Here's a problem you've probably met before (or something similar). You have a quadratic equation involving a variable x, say for example x2+4x+5=0, and you want to find out what x is.
So you write down the "quadratic formula" (see the Notebook on Quadratic Equations if you're not familiar with this):
and then you write down what the constants a, b and c are in this example. We see that a=1, b=4 and c=5 in this case.
Putting those into the formula we find that the quantity inside the square root is negative, in fact it's -4. In the past we have interpreted this as meaning that there's no solution for x. Now we can refine that conclusion to "there's no REAL roots for x".
Now that we know the square root of -4 is 2i, we can continue with
finding x. From the formula we find
x=-2+i or x=-2-i.
These are very strange-looking numbers, a combination of real and imaginary numbers put together. Such numbers are called "complex" numbers and it is with the manipulation, description and use of such numbers that the rest of this Notebook is concerned.
Every complex number has the following shape: a+ib.
The first part of that complex number is real: the real constant a. The second part is
imaginary: the real constant b multiplied by i.
The constant a is referred to as "the real part", not too controversially, and the constant b is referred to as "the imaginary part".
NOTICE: the imaginary part is just the constant b, not ib.
If the real part, a, is zero, then the complex number a+ib is just ib, so it's purely imaginary. If the imaginary part, b, is zero, then the complex number a+ib is just a, so it's purely real.
The above paragraph shows that all the real numbers and all the imaginary numbers are really part of the wider family of complex numbers, that have either a or b zero.
So now we can say that every quadratic equation has two roots (although it may not have two real roots).