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In this unit, we're going to[br]look at how to divide 2 complex

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numbers. Now, division of[br]complex numbers is rather more

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complicated than addition,[br]Subtraction, and multiplication.

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And Division of complex numbers[br]relies on two very important

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principles. The first is that[br]when you take a complex number

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and multiply by its complex[br]conjugate, you get a real

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number. The second important[br]principle is that when you have

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a fraction, you can multiply the[br]numerator and the denominator.

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That's the number on the top on[br]the number on the bottom of the

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fraction by the same value, and[br]not change the value of a

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fraction. So for example, if[br]you start with a fraction of

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half and you multiply the[br]top and bottom by 5, you get

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5/10 and the value of five[br]10s is the same as the value

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of 1/2. And that's really[br]going to be very important

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when we come into being able[br]to workout. How to divide 1

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complex number by another.[br]So let's look at an example.

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So we're going to take the[br]complex #4 + 7 I. I'm going to

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divide it by the complex number[br]1 - 3. I now remember the

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division is the same thing. It's[br]a fraction, so this complex

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number divided by this one. We[br]can just write a Swan complex

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number over another complex

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number. So now we have a[br]fraction we can say is that we

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won't change the value of this[br]fraction if we multiply the

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numerator and the denominator by[br]the same value.

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I'm going to choose to multiply[br]the denominator by 1 + 3 I.

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1 + 3 I is the complex conjugate[br]of 1 - 3 I and we choose this

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complex conjugate so that when[br]we do the multiplication, what's

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in the denominator will turn out[br]to be a real number.

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So for multiplying the[br]denominator by 1 + 3 I we've got

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to multiply the numerator by 1 +

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3 I. So that way we have[br]multiplied the numerator and

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denominator by the same value,[br]so we haven't changed the value

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of the answer.

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So let's now multiply these two[br]fractions together. We multiply

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out the two terms in the[br]numerator. We multiply out the

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two terms in the denominator, so[br]we get 4 * 1 is 4.

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4 * 3 I is 12 I.

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Seven 8 * 1 is 7 I.

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+78 times plus three. I is plus[br]21 by squares.

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So that's multiplied. The two[br]terms in the numerator. Now we

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multiply the two terms in the[br]denominator to get 1 * 1 one

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times plus 3I.

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Minus three items one.

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And minus three I times[br]plus three. I give this

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minus nine I squared.

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What time do this up?

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21 I squared is 21 times minus[br]one, so that's minus 21, so

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we've got 4 - 21 is minus 17.

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12 + 7 I is 99, so we've[br]got plus 99.

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And then in the denominator.

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I squared is minus one, so we've[br]got minus nine times minus one

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is plus nine, 1 + 9 is 10.

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And three I minus three. I is[br]nothing. So the management turns

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disappear. So we've ended up[br]with a real denominator so we

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could leave our answer like[br]this. Or we could split it up as

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minus 17 over 10 + 19 over 10

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I. And if we want we[br]could write as minus one point 7

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+ 10.9 I.

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So that's our answer. When we[br]divide 4 + 7, I buy 1 - 3.

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I get minus one point 7 + 1.9.

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Now let's do another example to[br]illustrate the principals again.

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Here are two more complex[br]numbers 2 - 5 I and minus 4 + 3

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i's going to divide the first[br]one by the second one.

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And we write those as a fraction[br]2 - 5 I over minus 4 +

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3. I now the way to do it[br]is to multiply. Want to multiply

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the denominator by its complex[br]conjugate, which is minus 4 - 3

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I. And because we're multiplying[br]the denominator by this value,

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we must multiply the numerator.

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By this value as well.

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Now we multiply out the[br]numerator and denominator.

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So we have two times minus four[br]is minus 8 two times minus

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three. I is minus six I.

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Minus 5I Times minus four is[br]plus 20I and minus 5I times

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minus three. I is plus 15[br]I squared.

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And then in the dominator we[br]have minus four times minus 4

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inches 16. Minus four[br]times minus three I,

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which is plus 12 I.

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Plus three I times minus 4[br]inches minus 12 I.

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I'm plus three I times[br]minus three I, which is

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minus nine. I squared.

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And now he tidies up.

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59 squared is minus 15, so we've[br]got minus 8 - 15 is minus 23.

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Minus six I plus[br]20I is plus 49.

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So that's the numerator[br]simplified, and then the

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denominator. We've got minus[br]nine. I squared, so that's plus

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nine. We got 16 + 9 is[br]25 and 12. I minus 12. I

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that disappears, leaving us with[br]a real denominator, which is

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what we wanted. So we can write[br]that as minus 23 over 25 +

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14 over 25 I.

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Which we could also write[br]us minus N .92 +

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.56 high.

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And so that's the result[br]of doing this division.

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Now in the next unit, we'll[br]look at something called the

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organ diagram, which is a way[br]of graphically representing

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complex numbers.