Imaginary numbers

Martin McBride
2022-01-30

In this article we will look at imaginary numbers.

The unit imaginary number, usually called i, is defined as a solution to the equation:

$$ x^2 = -1 $$

In other words:

$$ i = \sqrt{-1} $$

Of course there is no real number than can be squared to give a negative result. That is because any positive real number multiplied by itself will be positive, and any negative real number multiplied by itself will also be positive.

However, the mathematical concept of a number i, the square root of -1, is valid and useful. There is no real number that satisfies that description, but if we define the imaginary number i to have such a property, we can use mathematical rules to manipulate it. And it results in some interesting and useful maths.

The term imaginary number was originally coined by Descartes, to distinguish them from real numbers. It isn't necessarily a helpful name because ultimately all numbers are figments of our imagination, so i is no more imaginary than any other number. Still, that is what i is called.

Surds

You will most likely already be familiar with surds. A surd is an unresolved root of a number.

For example consider the square roots 2. It is an irrational number - it cannot be expressed exactly as a fraction, and therefore cannot be expressed exactly as a decimal number. It is approximately 1.41421356, but that is not its exact value.

If we want to express the square root of 2 exactly, we must use a surd. We simply call it $\sqrt{2}$.

We can manipulate surds. For example we can add them:

$$ \sqrt{2} + \sqrt{2} = 2 \sqrt{2} $$

We can sometimes simplify surds, for example:

$$ \sqrt{2}(1 + \sqrt{2}) = \sqrt{2} + \sqrt{2} \sqrt{2} = \sqrt{2} + 2 $$

In this case we use the fact that if we multiply root 2 by root 2 we get 2. However, we can't simplify the expression any further. We can't combine root 2 and 2 unless we resolve root 2 to an approximate number, to a certain number of decimal places. But as soon as we do that, our result is no longer exact.

Imaginary arithmetic

We can perform basic arithmetic on i, in a similar way to handling surds. The difference is that i is not a real number. So while it is possible to approximate root 2 by 1.41421356, we cannot do a similar thing with i. There is no approximate real number for i. We must always keep it as i.

Here are some examples. We can add terms in i:

$$ i + i + i = 3i $$

We can multiply i by a real number

$$ 2i + 1.5i = 3.5i $$

We can raise i to an integer power. Every i pair resolves to -1:

$$ \begin{align} i^2 = i \times i = -1 \newline i^3 = i \times i \times i = -1 \times i = -i \newline i^4 = i \times i \times i \times i = -1 \times -1 = 1 \newline i^5 = i \times i \times i \times i \times i = -1 \times -1 \times i = i \end{align} $$

We can go further, to handle expressions like these (but this will have to wait until we cover complex numbers):

  • $\sqrt{i}$ - we can find the square root of i.
  • Powers - can can find $i^x$ where x need not be an integer. We can also find $x^i$ or even $i^i$.
  • We can apply other standard functions to imaginary numbers, such as $\sin i$ or $\ln i$.