start":"10:37.740 ","End":"10:41.350","Text":"We\u0027ll do that in a moment. I\u0027ll just write that."},{"Start":"10:41.350 ","End":"10:50.880","Text":"On the numerator we get (1-2i*3)-(2i^2)."},{"Start":"10:50.880 ","End":"10:56.435","Text":"Perhaps, the only real computational I\u0027ll do is this one."},{"Start":"10:56.435 ","End":"11:04.240","Text":"This comes out as using the binomial expansion,"},{"Start":"11:04.740 ","End":"11:13.210","Text":"(3^2-2*3*2i+2i^2)"},{"Start":"11:15.090 ","End":"11:19.490","Text":"Close the other bracket."},{"Start":"11:20.240 ","End":"11:23.630","Text":"I\u0027ll change that to a square bracket."},{"Start":"11:23.630 ","End":"11:25.780","Text":"Bit at a time."},{"Start":"11:25.780 ","End":"11:29.255","Text":"Let\u0027s see where we go."},{"Start":"11:29.255 ","End":"11:32.040","Text":"I would simplify this."},{"Start":"11:32.040 ","End":"11:36.130","Text":"Here we get (1-2i)*,"},{"Start":"11:36.130 ","End":"11:40.390","Text":"now let us see, this is 9 and this is 4i^2."},{"Start":"11:40.390 ","End":"11:46.595","Text":"4i^2 is-4, so 9-4 is 5 and then the middle one,"},{"Start":"11:46.595 ","End":"11:48.670","Text":"2*3*2 is 12,"},{"Start":"11:48.670 ","End":"11:53.010","Text":"so we have -12i for the numerator."},{"Start":"11:53.010 ","End":"11:55.040","Text":"For the denominator,"},{"Start":"11:55.040 ","End":"11:59.565","Text":"what I get is here I have a number times its conjugate."},{"Start":"11:59.565 ","End":"12:05.230","Text":"Using the formula that we is longer on screen, the a^2+b^2,"},{"Start":"12:05.230 ","End":"12:12.295","Text":"on the denominator I have 3^2+2^2,"},{"Start":"12:12.295 ","End":"12:16.090","Text":"but there\u0027s also this squared here."},{"Start":"12:20.300 ","End":"12:28.100","Text":"Let me continue, equals dividing line."},{"Start":"12:28.100 ","End":"12:31.745","Text":"Let\u0027s do the (1-2i)*(5-12i)."},{"Start":"12:31.745 ","End":"12:34.140","Text":"1*5 is 5,"},{"Start":"12:34.140 ","End":"12:42.555","Text":"and 5-24i^2 is 5+24"},{"Start":"12:42.555 ","End":"12:46.690","Text":"is 29 for the real part."},{"Start":"12:48.140 ","End":"12:57.880","Text":"Sorry. No. Take 2 on the last bit."},{"Start":"12:57.880 ","End":"13:06.115","Text":"1-5 is 5,-2i*-12i is+24i^2,"},{"Start":"13:06.115 ","End":"13:16.190","Text":"5+24i^2 is like 5-24, so-19."},{"Start":"13:16.190 ","End":"13:18.660","Text":"The i term comes out to"},{"Start":"13:18.660 ","End":"13:29.410","Text":"be -10i-12i, altogether,-22i."},{"Start":"13:30.120 ","End":"13:33.525","Text":"The denominator, 3^2 is 9,"},{"Start":"13:33.525 ","End":"13:40.410","Text":"2^2 is 4, 13^2, 169."},{"Start":"13:40.410 ","End":"13:43.940","Text":"Finally, we just have to put it in standard form."},{"Start":"13:43.940 ","End":"13:48.960","Text":"It\u0027s -19/169, that\u0027s"},{"Start":"13:48.960 ","End":"13:58.210","Text":"the real part, -22/169i."},{"Start":"14:00.840 ","End":"14:03.455","Text":"Just highlight that."},{"Start":"14:03.455 ","End":"14:04.830","Text":"That\u0027s the last part,"},{"Start":"14:04.830 ","End":"14:06.940","Text":"so we are done."}],"ID":4926},{"Watched":false,"Name":"Exercise 5","Duration":"2m 12s","ChapterTopicVideoID":4923,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4923.jpeg","UploadDate":"2016-02-02T09:26:02.1570000","DurationForVideoObject":"PT2M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.845","Text":"In this exercise,"},{"Start":"00:01.845 ","End":"00:07.770","Text":"we have to write each of these 3 square root of negative numbers in terms of i."