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Is Zero a Perfect Square?

Date: 08/01/2001 at 15:46:43
From: Patricia Holmes
Subject: Is zero a perfect square?

Dear Dr. Math,

I have been told that zero is not considered a perfect square, yet 
the square root of zero is zero. According to the definitions of 
perfect square and square root:

Any number that can be written as an integer to the power of two is 
called a perfect square.

A square root is a number whose principal square root is a whole 

If zero is both an integer and a whole number, what part of these 
definitions, or a better definition you may know of, excludes zero 
from being a perfect square?

Thanks in advance for your help.

Patricia Holmes

Date: 08/01/2001 at 16:42:39
From: Doctor Ian
Subject: Re: Is zero a perfect square?

Hi Patricia,

Some mathematicians think that zero is a perfect square, and others 
think that it isn't. Personally, I think it makes perfect sense to 
think of zero as a perfect square. 

Note that some people would define 'perfect square' this way:

  Any number that can be written as a POSITIVE integer to the 
  power of two.

If you define it that way, then clearly 0 isn't a perfect square.  

Note that there isn't much riding on which definition you choose.  
That is, no important theorems stop being true if you decide that zero 
is or isn't a perfect square.  

Contrast that with something like saying that from now on, 1 is going 
to be considered a prime number. All kinds of theorems would suddenly 
become untrue.
You say tomato, I say tomahto, ... :^D

- Doctor Ian, The Math Forum   

Date: 08/01/2001 at 17:00:19
From: Patricia Holmes
Subject: Re: Is zero a perfect square?

Thanks so much for your quick and comprehensive reply, Dr. Ian.  I 
really appreciate your help!

Patricia Holmes

Date: 08/02/2001 at 14:14:44
From: Doctor Ian
Subject: Re: Is zero a perfect square?

Hi Patricia,

I'm glad I was able to help you. 

It turns out that yesterday, three people asked essentially the same
question you did. One wrote back to ask _why_ mathematicians disagree
on the answer. I thought you'd appreciate the follow-up that I sent to 

The issue goes to the history of the term 'square'.  A number N is a 
square if you can arrange N cookies into a square shape:

  1: @

  4: @ @
     @ @

  9: @ @ @
     @ @ @
     @ @ @

Now, can you arrange zero objects into a square shape? Some people 
have no trouble with that concept. Others find it absurd.  

The former tend to be the kinds of people who, like Lewis Carroll, 
enjoy saying things like 'All my sisters live in Antarctica', which is 
true (in mathematical logic) for anyone who has no sisters. 

There are people who like to play with math, and see where it takes 
them; and there are people who know where they want to go, and see 
math as a vehicle for getting them there. I think you can guess which 
ones think of zero as a perfect square. 

- Doctor Ian, The Math Forum   
Associated Topics:
High School About Math
High School Number Theory

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