www.mathcentre.ac.uk/.../03-What-is-a-Complex-NumberF61Mb.mp4

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

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

quadratic equation, and this
time we look at this quadratic

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

And we solve it using the
formula for solving a

quadratic equation.

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

is plus 10.

Plus or minus the square root of

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

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

All divided by two, A2 one or

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

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

Subtract 4 * 1 * 29, which is

116. All divided by two.

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

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

negative number appearing in

here. All divided by two.

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

And all that's divided by two.

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

denominator, all this will
simplify fly down to five plus

or minus two I.

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

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

of this quadratic equation.

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

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

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

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

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

imaginary part of the complex

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

complex number is going to look

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

were A&B are both real numbers.

And I is
the square root

of minus one.

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

as being the real part.

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

Let's have a look at some

more examples.
OK.

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

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

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

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

Suppose Zedd was minus
2 + 5 I.

Here the real part is minus 2.

And the imaginary part is 5.

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

this example the real part is
minus three and the imaginary

part is minus 9.

What about a number like this

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

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

is 5. This is a purely
imaginary complex number.

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

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

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

So in fact all real numbers
are complex numbers with

zero imaginary part.

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

we can start to add,
subtract, multiply, and

divide complex now.