WEBVTT
00:00:01.900 --> 00:00:08.620
Find the complex conjugate of the complex number one plus π and the product of this number with its complex conjugate.
00:00:10.920 --> 00:00:12.890
So, thereβre two parts to this question.
00:00:12.890 --> 00:00:17.620
Firstly, weβre asked to find the complex conjugate of this complex number one plus π.
00:00:19.270 --> 00:00:27.810
Well, we can recall that the complex conjugate of a complex number is the complex number we get when we simply change the sign of its imaginary part.
00:00:28.100 --> 00:00:37.790
So, in general, the complex conjugate of the complex number π§ equals π plus ππ is the complex number π§ star, which is equal to π minus ππ.
00:00:38.100 --> 00:00:40.460
Weβve changed the sign of the complex part.
00:00:40.490 --> 00:00:42.070
Itβs no longer positive π.
00:00:42.230 --> 00:00:43.720
Itβs now negative π.
00:00:45.370 --> 00:00:54.280
So if we let π§ be our complex number, one plus π, then to find its complex conjugate π§ star, we simply change the sign of the imaginary part.
00:00:54.450 --> 00:00:57.590
So previously, we had plus π, which is plus one π.
00:00:57.850 --> 00:01:01.150
And we change it to negative π or negative one π.
00:01:02.870 --> 00:01:06.910
The complex conjugate of one plus π is therefore one minus π.
00:01:08.640 --> 00:01:12.380
The second part of this question asks us to find the product of this number.
00:01:12.410 --> 00:01:16.640
So thatβs our original complex number with its complex conjugate.
00:01:16.820 --> 00:01:19.740
So weβre looking for the product of π§ and π§ star.
00:01:21.310 --> 00:01:28.860
As weβve just found the complex conjugate to be one minus π, weβre therefore looking for the product of one plus π and one minus π.
00:01:30.440 --> 00:01:32.620
We can go ahead and distribute the parentheses.
00:01:32.620 --> 00:01:34.960
One multiplied by one gives one.
00:01:35.200 --> 00:01:38.570
And then, one multiplied by negative π gives negative π.
00:01:39.210 --> 00:01:41.940
π multiplied by one gives positive π.
00:01:42.230 --> 00:01:45.900
And then, π multiplied by negative π gives negative π squared.
00:01:46.070 --> 00:01:49.690
So we have one minus π plus π minus π squared.
00:01:51.160 --> 00:01:55.960
Now, of course, in the centre of our expression, negative π plus π simplifies to zero.
00:01:55.960 --> 00:01:57.840
So these two terms cancel out.
00:01:58.090 --> 00:02:00.370
And weβre left with one minus π squared.
00:02:00.850 --> 00:02:04.330
We need to recall here that π squared is equal to negative one.
00:02:05.700 --> 00:02:10.510
We therefore have one minus negative one or one plus one, which is equal to two.
00:02:10.700 --> 00:02:15.450
And so we found that the product of our complex number with its complex conjugate is two.
00:02:17.070 --> 00:02:31.000
In fact, there is actually a general result that we couldβve used here, which is that, for the complex number π§ equals π plus ππ, the product of π§ with its complex conjugate π minus ππ will always be equal to π squared plus π squared.
00:02:32.450 --> 00:02:36.680
We can see that this is certainly the case for our complex number one plus π.
00:02:36.680 --> 00:02:39.400
Both the real and imaginary parts are equal to one.
00:02:39.690 --> 00:02:43.440
And one squared plus one squared is equal to one plus one, which is equal to two.
00:02:45.140 --> 00:02:52.010
To see why this is the case, we just need to distribute the parentheses in the product π plus ππ multiplied by π minus ππ.
00:02:52.280 --> 00:03:02.410
And we see that, in the general case, just as it did in our specific example, the imaginary parts of this expansion cancel, leaving π squared minus π squared π squared.
00:03:02.810 --> 00:03:07.900
Thatβs π squared minus π squared multiplied by negative one, which is π squared plus π squared.
00:03:09.990 --> 00:03:11.510
So weβve completed the problem.
00:03:11.560 --> 00:03:14.910
The complex conjugate of one plus π is one minus π.
00:03:15.090 --> 00:03:18.930
And the product of one plus π with its complex conjugate is two.