Paradox in the Zero Power
Date: 09/26/2000 at 11:13:38 From: Dina Zeliger Subject: Paradox in the zero power Hi, My question is not about solving a question, but more of proving the answer. I know that n^0 equals one, and there are a lot of ways to prove it. I understand those ways of proving the problem, but it doesn't seem logical, and I don't think it fits the definition of power. A power is the product of a number multiplied by itself a given number of times. i.e. n^k = n*n*...*n (k times). This definition in right when k > 0. But, if k = 0, why does the product of multiplying a number by itself zero times equal 1? Zero is basically nothing. Its value is not a quantity, i.e. zero times something is nothing - zero. in that case, how is it possible that n^0 = 1? Multiplying n by itself zero times equals nothing, since you don't multiply it at all. That theory also suggests that 0^0 = 0, and that this expression is not indefinite. If (n^0 = 0) and (0^n = 0), then (0^0 = 0). I hope you can help me solve my problem. It is very disturbing; I have been thinking about it for a week now. Thank you, Dina Zeliger
Date: 09/26/2000 at 14:32:29 From: Doctor Ian Subject: Re: Paradox in the zero power Hi Dina, If you haven't yet seen the explanations in the Dr. Math FAQ, you should take a look at them: http://mathforum.org/dr.math/faq/faq.number.to.0power.html In the end, you need to understand that the interpretation of n^0 isn't something that can be (or needs to be) proven. We're free to choose what we want it to be, so long as that choice is consistent with the choices we've already made. Basically, n^0 is _defined_ to be 1 for the simple reason that any _other_ definition would cause things to break. For example, if it's going to be true that n^a * n^b = n^(a+b) then n^0 can't be anything _but_ 1. Choosing any other value would cause this rule (and others like it) to stop working. It's true that this approach leads to some pretty goofy definitions, such as 0! = 1 and 0^0 = 1 But as Winston Churchill once said, democracy is the worst possible form of government... except for all the others. You could make a similar comment about these definitions. They are the worst possible choices... except for all the others. I hope this helps. Write back if you have more questions, about this or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 09/26/2000 at 17:01:42 From: Doctor Peterson Subject: Re: Paradox in the zero power Hi, Dina. I'd like to add something to what Dr. Ian wrote. He explained this the same way I would, in terms of defining the zeroth power to avoid breaking the rules we already have. I like to supplement that with an explanation which, while not mathematically profound, feels more emotionally satisfying, which may be what you need. Your complaint is that it just seems wrong that multiplying something zero times should give a non-zero result. Well, here's a way to think about that multiplication that makes this seem a little more reasonable: First, let's get some practice by trying to explain why anything TIMES zero is zero. We think of multiplication as repeated addition. If you add together 5 3's, you find that 5 * 3 = 3 + 3 + 3 + 3 + 3 = 15 \_______________/ 5 Notice that I'm not really "adding 3 to itself 5 times", as is commonly said; there are only four additions. In particular, adding together 1 * 3 uses NO additions. How can you use this concept to explain that 0 * 3 = 0? Would you have -1 additions?? Instead, I like to think about what you had BEFORE you started adding; you had nothing to start with, then added 3 to it five times: 5 * 3 = 0 + 3 + 3 + 3 + 3 + 3 = 15 \_________________/ 5 This is neater; now I can actually say "the product is the result of adding a number TO ZERO some number of times." And by starting with zero, we can better understand why 0 * 3 = 0; you just start with 0 and add nothing. 0 * 3 = 0 = 0 \_/ 0 I think you are sort of assuming that exponents work the same way: you start with 0, and if you multiply nothing, you end up with zero. But multiplication can't start with zero! If you multiply together 5 3's, you get 3 ^ 5 = 3 * 3 * 3 * 3 * 3 = 243 \_______________/ 5 But if we started with zero, and multiplied it by 3 five times, we would end up with zero. Instead, when we multiply we have to start with 1, and multiply that by 3 five times: 3 ^ 5 = 1 * 3 * 3 * 3 * 3 * 3 = 243 \_________________/ 5 So three to the fifth power is the result of multiplying 1 by 3 five times. That's a cleaner definition than "multiplying 3 by itself five times." Now, what do you get if you start with 1 and don't multiply it by anything? You get 1, which must be the value of 3^0! 3 ^ 0 = 1 = 1 \_/ 0 So the choice of 1 as the definition of a number raised to the zero power really fits well with the cleanest definition we can come up with of exponentiation in general. Again, this isn't a "proof," but a further reason to be satisfied with the definition. Now, what do you get if you multiply 1 by 3 -2 times? If division is the opposite of multiplication, then maybe you would start with 1 and divide it by 3 twice. Does that help you feel better about this? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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