},{"Start":"00:07.770 ","End":"00:12.165","Text":"It was once impossible before we had complex numbers and i."},{"Start":"00:12.165 ","End":"00:13.680","Text":"Now it becomes possible,"},{"Start":"00:13.680 ","End":"00:15.945","Text":"so we\u0027re going to simplify these."},{"Start":"00:15.945 ","End":"00:22.710","Text":"The main thing we\u0027re going to use is the fact that in general the square root of a,"},{"Start":"00:22.710 ","End":"00:26.580","Text":"b is the square root of a square root of b,"},{"Start":"00:26.580 ","End":"00:28.034","Text":"and in particular,"},{"Start":"00:28.034 ","End":"00:30.150","Text":"the square root of minus something,"},{"Start":"00:30.150 ","End":"00:33.180","Text":"say minus a, is going to be"},{"Start":"00:33.180 ","End":"00:38.745","Text":"the square root of a times the square root of minus 1, which is i."},{"Start":"00:38.745 ","End":"00:40.635","Text":"That\u0027s in general."},{"Start":"00:40.635 ","End":"00:44.220","Text":"That\u0027s the strategy."},{"Start":"00:44.220 ","End":"00:46.700","Text":"Let\u0027s get to the exercises."},{"Start":"00:46.700 ","End":"00:50.190","Text":"The first one,"},{"Start":"00:51.680 ","End":"00:54.395","Text":"I could even do this whole thing again."},{"Start":"00:54.395 ","End":"01:00.605","Text":"I\u0027ll write it as the square root of 4 times minus 1,"},{"Start":"01:00.605 ","End":"01:06.430","Text":"which is the square root of 4 times the square root of minus 1."},{"Start":"01:06.430 ","End":"01:09.855","Text":"The square root of 4 is 2,"},{"Start":"01:09.855 ","End":"01:12.570","Text":"square root of minus 1 is i,"},{"Start":"01:12.570 ","End":"01:14.660","Text":"so the answer is 2i,"},{"Start":"01:14.660 ","End":"01:19.950","Text":"and this one, very similar."},{"Start":"01:19.950 ","End":"01:24.170","Text":"I\u0027ll shorten it a bit."},{"Start":"01:24.170 ","End":"01:26.240","Text":"Whenever we have the square root of minus something,"},{"Start":"01:26.240 ","End":"01:31.290","Text":"we just take the square root of the positive part and write i."},{"Start":"01:31.490 ","End":"01:37.690","Text":"This is equal to 3i because square root of 9 is 3."},{"Start":"01:37.690 ","End":"01:42.965","Text":"Same here. This is square root of 5 times i."},{"Start":"01:42.965 ","End":"01:51.710","Text":"As opposed to previous cases where we could actually compute the square root,"},{"Start":"01:51.710 ","End":"01:54.470","Text":"here we can\u0027t, or at least we\u0027d need a calculator,"},{"Start":"01:54.470 ","End":"01:56.030","Text":"so to leave it precise,"},{"Start":"01:56.030 ","End":"01:58.675","Text":"we just leave the answer like this."},{"Start":"01:58.675 ","End":"02:02.870","Text":"Square root of 5i, it happens to be 2.26 something,"},{"Start":"02:02.870 ","End":"02:05.270","Text":"but don\u0027t bother doing it as a decimal."},{"Start":"02:05.270 ","End":"02:07.250","Text":"This is actually more precise."},{"Start":"02:07.250 ","End":"02:12.120","Text":"That\u0027s done with this exercise."}],"ID":4927},{"Watched":false,"Name":"Exercise 6","Duration":"12m 49s","ChapterTopicVideoID":4924,"CourseChapterTopicPlaylistID":45200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4924.jpeg","UploadDate":"2016-02-02T09:28:10.4070000","DurationForVideoObject":"PT12M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.441","Text":"In this exercise, there are six quadratic equations to solve."},{"Start":"00:05.441 ","End":"00:10.080","Text":"I should mention that we assume the variable to be a complex variable,"},{"Start":"00:10.080 ","End":"00:14.190","Text":"meaning it can be a complex number with a real and imaginary part."