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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
number.

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?

Sincerely,
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
http://mathforum.org/dr.math/
```

```
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

Sincerely,
Patricia Holmes
```

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

Hi Patricia,

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
her:

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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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