What is i?

Last week we spoke about the irrational, but very real, pi. Today we're going to take a look at the number i, where the i means 'imaginary'.

It is very likely that you won't have come across i before. It isn't generally taught at GCSE, being reserved for A-Level and above. Although I'd heard of it before, the first time I properly met this number was this year during my Chemistry degree.

Let's be curious and ask, 'what is i?'

Defining i

We all know the square root of 9 is 3. That's because 3 x 3 = 9. The square of any number is positive. Just as 4 x 4= 16, we also know that -4 x -4 = 16. Whenever two negative numbers are multiplied together, they make a positive number. So regardless of whether the number is positive or negative, its square will always be positive.

As squaring numbers always results in a positive number, that means we can only find the square root of positive numbers. For example we can't find the square root of -25 as there isn't one. Squaring either 5 or -5 gives the same answer of positive 25. The trouble is, in science and maths we sometimes end up with equations and such requiring that we find the square root of a negative number.

That's where i comes in. The definition of i is:

${\it i} =\surd-1$

Complex numbers

So, the square root of -1 is i. What does that mean the square root of -25 is? Quite simply it is the same as the square root of 25 multiplied by the square root of -1. In other words 5i. This term is called the imaginary term/part. The word imaginary just means it involves i.

We can use real numbers alongside imaginary terms to form what are known as complex numbers. Complex doesn't mean that they are difficult, but that they contain an imaginary and real part. For example if I wanted to add 3 to the square root of -36, I would write 3 + 6i.

This post isn't the place to go into the details regarding the mathematical rules governing addition/subtraction, multiplication, division, etc... of complex numbers. For those interested in learning more about mathematical manipulation of complex numbers refer to the further information section at the end of the post.

Imaginary?

It may, at first glance, seem that i cannot have any practical applications. After all it is imaginary, isn't it? In name, yes. In reality, not quite. Imaginary was a derogatory word, used by those who thought there was something wrong with them, or that they didn't really exist. Before we scoff those people, we should remember that at one point negative numbers were thought not to exist. Science and maths are constantly changing as we learn more about the world. That is why science is so successful: we change in light of new evidence and ideas. A lot of what we use imaginary numbers for now, hadn't even been thought of when they were first conceived.

But are they less 'real' than real numbers like 1, 10, pi, -1/12? For example you can't plot an imaginary or complex number on a number line, can you? Well, not a one dimensional line like you'd find adorning the walls of primary school classrooms. However, we can use what is called an Argand Diagram, also known as a complex plane. This looks like a typical set of axes. Only the x-axis (horizontal) maps the real part, and the y-axis (vertical) maps the imaginary part. This diagram represents 3 + 6i:

Practical Applications

As eluded to in the previous section, imaginary numbers do have practical applications. Sure, you're very unlikely to come across them in your everyday life, but you have definitely come across something that does use them. Electricity. Specifically with alternating currents (AC).

Direct Current (DC) is nice an simple. Everything can be described easily with one dimensional quantities. The trouble with AC is that the voltage is constantly changing. But it doesn't do so in a binary fashion where it flips directly from -1 to 1. It does so gradually (although very quickly). To simplify things, electrical engineers describe the quantities involved with AC using complex numbers (though they use j instead of i to avoid confusion with current, I). That's right. To make things simple they use complex numbers! Remember, though, that complex just means in multiple parts.

Another application of imaginary numbers is in the exciting field of quantum mechanics. Quantum mechanics is the study of phenomena at the very very small scales. Things like atoms, electrons, photons, etc... Complex numbers are required to make the maths work with concepts such as the wave function.

Beautiful Application

Beauty may be in the eye of the beholder, but I deny anyone to say this image doesn't have some intrinsic beauty:

Mandelbrot Set created using Fractal eXtreme (click for full size)

This may look like a weird picture made using the render settings on Photoshop, but that image is actually an Argand Diagram. This is the result of a series of a series of calculations using complex numbers. If you zoom in on this image, known as the Mandelbrot Set, you will find that it actually never ends. You can always zoom in farther. This is what's known as a fractal. In fact, Benoit B. Mandelbrot, who first created these images is world-famous in fractals and 'roughness' in mathematics.

In essence, the Mandelbrot Set runs a function, f(z) = z² + c where c is a complex number. The value is then re-input as z (which is 0 on the first run) and the cycle repeats. This explains the fractal nature of the images. For different complex numbers there are 2 possibilities: either the function will reach infinity, in which case it is unbound; or, it does not reach infinity, in which case it is bound. Bound values of c are represented as black. Unbound values are assigned a colour depending on how rapidly the function reaches 0.

I realise that explanation could be confusing, so don't worry if you don't understand. Either way you can appreciate the Mandelbrot Set's beautiful aesthetic. I didn't mention the Julia Set which the Mandelbrot Set is derived from. Here is an image from a Julia Set. As with the Mandelbrot set it is a fractal. The black and colour represent bound and unbound functions, but the calculation is a bit different.

Julia Set - Image created using Fractal eXtreme

Curious Joke

What does the B in Benoit B. Mandelbrot stand for? Benoit B. Mandelbrot!

Further Information

For more information on the mathematical manipulation of complex numbers visit mathsisfun.com.

Here is a video explaining the Mandelbrot Set. The channel also has an explanation of the Julia Set.

You can download the Fractal eXtreme software for free from cygnus-software.com

Do you know a curious fact about imaginary numbers? Perhaps you found something particully awesome with the Fractal eXtreme software. Have something to say about this post? If so comment below.

As always you can share this post using the links to the left, and follow It Is All Science with the links to the right.

Remember, it is all science. So, let's be curious!