Why is Zero Not a Natural Number?Date: 03/14/2001 at 00:28:33 From: Laura Subject: Why is Zero not considered a natural number? Hello! I don't understand why zero is not considered to be a natural number. Natural numbers are our typical counting numbers and I have learned that in mathematics we count beginning at zero. Correct? Can you explain why the natural numbers are {1, 2, 3.......}? Also, is there a way to prove that zero is an integer? Thanks! Laura Date: 03/14/2001 at 13:14:09 From: Doctor Ian Subject: Re: Why is Zero not considered a natural number? Hi Laura, Here are a couple of Dr. Math references for you to check out before I go on: Natural Numbers, Positive Integers - Dr. Math archives http://mathforum.org/library/drmath/view/55753.html Integers, Rational and Irrational Numbers - Dr. Math FAQ http://mathforum.org/dr.math/faq/faq.integers.html Actually, counting from zero is _not_ a very natural thing to do! It takes a certain amount of sophistication to realize that you can have a set containing zero elements, or that you need a number to describe such a set in the first place. Consequently, most numeral systems from history (e.g., Roman numerals) don't contain a symbol for zero, or any way to represent the concept. The most 'natural' way to count is by making a mark for every item counted. If you have seven sheep, you write ||||||| If you have two sheep, you write || If you have no sheep at all, what do you write? Nothing at all! If there's nothing to keep track of, why would you need a number to keep track of it? The concept of 'zero' may seem natural to you, but that's only because you're growing up surrounded by people who already know about it. Similarly, it probably seems natural to you to think of the earth as being round, but it took a long time for people to figure that out, too. As for 'proving' that zero is an integer, it's not really necessary, since the set of integers is whatever we say it is. If we say that the set includes zero, then it includes zero. (Proving that zero is an integer would be like proving that people are members of a club. Are they on the membership list? Okay, then they're members.) And anyway, all the other integers are _defined_ in terms of zero and one, so zero had better be an integer, or we're in big trouble! Having said all that, I should point out that agreement on this is not universal. One of our other math doctors, Doctor Floor, points out that in the Netherlands, it is generally taught that the natural numbers are {0, 1, 2, ...}. Probably the best thing to do is to avoid the term 'natural' altogether, and use the term 'positive' or 'non-negative' to specify unambiguously which set you're referring to in any given context. I hope this helps. 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/ |
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