Perfect Squares and Natural Numbers

I just posted the following message in my Module Discussion Forum for my First-Year module G11FPM Foundations of Pure Mathematics.

Hi everyone,

In the first lecture, the question arose as to whether or not 0 is a “perfect square”. In fact, it turns out that mathematicians don’t all agree on the answer to this.

Part of the problem is the question of whether or not 0 is a natural number. In fact there is probably a 50-50 split in the literature on this issue. However, in Nottingham I believe that 0 is usually NOT included in the set of natural numbers. But you may need to confirm this with your lecturers on a module-by-module basis.

In G11FPM the natural numbers will start from 1, and so (for us) 0 is not the square of a natural number. However, 0 is the square of an integer, and this leads on to the other reason mathematicians disagree. Are “perfect squares” the squares of natural numbers or the squares of integers? Wikipedia, at least, appears to go along with my “squares of integers” approach. [See] but this is far from decisive evidence!

This will not be an issue in the module G11FPM: from now on, if the term “perfect square” (or equivalent) arises, I will always clarify what I mean at the time. However, it does give yet another illustration of the fact that your answers to questions depend  crucially on the definitions you are using. Since most terms in this module will have official and unambiguous definitions, you will need to know and understand the official definitions in order to obtain the correct answers in this module.

Best wishes,

Dr Feinstein


One response to “Perfect Squares and Natural Numbers

  1. Who uses the phrase “perfect square” anyway? I think most square number theorems talk about square numbers, of which 0 is one of my favourites! 😉

    In fact for results like:
    “All positive integers not of the form 4^n(8m+7) for some m,n\geq 0 can be written as the sum of three squares.” It’s rather necessary.


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