Is Zero a Square Number?

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.

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/