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In this unit I'm going to define
formally what we mean by a

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complex number. To do that,
let's revisit the solution of a

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quadratic equation, and this
time we look at this quadratic

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equation. Here X squared minus
10X plus 29 is 0.

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And we solve it using the
formula for solving a

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quadratic equation.

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Off we go X equals minus B,
which is minus minus 10, which

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is plus 10.

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Plus or minus the square root of

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be squared. B squared is minus
10 squared, which is 100.

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Minus four times a which is one
times C which is 29.

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All divided by two, A2 one or

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two. Let's tidy it. What we've
got? We've got 10 plus or minus

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the square root and under the
square root sign, we've got 100.

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Subtract 4 * 1 * 29, which is

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116. All divided by two.

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Which is 10 plus or minus the
square root and 100 - 116 is

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minus 16 and you'll see we've
got now the square root of a

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negative number appearing in

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here. All divided by two.

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Now the square root of minus
16. We can write as 4I.

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And all that's divided by two.

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And if we cancel the factor of
two in the numerator and

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denominator, all this will
simplify fly down to five plus

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or minus two I.

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And we've got 2 numbers here,
really. We've got one which is 5

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+ 2 I and one which is 5 - 2 I.
And these are the two solutions

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of this quadratic equation.

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Now a number like this one,
which has got a real part, which

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in this case is 5 and an
imaginary part, which is the bit

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that is multiplying the eye,
which in this case is either 2

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or minus two. A number like this
is called a complex number. We

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say that five is the real part
and either 2 or minus two is the

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imaginary part of the complex

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number. We have a formal way of
writing this down. In general, a

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complex number is going to look

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like this. It's going to take
the form zed equals A plus, BI

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were A&B are both real numbers.

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And I is
the square root

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of minus one.

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And this is the general form of
a complex number. We refer to a

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as being the real part.

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And to be as being the imaginary
part of the complex number.

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Let's have a look at some

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more examples.
OK.

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First example, suppose we write
down zed equals 3 + 4 I

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this is a complex number where
the real part is 3.

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And the imaginary part, which is
the number multiplying I.

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Is 4 so the real parts 3 and the
imaginary part is full.

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Suppose Zedd was minus
2 + 5 I.

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Here the real part is minus 2.

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And the imaginary part is 5.

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Both the real and imaginary
parts might be negative, so in

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this example the real part is
minus three and the imaginary

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part is minus 9.

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What about a number like this

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one? There isn't a real part
to this complex number.

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That's purely an imaginary
part, and the imaginary part

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is 5. This is a purely
imaginary complex number.

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Finally. If we
look at say, zed is 17.

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This is a purely real complex
number. If we wanted to do, we

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could write on an imaginary
part, but it will be 0 I.

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So in fact all real numbers
are complex numbers with

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zero imaginary part.

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In the following unit,
we're going to look at how

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we can start to add,
subtract, multiply, and

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divide complex now.