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? Thanks in advance for your help. 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 really appreciate your help! Sincerely, 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 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/
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