WEBVTT
00:00:01.320 --> 00:00:07.240
Simplify negative 12 minus four π all over two π.
00:00:08.200 --> 00:00:13.880
We have two ways in which we can divide a complex number by a purely imaginary number.
00:00:14.480 --> 00:00:16.440
Letβs think about both of those methods.
00:00:17.000 --> 00:00:19.960
The first of these methods involves rewriting the fraction.
00:00:20.800 --> 00:00:35.000
We essentially reverse the rules for adding and subtracting fractions to write negative 12 minus four π over two π as negative 12 over two π minus four π over two π.
00:00:35.760 --> 00:00:41.480
And now we see that if we look at the second fraction, we can simplify it quite nicely.
00:00:42.000 --> 00:00:44.120
π divided by π is one.
00:00:44.640 --> 00:00:53.200
Then, four divided by two is two, so this second fraction simplifies to simply negative two.
00:00:53.800 --> 00:00:56.600
We can also do a little bit of simplifying on our first fraction.
00:00:57.080 --> 00:00:59.640
We divide both the numerator and the denominator by two.
00:01:00.000 --> 00:01:04.760
And so our expression becomes negative six over π minus two.
00:01:05.560 --> 00:01:10.640
But we still have negative six over π causing us some problems.
00:01:11.000 --> 00:01:16.200
And so weβre going to use the key facts that π squared is equal to negative one.
00:01:16.800 --> 00:01:23.920
This means if we can multiply the denominator of our fraction by π, we will get a purely real number.
00:01:24.360 --> 00:01:26.280
Weβll get negative one.
00:01:26.800 --> 00:01:30.120
But of course, we have to do the same to the numerator.
00:01:30.640 --> 00:01:34.840
So weβre going to multiply both the numerator and the denominator of this fraction by π.
00:01:35.520 --> 00:01:39.920
That gives us negative six π over π squared minus two.
00:01:40.560 --> 00:01:43.560
But we now know that π squared is negative one.
00:01:43.920 --> 00:01:48.920
So it in fact gives us negative six π over negative one minus two.
00:01:49.480 --> 00:01:59.280
We can then divide negative six π by negative one, remembering that a negative divided by a negative gives a positive result.
00:01:59.920 --> 00:02:06.360
And we get simply six π minus two or negative two plus six π.
00:02:07.040 --> 00:02:09.480
So, thatβs method one.
00:02:09.880 --> 00:02:11.960
Letβs now consider method two.
00:02:12.480 --> 00:02:16.840
In method two, weβre going to rewrite our purely imaginary denominator.
00:02:17.480 --> 00:02:25.160
Weβre going to write it as zero plus two π so that it essentially looks like the general form of a complex number.
00:02:25.880 --> 00:02:35.400
And then we recall that to divide by a complex number, we write it as a fraction and then multiply both the numerator and the denominator of the conjugate of the denominator.
00:02:36.160 --> 00:02:39.800
And to find the conjugate, we simply change the sign of the imaginary part.
00:02:40.280 --> 00:02:50.760
So, for a complex number of the form π plus ππ, its conjugate π§ bar is π minus ππ.
00:02:51.400 --> 00:02:57.320
And this means the conjugate of zero plus two π is zero minus two π.
00:02:57.960 --> 00:03:03.840
Now that weβve seen where this negative two π comes from, weβre going to get rid of all the zeros.
00:03:04.560 --> 00:03:08.200
And so weβre simply going to multiply the numerator and denominator of our fraction by negative two π.
00:03:08.760 --> 00:03:09.920
Letβs begin with the numerator.
00:03:10.360 --> 00:03:16.120
Weβre going to work out negative two π times negative 12, which is 24π.
00:03:17.160 --> 00:03:24.480
And then weβre going to work out negative two π times negative four π, which is eight π squared.
00:03:25.080 --> 00:03:28.760
Once again, though, we know that π squared is negative one.
00:03:29.360 --> 00:03:40.840
So we get 24π plus eight times negative one, which is 24π minus eight.
00:03:41.400 --> 00:03:43.520
Then we work out the value of the denominator.
00:03:44.160 --> 00:03:46.440
Itβs two π times negative two π.
00:03:46.880 --> 00:03:56.400
Thatβs negative four π squared, which can, of course, be written as negative four times negative one, which is simply four.
00:03:56.960 --> 00:04:07.720
And so when we multiply the numerator and the denominator of our earlier fraction by negative two π, we get 24π minus eight all over four.
00:04:08.280 --> 00:04:12.040
And then we know we can divide both parts of our numerator by four.
00:04:12.880 --> 00:04:20.520
So we get six π minus two, which once again is equal to negative two plus six π.
00:04:21.480 --> 00:04:24.720
Of course, both methods are perfectly valid here.
00:04:25.200 --> 00:04:30.520
Either way, we see that we simplified negative 12 minus four π over two π.
00:04:31.160 --> 00:04:34.960
And we get negative two plus six π.