0:00:02.800,0:00:05.880
In this unit, we're going to[br]look at the complex conjugate.
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Every complex number as[br]associated with it, another
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complex number, which is called[br]its complex conjugate.
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And you find the complex[br]conjugate of a complex number
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simply by changing the imaginary[br]part of that number.
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This is best illustrated by[br]looking at some examples.
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So here in this table we've got[br]three different complex numbers,
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and we're going to do is going[br]to find the complex conjugate of
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each of these three numbers.
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So we start by looking at the[br]complex #4 + 7 I.
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On the way to find the complex[br]conjugate is to change the sign
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of the imaginary part. So that[br]means that the plus sign changes
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to a minus sign, so the complex[br]conjugate is 4 minus.
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Seven I.
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Here's another complex number 1[br]- 3. I defined its complex
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number. We change the sign of[br]the imaginary part. In other
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words, we change this minus sign[br]to a plus. So we get the complex
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number 1 + 3 I.
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As another complex number minus[br]4 - 3 I.
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And defined its complex[br]conjugate. Again we change the
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sign of the imaginary part. We[br]don't need to be worried about
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what the sign of the real part[br]is. We just changing the sign of
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the imaginary part and so we get[br]minus 4 + 3 I.
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So whenever we start with any[br]complex number, we can find
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its complex conjugate very[br]easily. We just change the
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sign of the imaginary[br]partners.
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Now the complex conjugate has a[br]very special property and we'll
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see what that is by doing an
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example. OK, what we're going to[br]do is we're going to take a
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complex #4 + 7 I I'm going to[br]multiply it by its own complex
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conjugate, which is 4 - 7 I, and[br]we're going to see what we get.
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So we do. 4 * 4 is[br]16 four times minus Seven. I is
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minus 28 I.
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Plus Seven I times four is
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plus 28I. And plus Seven[br]I minus Seven I is minus
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49 I squared.
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Now when we come to tidy this
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up. The 16 stays there.
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We have minus 28I Plus 28I, so[br]they cancel each other out, so
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we're left with no eyes.
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So there's nothing coming from[br]those two terms, and from this
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term on the end, we've got minus[br]49. I squared. We remember that
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I squared is minus one, so we[br]got minus 49 times minus one, so
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that's plus 49.
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And 16 +[br]49 is 65.
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So when we multiply the two[br]complex numbers together 4 + 7 I
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and its complex conjugate 4 - 7[br]I we find that the answer we get
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is 65. There was the answer is a[br]purely real number, it has no
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imaginary part or an imaginary[br]part of 0.
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That is quite important. So two[br]complex numbers multiplying
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together to give a real number.
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Let's see if it's always[br]happens. Let's try another pair
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and complex number and its[br]complex conjugate and see what
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happens then. OK, in this[br]example we're just going to take
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another complex number and its[br]complex conjugate and multiply
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them together. So what we've got[br]is 1 - 3 I. Its complex
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conjugate is 1 + 3 I let's[br]multiply them together. 1 * 1 is
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one. One times plus three. I[br]is plus 3I.
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Minus three items, one is minus
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three I. And minus three I times[br]plus three I is minus 9.
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I squat.
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Always do now is tidy this up.[br]That means we combined together
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are terms in I and we use the[br]fact that I squared is equal to
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minus one. So we get one start[br]plus three. I minus three I, so
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that's no eyes and then minus[br]nine isquared. Remembering that
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I squared is minus one, we've[br]got minus nine times minus one,
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giving is plus 9, which is an[br]answer of text.
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So once again we've[br]multiplied complex number by
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its complex conjugate and[br]we've got a real number.
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Now this is a very important[br]property and it doesn't just
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happen in the two examples that[br]I've picked, it happens that
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every complex number. If you[br]pick any complex, then be like
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and multiply it by its complex[br]conjugate, you will get a real
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number and that turns out to be[br]very important when we come to
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learn how to divide complex[br]numbers, which is what will be
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doing in the next unit.