},{"Start":"00:14.190 ","End":"00:21.375","Text":"It isn\u0027t clear until you see Part 6 that there are actually complex numbers here."},{"Start":"00:21.375 ","End":"00:26.910","Text":"Having said that, let\u0027s start with the first one."},{"Start":"00:26.910 ","End":"00:30.930","Text":"We could use the quadratic formula but in this case,"},{"Start":"00:30.930 ","End":"00:33.435","Text":"there\u0027s a missing x."},{"Start":"00:33.435 ","End":"00:38.090","Text":"We can solve it more simply by saying x^2 equals minus"},{"Start":"00:38.090 ","End":"00:46.415","Text":"36. x equals plus or minus the square root of minus 36."},{"Start":"00:46.415 ","End":"00:50.600","Text":"Like I said, we allow complex numbers."},{"Start":"00:50.600 ","End":"00:57.625","Text":"This is equal to plus or minus 6i."},{"Start":"00:57.625 ","End":"01:03.470","Text":"Because the square root of minus"},{"Start":"01:03.470 ","End":"01:13.140","Text":"36 is the square root of 36 times the square root of minus 1 which is 6i."},{"Start":"01:14.360 ","End":"01:16.685","Text":"Onto the next one."},{"Start":"01:16.685 ","End":"01:20.815","Text":"These are the two answers for the first highlighted."},{"Start":"01:20.815 ","End":"01:23.388","Text":"Anyway, next one."},{"Start":"01:23.388 ","End":"01:26.100","Text":"x^2 minus 2x plus 5 is 0."},{"Start":"01:26.100 ","End":"01:28.940","Text":"Let\u0027s use the formula here."},{"Start":"01:28.940 ","End":"01:33.350","Text":"x equals minus b which is 2 plus or"},{"Start":"01:33.350 ","End":"01:39.020","Text":"minus the square root of b^2 is 4 minus 4,"},{"Start":"01:39.020 ","End":"01:42.470","Text":"a is 1 times c,"},{"Start":"01:42.470 ","End":"01:46.900","Text":"and all this over 2a."},{"Start":"01:46.900 ","End":"01:51.165","Text":"What we get is 2 plus or minus."},{"Start":"01:51.165 ","End":"01:58.135","Text":"Let\u0027s see. 4 minus 20 is minus 16."},{"Start":"01:58.135 ","End":"02:01.370","Text":"Because we\u0027re working with complex numbers,"},{"Start":"02:01.370 ","End":"02:05.195","Text":"we can say that this is 2 plus or minus."},{"Start":"02:05.195 ","End":"02:08.720","Text":"The square root of minus 16 is 4i."},{"Start":"02:08.720 ","End":"02:17.405","Text":"Square root of minus 1 is i and square root of 16 is 4/2 which gives us the two answers,"},{"Start":"02:17.405 ","End":"02:23.625","Text":"either 2 plus 4i/2 which is 1 plus 2i."},{"Start":"02:23.625 ","End":"02:26.565","Text":"Dividing by 2 is easy."},{"Start":"02:26.565 ","End":"02:31.935","Text":"2 minus 4i divided by 2 so it\u0027s 1 minus 2i."},{"Start":"02:31.935 ","End":"02:34.800","Text":"These are the two solutions."},{"Start":"02:34.800 ","End":"02:38.610","Text":"For this one, we could combine them if you want to,"},{"Start":"02:38.610 ","End":"02:44.720","Text":"you could say 1 plus or minus 2i and use the plus-minus optional."},{"Start":"02:44.720 ","End":"02:48.230","Text":"Next one. Again, the formula."},{"Start":"02:48.230 ","End":"02:55.910","Text":"This time the variable is t. We got minus b plus or minus the square root of"},{"Start":"02:55.910 ","End":"03:05.615","Text":"b^2 minus 4 times 1 times 5/2."},{"Start":"03:05.615 ","End":"03:07.550","Text":"This time under the square root sign,"},{"Start":"03:07.550 ","End":"03:12.110","Text":"it\u0027s 16 minus 20 which is minus 4."},{"Start":"03:12.110 ","End":"03:19.260","Text":"What we get is 4 plus or minus the square root of minus 4/2."},{"Start":"03:19.660 ","End":"03:25.205","Text":"The square root of minus 4 is 2i."},{"Start":"03:25.205 ","End":"03:29.520","Text":"4 plus or minus 2i/2."