Finite NumbersDate: 05/13/2003 at 18:03:51 From: Bola Subject: Finite numbers Dr. Math, I was wondering if you could tell me about finite numbers. Date: 05/14/2003 at 14:20:03 From: Doctor Ian Subject: Re: Finite numbers Hi Bola, This has been an interesting question to think about. Informally, the definitions of 'finite' and 'infinite' are somewhat circular: Something is infinite if it's not finite, and it's finite if it's not infinite. But that doesn't shed a lot of light on the subject, does it? My dictionary has this definition: 3. Mathematics. a. Bounded in an interval. Said of a quantity defined in an interval. b. Incapable of being put into one-to-one correspondence with a part of itself. Said of a set. c. Real or complex, as distinguished from ideal. Said of a number. But it's not clear that these terms are going to be meaningful to you. There are a lot of subtleties to understanding this. I guess I'd define a finite number this way: If it's possible to reach a number by doing arithmetic operations (+, -, x, /) starting with 1, in a sequence that will come to an end, the number is finite. So this handles any integer that you can think of, since you can get to any integer by starting from 0 and adding or subtracting 1 some number of times. (You can get there in fewer steps if you use multiplication.) You may need a _lot_ of steps, e.g., if the number you've selected is a million billion trillion; but eventually you reach a final step, and that's what makes even a huge number finite. You can get to any rational number by making two integers and dividing one by the other (so long as you don't divide by zero). Unfortunately, this definition doesn't cover irrational numbers, like pi or e. However, for any irrational number, we can always find a pair of rational numbers that surround it. So we can say that a number is finite if we don't have to do an endless sequence of operations to generate it; or if it lies between two finite numbers. There are other subtleties, like how to deal with 'numbers' that have components, like complex numbers or vectors. But if you ignore those, does this definition make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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