Posted tagged ‘complex’

The Imaginary

July 12, 2010

Our imagination is a wonderful thing. It allows us to conjure up wondrous, impossible events and entertain ourselves with an alternate, exciting reality. It allows us to dream. We can be whatever, whoever, we want to be in our own little world and all because of our amazing ability to create the unbelievable. This all sounds a bit poetic for maths though, doesn’t it? And yet, even in the world of absolute truths we find a bit of the imaginary.

Well, imaginary numbers to be precise. Imaginary numbers, as you may have guessed, aren’t real. This may be a step too far for some of you. I’ll be trying to convince you about invisible numbers next, won’t I? Or inventing the God number and explaining how it controls the rest of the numbers (I’d go with 2, incidentally. It’s always had an air of superiority about it). But I kid you not with this one, the imaginary numbers definitely exist (though sentences like that make my head hurt).

I do wish they had been named differently, though. There is nothing quite so confusing as explaining that a number is imaginary and exists but is not real. People begin to look at you funny. Saying that, people tend to accept negative numbers fairly easily. Obviously there is the analogy with debt, but have you ever seen -2 cars, or walked past a group of -5 people? Still, I’ll let you make your mind up about how ridiculous imaginary numbers are. Time for a bit of an explanation.

A very different kind of explanation

Now, you may have been told that you cannot square root a negative number, even if you try really hard. This is because no number multiplied by itself is negative, so you can’t take the root of a negative number. All sounds pretty straightforward so far. You can even try it on your calculator if you like. Done? It didn’t like that, did it (some calculators are capable of this, but they’re always a little too smug about it for my liking)? If your calculator can’t find an answer then what hope do you have?

Ok, let’s not panic. We can do this. We just need to think about the problem a bit differently, use a little imagination. Imagination! That’s it! If there’s no real answer to square rooting a negative number, why don’t we create an imaginary one? All we have to do is define \sqrt{-1} as i and we’re done, imaginary numbers to the rescue. Simple, eh? All of our problems solved with one little trick. Not seeing it?

Consider \sqrt{-64}

\sqrt{-64}=\sqrt{64*-1}

\sqrt{64*-1}=\sqrt{64}*\sqrt{-1}

Therefore \sqrt{-64}=8i

Gotta love math humour

How easy was that? All it took was the addition of one letter and we have our answer. What we see is that the root of negative numbers produces an imaginary number. This is, of course, a great lesson for life: if you can’t work out the answer, just make it up.

What happens if we want to add some real numbers to our imaginary mix? Here things get a little complex. Literally. Complex numbers are numbers consisting of a real and imaginary part (in the same way Christmas has Santa and awkward family reunions) , such as 4+8i. However, it’s difficult to visualise that number, isn’t it? It’s like trying to imagine what love looks like, or a social life.

This is where Argand diagrams come in (in visualising complex numbers, not a social life). An Argand diagram is a graph with one real axis and one imaginary axis. Our complex number then becomes a point on the graph, making things much easier.

An Example of an Argand Diagram

Not so complex now

There you have it, the wonderfully weird world of imaginary numbers. Throw out your calculator, play with the imaginary and question your very perception of reality. What else are you going to do on a Monday?

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