},{"Start":"03:30.070 ","End":"03:35.060","Text":"This time I\u0027ll leave it as plus-minus and it\u0027s 2 plus or"},{"Start":"03:35.060 ","End":"03:42.095","Text":"minus i. I\u0027ll highlight this."},{"Start":"03:42.095 ","End":"03:44.620","Text":"For those that don\u0027t like the plus or minus,"},{"Start":"03:44.620 ","End":"03:50.810","Text":"we can say 2 plus i and 2 minus i are the solutions."},{"Start":"03:51.410 ","End":"03:54.977","Text":"Proceeding. Number 4,"},{"Start":"03:54.977 ","End":"03:57.250","Text":"let\u0027s see the rest of them."},{"Start":"03:57.250 ","End":"04:02.710","Text":"We have 4, 5, and 6. z or z"},{"Start":"04:02.710 ","End":"04:06.925","Text":"is actually a very common letter to use when"},{"Start":"04:06.925 ","End":"04:11.570","Text":"using complex numbers just like x for real numbers."},{"Start":"04:11.760 ","End":"04:14.910","Text":"Let\u0027s see. If I say z,"},{"Start":"04:14.910 ","End":"04:19.060","Text":"I mean z. The British way."},{"Start":"04:19.280 ","End":"04:30.226","Text":"z equals minus b plus or minus the square root of b^2,"},{"Start":"04:30.226 ","End":"04:37.310","Text":"that\u0027s 144 minus 4 times 4 times 25."},{"Start":"04:37.310 ","End":"04:41.730","Text":"All this over 2a is 8."},{"Start":"04:41.730 ","End":"04:44.355","Text":"Now, under the square root sign,"},{"Start":"04:44.355 ","End":"04:51.384","Text":"what we have is 4 times 4 times 25 is 400."},{"Start":"04:51.384 ","End":"04:56.570","Text":"144 minus 400 is 256."},{"Start":"04:56.570 ","End":"05:04.380","Text":"What I get is minus 12 plus or minus the square root of, did I say 256,"},{"Start":"05:04.380 ","End":"05:07.330","Text":"I meant minus 256/8"},{"Start":"05:09.380 ","End":"05:15.320","Text":"and this is equal to minus 12, plus or minus."},{"Start":"05:15.320 ","End":"05:21.110","Text":"Now, this comes out to be 16i because we\u0027ve got the square root of 256 is 16 and"},{"Start":"05:21.110 ","End":"05:26.950","Text":"the square root of minus 1 is i over 8."},{"Start":"05:26.950 ","End":"05:30.930","Text":"If I write it in standard form,"},{"Start":"05:30.930 ","End":"05:34.730","Text":"12/8 is 1/2 or 3/2."},{"Start":"05:34.730 ","End":"05:37.175","Text":"Let\u0027s write it as an improper fraction."},{"Start":"05:37.175 ","End":"05:41.060","Text":"We got minus 3/2."},{"Start":"05:41.060 ","End":"05:43.140","Text":"You could write this as 1 1/2 if you feel like,"},{"Start":"05:43.140 ","End":"05:51.330","Text":"plus or minus 16/8 is 2i."},{"Start":"05:51.330 ","End":"05:53.845","Text":"You can write the answer like this."},{"Start":"05:53.845 ","End":"05:56.440","Text":"Or if you prefer to write it separately,"},{"Start":"05:56.440 ","End":"06:05.390","Text":"then minus 3/2 plus 2i is one solution and the other solution is minus 3/2 minus 2i."},{"Start":"06:05.390 ","End":"06:11.140","Text":"Whatever you prefer. In the next one, again,"},{"Start":"06:11.140 ","End":"06:17.455","Text":"the formula Z equals minus b plus or minus"},{"Start":"06:17.455 ","End":"06:25.060","Text":"the square root of b^2 is 256 minus 4 times 4 times 25,"},{"Start":"06:25.060 ","End":"06:31.736","Text":"that looks familiar, over 8."},{"Start":"06:31.736 ","End":"06:35.990","Text":"Very similar to the previous one."},{"Start":"06:35.990 ","End":"06:42.480","Text":"What we get is minus 16 plus or minus the square root."},{"Start":"06:43.300 ","End":"06:45.950","Text":"This is 400."},{"Start":"06:45.950 ","End":"06:54.165","Text":"We computed that before and this minus this gives us the 144, 256 minus 400."},{"Start":"06:54.165 ","End":"06:58.570","Text":"All this over 8."