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Can a perfect square only be an integer number? Can this idea be extended to real numbers or rational numbers?

Niraj Raut
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  • Cf. this question – J. W. Tanner May 14 '20 at 06:35
  • @John Omlean So 0.0625 is not a perfect square, right? – Niraj Raut May 14 '20 at 06:38
  • I don't see exactly what "idea" you are referring to. The definition "square of some element" makes sense in all rings, of course. Whether or not this is interesting, or if it comes from some "idea", is a whole different story. –  May 14 '20 at 06:39
  • The usual definition of perfect square is that is a square of an integer, which must be positive (or zero.) As to whether this idea could be extended - well, sure, what's stopping you? You can talk about the squares of $\mathbb{R}$, which are all nonnegative real numbers, the squares of complex numbers (which are just all complex numbers), or the squares of the Gaussian integers $\mathbb{Z}[i]$, or the squares in whatever other weird number system you want to play with or invent. – Jair Taylor May 14 '20 at 06:42
  • @NirajRaut As I stated, it depends on the definition you use. According to Wikipedia's Perfect square article, it's "an element of algebraic structure that is equal to the square of another element". As such, $0.0624$ could be a "perfect square" if you're referring to squares of rational values. However, a "perfect square integer" is a square integer, i.e., a square of an integer, with this being, as far as I know, the usual default meaning of just "perfect square" with no other appropriate context. – John Omielan May 14 '20 at 06:43
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    The title and body of the question don't match. – joriki May 14 '20 at 06:51

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Let $X$ be some set with multiplication $\cdot$. An element $x \in X$ will be called a perfect square if there is an element $r \in X$ such that $r \cdot r = x.$

For $X = \mathbb{N}_0$ the perfect squares are $\{ 0, 1, 4, 9, 16, 25, \ldots \}.$

For $X = \mathbb{R}$ every non-negative numbers is a perfect square. For example, $\pi = \sqrt{\pi} \cdot \sqrt{\pi}.$ Therefore, a notation of perfect square is not very interesting for the reals.

For $X = \mathbb{Q}$ however, $\frac{4}{9} = \left(\frac{2}{3}\right)^2$ is a perfect square, while $\frac{2}{3}$ is not a perfect square. So here we might speak of perfect squares, although I do not think that it is very common.

md2perpe
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