Zero and Empty Set
Name: Dan G.
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?
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
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!!
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.
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Update: June 2012