Complex NumbersDate: 04/11/2001 at 03:56:59 From: Rosie Subject: Complex numbers What exactly is the complex number system comprised of? Thank you. Date: 04/11/2001 at 18:01:53 From: Doctor Ian Subject: Re: Complex numbers Hi Rosie, Let's start with the square root function. The most 'natural' square roots are the square roots of perfect squares, e.g., 2 = sqrt(2*2) = sqrt(4) 3 = sqrt(3*3) = sqrt(9) and so on. For any positive number x, we can write sqrt(x) = ? and we can compute either an exact value or an approximation for ? . Once a notation exists, there is always somebody who will try to use it for something it wasn't intended to handle. For example, what do we do with sqrt(-4) = ? The answer can't be 2, since 2^2 = 4, not -4. And it can't be -2, since (-2)^2 = 4, not -4. For a long time, the answer was that the operation simply didn't make sense, much like dividing by zero. But it turns out that if you invent a single number, i, which is defined such that i*i = -1 then suddenly you can write things like sqrt(-4) = sqrt(4) * sqrt(-1) = 2 * i or just '2i'. Now, what if we want to do something like 3 + sqrt(-4) = ? Well, we can make some progress: 3 + sqrt(-4) = 3 + 2i but now we're stuck. We can't add a multiple of i to something that isn't a multiple of i, any more than we can add fractions with different denominators. What we _can_ do is consider 3 + 2i the reduced form of a new kind of number, called a 'complex' number. We can plot the real numbers on a line, <------------------+----------------------> -2 -1 0 1 2 3 But we have to plot the complex numbers on a plane: | * 2 + 3i | -2 + i * | <------------------+------------------------> -2 -1 0 1 2 3 | | * 1 - 2i | | * 1 - 4i This plane is called the 'complex plane', and to locate the complex number a + bi you move a units along the real (horizontal) axis, and b units along the complex (vertical) axis. That is, plotting the point a + bi on the complex plane is just like plotting the point (a,b) on a regular x-y plane. Note that we can think of the real numbers as the 'shadows' of the complex numbers. For every real number, like 2.5, there is an entire line of complex numbers, 2.5 + 68.92232i 2.5 + 2i 2.5 + 1i 2.5 + 0i 2.5 - 1i 2.5 - pi*i If you think of a light shining down from above the line, the line would cast a shadow on the real number line at 2.5. So, to answer your question, the complex numbers system is composed of the complex plane, the rules for locating points of the form a + bi on that plane, and the definition that i^2 = -1. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 04/12/2001 at 04:20:02 From: Rosie Subject: Re: Complex numbers Thank you very much for the help. I now understand complex numbers! Rosie |
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