Can a perfect square be only an integer number?

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

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1 Answer

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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.

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