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### Is Zero a Square Number?

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Date: 02/16/2000 at 10:47:33
From: Janie Cates
Subject: Is zero a square number?

I would like to know whether or not zero is considered to be a square
number. I have found some sources that say it is. and others that say
it isn't. Please let me know so that I can tell my students.
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Date: 02/16/2000 at 11:11:18
From: Doctor TWE
Subject: Re: Is zero a square number?

Hi Janie - thanks for writing to Dr. Math.

This depends on the way you (or your source) define the term "square
number." Some sources define a square number as any number that is the
square of an integer. By this definition, zero is indeed a square
number, because 0 = 0^2 and zero is an integer.

Others define a square number as any number that is the square of a
natural (or counting) number. The natural numbers are the set of all
integers greater than or equal to one. Since zero is not considered a
natural number (and no other natural number squared equals zero), zero
would not be considered a square number using this definition.

This second definition has more of a "geometric" interpretation.
Imagine measuring the area of a square that is n units on a side.
If n = 0, you don't really have a square to begin with - you have
nothing. In ancient times mathematicians didn't count zero (or even
have a symbol for it) because they used numbers to represent physical
measurements, and a physical object couldn't have a measurement of
zero.

I hope this helps - if you have any more questions, write back.

- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/
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