Something of a place-holder entry this week, I'm afraid. With more questions than answers.
First question: What are imaginary numbers? There's a fascinating introduction to the subject available as part of the BBC's In Our Time archive, but I'll précis the basics for you now:
An imaginary number is that which gives a negative result when multiplied by itself. The square root of minus one - also known as i - is an example. The most astonishing thing about imaginary numbers (though perhaps their name ought to have given us fair warning) is that they don't 'exist' in the real world. One cannot count or measure with them. And yet - when embedded in equations - they have proven extraordinarily helpful in providing verifiably accurate solutions to real world problems. Imaginary numbers are crucial conceptual tools in contemporary scientific models of electromagnetism, fluid dynamics and quantum mechanics for example.
How can this possibly be? How can something that doesn't exist describe something that does?
A helpful analogy - though not really a solution - can be found in negative numbers. After all, negatives don't really 'exist' either. And that doesn't forestall their use in equations that come out with positive results. Imagine a healthy balance sheet. So long as your income (modelled by 'real' positive numbers) outweighs your debts (modelled by 'conceptual' negative numbers), then your bottom line will be a 'real' number insofar as you could convert it into tangible purchases if you so wished. It doesn't matter that you've used unreal negative numbers to get there. The only difference between negative numbers and imaginary numbers, then, is that the former may be attached to an intuitively graspable concept: debt.
Perhaps a more comprehensive explanation could be found by going one stage further and admitting that even positive numbers are, in a sense, unreal. Mathematical constants are just like nouns in any spoken language: they divide a continuous universe up into discrete chunks that may be talked about. That some mathematical concepts (such as positive numbers) 'make more sense' to us as human beings is interesting, but this should have no bearing upon whether they (or any other concept) should be considered 'real'. All concepts are representational, their definitions man-made. The question is whether they are useful, by which I mean that they aid our ability to comprehend and/or predict observable phenomena.
I apologise if I'm bordering on incomprehensibility here. This idea that scientific concepts ought to be judged on their usefulness rather than their essence is one that has been intriguing me for a while now. I'm going to try and write about it with more clarity (and perhaps some nice pictures) before too long.