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According to Wikipedia: ...a square number or perfect square is an integer that is the square of an integer...

Is the statement strictly true? And if it is, why are only Integers considered to be square numbers?

For example, if I have a square in the real world, with all sides being 1.5 units. Why is 2.25 not considered a square number? As: $(1.5)^2$ = 2.25

Consider the square root of 2.25. As the result is a Rational number.

$\sqrt{2.25}$ = 1.5

Not a square?

Bill Dubuque
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Divan
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    You could call them squares I suppose, but not "perfect squares". The definition wouldn't be very interesting because every real $x \geq 0$ is a square in that sense, and we already have the term "nonnegative" for that. – Jair Taylor May 12 '20 at 20:22
  • Taken as a reduced-form fraction, the numerator and denominator are each integer squares in your example. There are no integers which are squares of non-integer rational numbers. Separately, note that the Wikipedia article is stating a definition, not telling you that a square having some length of sides isn't actually square... – abiessu May 12 '20 at 20:23
  • It is fine to call $2.25$ a square (or even perfect square, I guess) if you are in some context where the notion of square of a rational number is relevant. However, the case where you are interested in the squares of integers is more common, all the more because a rational number is a square if and only if it can be written as the quotient of to perfect squares (integers). –  May 12 '20 at 20:27
  • @JairTaylor I would not imagine that all Real numbers are squares. But rather, that all numbers, when squared, result in a Rational number. As the Rational numbers are not repeating and can be measured in the real world. Which is a much smaller subset of x > 0. – Divan May 12 '20 at 20:29
  • @Divan I see. Given any ring $R$ (roughly, a set of number-like objects that you can add and multiply) you can define the squares of $R$ to be ${x^2 | x \in R}$. So the squares in the integers are the perfect squares, while the squares in the rationals are $x^2/y^2$ for $x,y \in \mathbb{Z}$ ($y \neq 0$), and the squares in $\mathbb{R}$ are ${x \in \mathbb{R} | x \geq 0}$. It's just a matter of context. – Jair Taylor May 12 '20 at 20:38
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    As with many definitions in mathematics they are chosen because they capture an interesting idea worth studying. There is often no other motivation than this. – CyclotomicField May 12 '20 at 20:42
  • THe key is not is "square" but in "perfect". Every positive number $x$ is the square of $\sqrt x$ so that would be useless and meaninglessness. No-one can possibly give a flying donut about numbers that are squares of other numbers (because, !hello!, every number) but squares of integers are fascinating and important and worth getting out of bed over, we really need a way to distinguish the idea that they are squares... OF INTEGERS. So we call the perfect squares. well, squares of rationals are interesting too... but if you just extracte the denomr they are, in theory the exact same thing. – fleablood May 12 '20 at 20:45
  • What about the downtown area called "market square"? Is that not a "square". Everyone says it is? Is everyone wrong? Are mathematicians elitist saying they are wrong. the wikipedia article is defining a square NUMBER or a PERFECT square. Your figure is a GEOMETRIC square or a square FIGURE. And the market square is a MUNICIPAL square. There's no contradiction in these definitions. They are terms used for specific purposes. Also notice. A rectangle with side $2$ is a SQUARE. And $4=2\times 2$ is a SQUARE. And one is the AREA of the other. But they are NOT the same thing. – fleablood May 12 '20 at 20:51
  • There are square rationals, reals, polynomials, power series, etc.. But if someone writes square number in a context where "number" denotes an integer then it denotes a square integer. It's as simple as that. – Bill Dubuque May 12 '20 at 20:59
  • "I would not imagine that all Real numbers are squares." Of course all non-negative one are. $x=(\sqrt x)^2$. "But rather, that all numbers, when squared, result in a Rational number" which is NOT true. $\sqrt[3] 2$ when squared is $\sqrt[3]4$ which is not rational. And $\sqrt \pi$ when squared is $\pi$ which is not rational. "As the Rational numbers are not repeating" um the rational numbers are repeating "and can be measured in the real world" $\pi$ and $e$ can be measured in the real world but half of planks constant (which is rational) can not. – fleablood May 12 '20 at 20:59
