History and Discovery of Imaginary Numbers
Date: 02/28/2006 at 18:41:30 From: Lisa Subject: Who thinks up stuff such as "imaginary numbers" How do mathematicians decide on creating imaginary numbers such as the square root of -1 or (i)? It just seems weird to have a number that is the square root of a negative one. Is it not easier to say that we deal with only "real" numbers instead of these "imaginary" ones? It would make life, and math class, a lot easier!
Date: 02/28/2006 at 22:50:07 From: Doctor Peterson Subject: Re: Who thinks up stuff such as Hi, Lisa. There are two things that drive mathematicians to invent something new, and then to accept it as worth keeping: curiosity and usefulness. Curiosity is what makes us wonder, "What if there were a square root of -1?", and then go ahead and see what happens. We're interested in new possibilities, because much of mathematics today is really a big game of "what if". Usefulness is what makes us say, "I'm stuck trying to solve this, so I'm willing to try anything!" It's also what later leads us to say, "Weird as it seems, this 'i' business actually works, so we'd better keep studying it and see if we can make sense of it." Both are involved in the history of imaginary numbers, but especially the second, to begin with. I don't know that anyone tried out the idea from mere curiosity at first; that attitude is a more recent development. What started it all is that, in the 1500's, people were trying to work out ways to solve more complicated kinds of equations, having completely tamed quadratic equations. They wanted a formula for cubic equations. In the process, they found that for some equations (which they knew had solutions), they ran up against a brick wall: they had to take the square root of a negative number. Some of them decided to go ahead and pretend that there was such a number; they just wrote sqrt(-16) as 4 sqrt(-1) and kept going as if sqrt(-1) were a number. To their surprise, they found that the weird numbers canceled out and left them with the correct solution! Pretending the number existed turned out to work. It took a couple hundred years before mathematicians got comfortable with thinking of the so-called "imaginary numbers" as really existing; they thought of them as a cheap trick to get to the right answer. (Oddly enough, during the same period, and even beyond, negative numbers were likewise treated with suspicion, as if using them were cheating.) As they tried out more and more things with complex numbers, they found that they were useful in many more ways: that complex numbers could be solutions to polynomial equations, and that including them actually simplified that subject, rather than making it more complicated; that they unified the subjects of trigonometry and exponents, so that the rules of one turned into mere applications of the rules of the other; and so on. They are now routinely used to describe very real things like alternating electric currents. Also, mathematicians became more comfortable, along the way, with the idea that they could make any definition and see what would happen; they found that complex numbers were not only useful, but also interesting in themselves. Having done this with complex numbers, they found themselves inventing many other new kinds of objects that behaved sort of like numbers, and had interesting properties worth studying. Some of these, eventually, turned out also to be useful in solving real problems. So as it turns out, complex numbers actually make things easier rather than harder, in the long run; and they make math beautiful in some suprising ways. Yes, it takes some getting used to, and some effort to learn how to work with them, but once you have them, you can do things you couldn't have done without them, or do familiar things much more easily. Mathematicians were almost forced into inventing them in the first place, but now consider them one of their most valuable creations (or discoveries, if you think of them as a world we didn't know existed, but found ourselves in by accident). Here is a page with some of the names and dates I've left out of this summary: History of Imaginary Numbers http://mathforum.org/library/drmath/view/52584.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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