1.3 Complex Numbers

Among the operations of multiplication is that of squaring a number. This is the operation of multiplying a number by itself. Thus \(5\) times \(5\) is \(25\). We can ask for the inverse of this squaring operation. This is an operation that acting on \(25\) should give back \(5\). This operation has a name: it is called the square root. A square root of \(25\) is \(5\).

There are two wonderful complications here. The first is that \(-5\) times \(-5\) is also \(25\), so \(25\) has two square roots, \(5\) and \(-5\). And the same thing holds for any positive real number. Any positive real number has two square roots.

The second complication is: what on earth is the square root of a negative number?

Well no real number has square that is \(-2\) or \(-1\) or minus anything positive.

When we found that subtraction, which is something of an inverse operation to addition, among natural numbers led to non-natural numbers, we extended the natural numbers by defining the integers to include both the natural numbers and their negatives and zero as well.

When we considered division, which is an inverse operation to multiplication, we extended our numbers again to include fractions.

Well, to accommodate the inverse operation to squaring a number, we can also extend our numbers to include new entities among which we can find square roots of negative numbers.

It turns out to do this we need only introduce one new number, usually designated as i, which is defined to have square given by \(-1\). In other words, we define the new number i to obey the equation \(i * i = -1.\) We can get numbers whose squares are any other negative number, say \(-5\), by multiplying \(i\) by an appropriate real number, here by the square root of \(5\). The number \(i\) is definitely not a real number, so we call it an imaginary number; this nomenclature is in fact silly. Imaginary numbers have just as much existence in our imaginations as real numbers have. Of course they are not natural numbers or integers or even fractions, or real numbers at all.

It turns out that if we look at numbers of the form \(a + bi\) where \(a\) and \(b\) are real numbers, we get what are called the complex numbers, and we can define addition, subtraction multiplication, division for these just as we can for rational or real numbers.

If you want to see what these rules are, click here.

So by numbers we will mean things like the rational numbers, the real numbers or complex numbers, among which the operations of addition, subtraction multiplication and division are defined and have all the standard properties.

By the way, we often represent complex numbers by points in the plane. Real numbers correspond to points on the x-axis, and imaginary numbers can be considered points on the y-axis. The number \(i\) is a distance \(1\) above the origin on the y-axis. A general complex number has real part that is described by its \(x\) component and complex part described by its \(y\) component.