Why Do Imaginary Numbers Exist?Date: 02/17/2001 at 00:43:21 From: Cara Subject: Imaginary numbers Why are there imaginary numbers? I don't understand why we need to have imaginary numbers or what their purpose is. Date: 02/17/2001 at 06:01:52 From: Doctor Mitteldorf Subject: Re: Imaginary numbers Dear Cara, I remember thinking just the way you do when I first came on the more abstract forms of math. And the name "imaginary" just adds to the sense that they don't really exist; they're just ideas. So there are two sorts of answers to your question. One is that mathematicians are just like children at Christmas. We love new toys, and we want to try this and that to see what happens. The mathematics of imaginary numbers exists just because you can do so much with it that's fun - perhaps the same reason that people love to play chess. Did you know that even though i was invented as a solution to equations like x^2 = -1, you could pick any other equation that doesn't have a real solution and define the solution to that as our new "i" and end up with the same mathematics? You could start, for example, with the equation sin(x) = 2, and define the number x that solves this equation as the fundamental imaginary number "i," and by the time you'd finished trying to solve all the other equations, the math you'd end up with is the same as what we've got with the standard "i." You'd think that once you started inventing numbers to solve equations you'd have to keep going and going. We'll let i be the solution of x^2 = -1. But then, what's the square root of -i? We'll call that j. And what's the square root of -j? We'll call that k.... Well, it turns out you don't have to do that. There's a combination of real and imaginary numbers that satisfies the equation x^2 = -i. In fact, there are two such combinations - just as every positive real number has two square roots, one the opposite of the other, every negative and imaginary number also has two square roots. Can you think of how you would go about finding the square root of -i? I promised you there were "two sorts" of answers, but I haven't gotten to the second one yet. The second is that there are areas of science where imaginary numbers turn out to be enormously useful. That may be hard to imagine, because it's obvious that when you ask questions like "how far does the ball bounce" or "how much energy does it have," etc., the answer must always be a real number. But there are situations where, even though you start with real physical quantities and what you are trying to compute is another real physical quantity, the shortest, easiest way to compute that real number takes you through imaginary numbers along the way. The reason for this has to do with the fact that so many trig identities are complicated and unexpected. Exponential equations are comparatively simple, and obey simple laws. When you get deep into imaginaryland, you find a close, unexpected relation between trig and exponentials. It turns out that an angle is a lot like an imaginary logarithm, and a logarithm is like an imaginary angle. Until you do the math, I'm sure it's hard to even imagine what that could mean. But it's an exciting chapter for you to look forward to. So my advice is to keep playing with those imaginary toys. And you can read more, if you like, at our FAQ on imaginary numbers: http://mathforum.org/dr.math/faq/faq.imag.num.html - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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