},{"Start":"06:58.850 ","End":"07:05.420","Text":"We get minus 16 plus or minus."},{"Start":"07:05.420 ","End":"07:09.015","Text":"Did I forget to write the minus? Silly me."},{"Start":"07:09.015 ","End":"07:10.950","Text":"There it is, minus."},{"Start":"07:10.950 ","End":"07:13.980","Text":"Square root of 144 is 12,"},{"Start":"07:13.980 ","End":"07:16.425","Text":"square root of minus 1 is i,"},{"Start":"07:16.425 ","End":"07:18.790","Text":"all this over 8."},{"Start":"07:18.790 ","End":"07:21.485","Text":"This time again, similar to this,"},{"Start":"07:21.485 ","End":"07:24.215","Text":"it comes out to be minus 2,"},{"Start":"07:24.215 ","End":"07:28.080","Text":"plus or minus 3/2i."},{"Start":"07:28.300 ","End":"07:31.550","Text":"I\u0027ll leave the answer like this."},{"Start":"07:31.550 ","End":"07:33.890","Text":"Although like before, you can split it up with"},{"Start":"07:33.890 ","End":"07:37.170","Text":"a plus separately and the minus separately."},{"Start":"07:37.480 ","End":"07:41.480","Text":"Last one. This time"},{"Start":"07:41.480 ","End":"07:45.490","Text":"the coefficients are also complex numbers or at least some of them are."},{"Start":"07:45.490 ","End":"07:51.360","Text":"In other words, a is 1 plus i,"},{"Start":"07:51.360 ","End":"07:54.225","Text":"b is 2,"},{"Start":"07:54.225 ","End":"07:59.119","Text":"and c is 1 minus i. I\u0027m emphasizing"},{"Start":"07:59.119 ","End":"08:04.250","Text":"it because we haven\u0027t so far encountered one with complex number coefficients."},{"Start":"08:04.250 ","End":"08:14.740","Text":"We get that Z equals minus b plus or minus the square root of"},{"Start":"08:14.740 ","End":"08:21.075","Text":"b^2 is 4 minus 4 times a"},{"Start":"08:21.075 ","End":"08:28.620","Text":"which is 1 plus i times c which is 1 minus i,"},{"Start":"08:28.620 ","End":"08:37.240","Text":"all this over 2a twice 1 plus i."},{"Start":"08:38.870 ","End":"08:44.375","Text":"We\u0027ll need to do some maybe side computations."},{"Start":"08:44.375 ","End":"08:48.575","Text":"I\u0027d like to just do the bit under the square root sign at the side."},{"Start":"08:48.575 ","End":"08:52.355","Text":"What I have is 4 minus 4."},{"Start":"08:52.355 ","End":"08:55.129","Text":"Now, this is a number times its conjugate,"},{"Start":"08:55.129 ","End":"08:57.565","Text":"the 1 plus i times 1 minus i."},{"Start":"08:57.565 ","End":"09:04.085","Text":"It\u0027s 1^2 plus 1^2 using the rule for conjugates."},{"Start":"09:04.085 ","End":"09:06.230","Text":"But you can just do it directly."},{"Start":"09:06.230 ","End":"09:13.760","Text":"Difference of squares formula 1^2 minus I squared is 1 plus 1 is 2."},{"Start":"09:13.760 ","End":"09:21.990","Text":"Either way, what we get here is 4 minus 4 times 2 is 4 minus 8 is minus 4."},{"Start":"09:22.130 ","End":"09:26.170","Text":"Also note that the square root of minus 4"},{"Start":"09:26.170 ","End":"09:30.225","Text":"is square root of 4 times square root of minus 1."},{"Start":"09:30.225 ","End":"09:38.290","Text":"I\u0027ll just write that, square root of 4 and square root of minus one which is 2i."},{"Start":"09:38.290 ","End":"09:43.800","Text":"Back here, we get minus 2 plus or"},{"Start":"09:43.800 ","End":"09:50.770","Text":"minus 2i over twice 1 plus i."},{"Start":"09:51.610 ","End":"09:59.330","Text":"Now here we have a division of complex numbers but we can first of all,"},{"Start":"09:59.330 ","End":"10:01.895","Text":"divide top and bottom by 2."},{"Start":"10:01.895 ","End":"10:05.705","Text":"What we need actually are two computations."