Sometimes (often?) we learn about things at school or university the relevance of which isn't immediately obvious. 'Why are we learning this?' we cry, 'We're never going to need to know about this in the real world!' For example, anyone who has studied maths to more than a basic level may well have come across the concept of imaginary
numbers, which take as their starting point a quantity called 'i', defined as the square root of -1. In the 'real world', we're all taught that you can't find the square root of a negative number, hence the name 'imaginary'. So, if they're imaginary, and don't exist in the real world, wondered Apollyon...
Does i, the square root of -1, have any practical purpose?
Well, as it happens, yes it does:
The mathematics of alternating current, radio transmitters, stability of control systems, quantum mechanics - anything that requires two dimensions to describe it and is subject to oscillations and waves - all these are greatly simplified by the use of complex numbers.
It also enables you to solve all quadratic equations.
So, pretty useful little items, these imaginary numbers. As to why they're so useful in these applications, we turn to Arnie Appleaide for this excellent explanation:
The exponential function, which is the number e raised to the power x (ex), increases rapidly as x is increased, and decreases rapidly as x becomes negative.
However, if you multiply x in the above function by i:
You get a function which oscillates. In fact:
e(i*x) = cos(x) + i*sin(x)
This is extremely useful in physics and engineering because it is very easy to work with the function e(i*x) and this function can represent waves of all kinds — something that crops up all the time in science.
So, for example, in quantum mechanics, the central equation is the Schrodinger equation, and for the most basic application of this equation, the solution is e(i*x).
Perhaps this is something that teachers could bear in mind when presenting what appear to be abstract concepts. Although students obviously have to be taught the basics before progressing to more complex applications1, that doesn't mean they can't be told why they're learning the basics. As Lizard King bemoans of her teachers:
I've had my first-year maths course at uni dealing with these imaginary numbers, but I can't remember anyone ever actually pointing out why we need them. I learned the formulas and how to do all the maths with them without understanding what the practical use of all this knowledge would be.
But, in closing, the clinching evidence of the importance of i is in this little nugget, provided by TRiG:
Electricians call it j, because i is current. Electricians wouldn't call it anything if it didn't make their lives significantly easier.