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Name: Dan G.
Status: student	
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Question:
I was helping my daughter with her homework and told her that zero was not an even number. She said that her teacher told her that it was. Doesn't zero represent an empty set and would not be positive or negative?



Replies:
Dan,

Zero is not an empty set. Zero is the number of items in an empty set. Zero can be confusing in that sense, which is why a wide variety of ancient civilizations had number systems that started at one. They never discovered zero. The definition of an even number is a multiple of two. If you can write a number as two times an integer, the number is even. Since 0=2*0, and zero is an integer, zero is even. Decimals and fractions are the numbers that are neither even nor odd.

Dr. Ken Mellendorf
Physics Professor
Illinois Central College


An even integer, N, is one where: N = 2*n, where n is an integer. So strictly speaking, 0, meets this test. However, I also see your point that it is rather a trivial example. There are much more important and interesting things to look at in number theory. For example, think about this oddity. Every rational number is a repeating decimal. For example, 1/11 = 0.09090909... and the converse, every repeating decimal is a rational number, is also true. Both of these theorems are pretty easy to prove even for a young student. In fact there are several proofs for each. Now a prime number, P, is one that is divisible only by itself and by "1". I find it rather almost mystical that the reciprocal, 1/P, is always rational, and hence a repeating decimal!! Ain't numbers strange!!

Vince Calder


If you are helping your daughter with homework, then you should /help/, by making as much sense as possible out of whatever her teacher says. You do not want to be bringing notions from set theory into the discussion unless your daughter is studying set theory.

Zero is an even number because it fits the simple definition of evenness used in elementary arithmetic: there is no remainder after division by two. Also, zero's nearest integer neighbors are 1 and -1 -- numbers which unquestionably are odd -- and everywhere else along the number line, an integer whose nearest integer neighbors are odd is an even integer.

Tim Mooney



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