},{"Start":"10:05.705 ","End":"10:08.854","Text":"I mean, this thing itself is,"},{"Start":"10:08.854 ","End":"10:10.640","Text":"well, I could write it down here,"},{"Start":"10:10.640 ","End":"10:14.705","Text":"it\u0027s minus 1,"},{"Start":"10:14.705 ","End":"10:19.400","Text":"plus or minus i/1 plus I."},{"Start":"10:19.400 ","End":"10:23.020","Text":"That\u0027s after I got rid of the 2."},{"Start":"10:23.020 ","End":"10:24.410","Text":"I\u0027ll continue here anyway."},{"Start":"10:24.410 ","End":"10:26.090","Text":"If I take the plus,"},{"Start":"10:26.090 ","End":"10:33.610","Text":"I have minus 1 plus i over 1 plus i."},{"Start":"10:33.610 ","End":"10:35.760","Text":"That\u0027s taking the plus here."},{"Start":"10:35.760 ","End":"10:37.745","Text":"If I take the minus,"},{"Start":"10:37.745 ","End":"10:40.820","Text":"then it\u0027s minus 1,"},{"Start":"10:40.820 ","End":"10:46.435","Text":"minus i/1 plus i."},{"Start":"10:46.435 ","End":"10:52.325","Text":"I did the equal here but then I\u0027m moving over to the branch over here."},{"Start":"10:52.325 ","End":"10:55.715","Text":"This is the plus and this is the minus."},{"Start":"10:55.715 ","End":"10:58.675","Text":"One of them turns out to be easy."},{"Start":"10:58.675 ","End":"11:01.520","Text":"This one is the easy one because if you look at it,"},{"Start":"11:01.520 ","End":"11:04.970","Text":"the numerator is just minus 1 times the denominator."},{"Start":"11:04.970 ","End":"11:07.175","Text":"I mean just the same thing with opposite signs."},{"Start":"11:07.175 ","End":"11:09.110","Text":"This is minus 1."},{"Start":"11:09.110 ","End":"11:12.620","Text":"There\u0027s only one more thing left to compute and that\u0027s this."},{"Start":"11:12.620 ","End":"11:15.154","Text":"When we divide complex numbers,"},{"Start":"11:15.154 ","End":"11:19.580","Text":"then we multiply top and bottom by the conjugate of the denominator."},{"Start":"11:19.580 ","End":"11:23.870","Text":"Remember that 1 plus i so 1 minus i but top and"},{"Start":"11:23.870 ","End":"11:28.765","Text":"bottom to make it valid and legal so we\u0027ll multiply it by 1 effectively."},{"Start":"11:28.765 ","End":"11:31.885","Text":"Now, on the denominator,"},{"Start":"11:31.885 ","End":"11:35.540","Text":"the product of the conjugates 1 plus i,1 minus i. I"},{"Start":"11:35.540 ","End":"11:38.855","Text":"think we\u0027ve done this before already here and we got that as 2."},{"Start":"11:38.855 ","End":"11:43.055","Text":"It\u0027s 1^2 plus 1^2 is 2 here."},{"Start":"11:43.055 ","End":"11:50.260","Text":"On the numerator what we have is, let\u0027s see."},{"Start":"11:51.290 ","End":"11:53.745","Text":"Let\u0027s do the real part."},{"Start":"11:53.745 ","End":"12:02.385","Text":"Minus 1 times 1 is minus 1 and i times minus i"},{"Start":"12:02.385 ","End":"12:13.510","Text":"is minus i^2 so that plus 1."},{"Start":"12:14.890 ","End":"12:18.440","Text":"There\u0027s no real part, only an imaginary."},{"Start":"12:18.440 ","End":"12:24.535","Text":"Let\u0027s see, the imaginary part is i times 1,"},{"Start":"12:24.535 ","End":"12:30.080","Text":"and then minus 1 times minus i is also plus i."},{"Start":"12:30.080 ","End":"12:33.210","Text":"So it\u0027s i plus i so it\u0027s 2i/2."},{"Start":"12:34.390 ","End":"12:37.130","Text":"That is equal to just i,"},{"Start":"12:37.130 ","End":"12:39.085","Text":"the 2 is canceled."},{"Start":"12:39.085 ","End":"12:48.640","Text":"Our two answers are minus 1 and i for part 6 and we are done."}],"ID":4928}],"Thumbnail":null,"ID":45200}]

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