Is Zero Really a Number or Just a Concept?
Date: 09/15/2005 at 20:24:42 From: Swapnil Subject: Zero: Just a concept like infinity? Hi, for a long time my teachers have hammered into my brain that infinity is not a number, it's just a concept. Now that I understand that, I wonder if zero is just a concept too? I guess this is both a philosophical and a mathematical question. Philosophically, you can't really have a zero of things, because the thing which is zero doesn't really exist. In fact, if you do count things that don't really exist, you will have an infinite number of those things since there is no limit to things that you can imagine existing (which is rather an interesting fact). In mathematics, many definitions and rules go haywire when they try to deal with zero (such as 0^0, 1/0, 0/0). The two most used bounds for evaluating the limit of a function are in fact 0 and infinity. So it seems obvious that there is a deep intricate relation between zero and infinity. And like infinity, it seems that zero should just be considered a concept. Furthermore, by making zero a concept we would be able to avoid writing those exceptions to rules where we have to specify that the equation is true as long as something is not zero. I just thought of few more examples: 1) If in a game, your opponent has 1 point and you have two points then you can say that you have twice as many points as him. Whereas if your opponent has zero points and you have two points, you would technically have to say that you have infinite times as many points as him. 2) You can't really have a circle with radius zero. 3) Zero is neither positive nor negative and neither it is even or odd.
Date: 09/15/2005 at 20:56:31 From: Doctor Peterson Subject: Re: Zero: Just a concept like infinity? Hi, Swapnil. It's true that infinity and zero have many interesting relationships. But zero is far less troublesome than infinity, and far more important. We couldn't do much algebra without zero, because we couldn't define the additive inverse (negative) without the additive identity. For example, in solving x+5=1, we add -5 to each side, getting x+5-5 = 1-5. The left side becomes x because 5-5=0, and x+0=x. We couldn't do that without zero! There would be ways to get around it, I suppose, but they would be very awkward. You'd lose a lot if you refused to do any arithmetic with zero, which is what we mean when we say that infinity is not a number. Now, it's true that zero is "just a concept"--but the same can be said of all numbers. You've never seen a three, have you? You see three boys, three sticks, and so on, and you generalize from that to the concept of "three". You can't have zero apples--but you also can't have -3 apples. (Actually, you can have both, if you think of it the right way.) You can't have a circle with radius 0, or with radius -3, either--so do negative numbers not exist? Zero and negative numbers are just extensions of the concept of counting, taking the idea of "number" beyond where it began. The reason zero is neither positive nor negative is that positive MEANS greater than zero, and negative means less than zero. Zero is not an exception there--it's what the whole idea of positive and negative revolves around (almost literally!). Zero IS even--I hope no one taught you that it isn't! And the fact that it DOES fit neatly into the integers in ways like that is one reason it is obviously one of them. In general, zero works in MOST settings, while infinity works in almost none of them. You can't divide by zero, but you can add it; you can't even add infinity without getting in trouble! So it makes good sense to work with the few peculiarities of zero, but to treat infinity as something else, just a direction to go when you take a limit, rather than an actual destination. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 09/15/2005 at 21:20:14 From: Swapnil Subject: Thank you (Zero: Just a concept like infinity?) I would like to thank you, Dr. Peterson, for responding to my question in such a short period of time and with such a good answer. I really appreciate your help. Keep up the good work! :) -Swapnil
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