  • @fleablood I understand that there is a difference between a "geometric" and "number" square. I am interested in the number theory definition. What I am asking is why rational numbers are not part of the definition for a square number. Why only Integers. In the sources I have read online, and what I hear in class, people seem to only regard Integers as squares. – Divan May 12 '20 at 20:59
  • Because $r=q^2$ when $r=\frac ab$ and $q=\frac cd \iff \frac ab=(\frac cd)^2 \iff ad^2 = bc^2 \iff c^2 = r*d^2$. So there is nothing fundamentally different or interesting about rational squares. – fleablood May 12 '20 at 21:02
  • @fleablood I see you have misunderstood a part of my first comment. I will try to rephrase it now: "But rather, that all numbers, which when squared, result in a Rational number, are squares". I can see where the misinterpretation has occurred as it was meant to be read in conjunction with the prior sentence. I am very aware that not all numbers squared, are rational. – Divan May 12 '20 at 21:05
  • It is similar to asking if nonintegers are even or odd or prime etc. Such notions do have extensions to more general rings, and using such terms in these more general contexts does not imply that the "number" need be natural or integral. There are prior questions here on such extended terminology. – Bill Dubuque May 12 '20 at 21:07
  • Similarly $\alpha$ is irrational need not imply $\alpha$ is real, e.g. see Is $i$ irrational? – Bill Dubuque May 12 '20 at 21:10
  • Based on what I have read here, it seems that it is a matter of definition. Is there a reliable source on the internet where one can find the accepted definitions in mathematics? Ideally one which addresses extended definitions based on context, like my example. Or, is the search for a universally accepted true definition for each term in math, not a good course of action? Does that even exist? – Divan May 12 '20 at 21:15
  • @ Divan Definitions are just conventions of language, they have no intrinsic mathematical truth or philosophy to them. In the case of (perfect) squares, you are right that certain rationals are squares of other rationals, and that might be interesting enough to you to want a special name for them, but since they amount to simply ratios of perfect square integers, there's little to note about them beyond the squareness of the integers in the numerator and denominator -- so no name for them has taken hold in the community. The interesting "squareness" questions are about the integers. – Ned May 12 '20 at 21:24
  • @ned No, if one is working in $\Bbb Q$ then a (perfect) square denotes a rational square. Similarly for other rings and (semi)groups etc. The definitions are restricted only in elementary (grade school) contexts (where the (structural) scope is restricted) – Bill Dubuque May 12 '20 at 21:28
  • No, there is no rigid "bible" of math terminology. Terminology is always evolving. E.g. in some cases "numbers" can be "functions" and vice versa (which is crucial in analogies between number fields and function fields in number theory). And the field with one element is not even a field! If definitions were too rigid it would greatly encumber abstraction and generalization. Welcome down the rabbit hole... – Bill Dubuque May 12 '20 at 21:34

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Non-Integer numbers can be considered squares, in certain contexts. Although a square number is usually meant to be an Integer. Especially at elementary level, and when it is stated to be a "perfect square".

There is no rigid source of definitions for math terminology.

These statements are based on my understanding, of the comments by other users, on the question above.

Divan
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  • The point of making the difference is that perfect square in the integers context leads to an interesting theory. If you consider all positive rational numbers then everything is a perfect square. So in the rational context there is nothing interesting to search with this definition. – EQJ May 13 '20 at 16:24
  • Why the requirement for limiting to rational numbers? If you consider 3.14 to be a square in some context (the square of sqrt(3.14)), why wouldn't you consider pi to be a square (the square of sqrt(pi))? I don't see anything fundamentally different about squaring an irrational number versus a rational one. I'm not sure what that useful context is, though, since this interpretation of "square number" is equivalent to "non-negative number". – Nuclear Hoagie May 13 '20 at 16:29
  • @NuclearWang I think it depends on the context as suggested in my answer. Are you implying that the definition could also be extended to Real numbers? – Divan May 13 '20 at 16:43
  • @YTS I am not sure what you mean by "everything"? As there are numbers that are not positive rational numbers. Or do you mean everything that can be measured in the physical world? – Divan May 13 '20 at 16:45