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.