What is the difference between, a "square" and a "perfect-square", number?

Question

Is, "36", a perfect square?

I know that, "4" is a perfect square.

Similarly, "1","9","25", are "perfect-square"s.

Answer 1

A square number can also be called a perfect square. So yes, the number $36$ is a perfect square, because it is the product of $6$ and itself.

Answer 2

It's interesting how a discipline that is so reliant on the precision of its language can still have terms that are fuzzy.

Anyhow, for what it's worth, my interpretation of "square number" is any number that is formed by squaring a natural number (or maybe an integer at a push).

Many people refer to $a^2-b^2$ as a difference of perfect squares and that informs my interpretation of "perfect square number" as one whose square root is a rational number (remembering that this includes integers).

Answer 3

A perfect square is when two identical integers are multiplied together to form a number.

For example, $\quad\quad2\times 2=4,\quad 3\times 3=9,\quad 4\times 4=16.$

So $4, 9, 16,$ and many others are perfect squares.

$10$ is not a perfect square, because for us to get it as a result, we need to multiply $3.16227766017$ with $3.16227766017$, which are Not integers.

Answer 4

The term perfect square originates outside of the community of rigorous professional mathematicians. It originates from the history of educators of mathematics writing elementary-school through high-school textbooks without themselves being practicing mathematicians. Typically those textbook authors would more likely have had doctor-of-education degrees more than PhDs in mathematics. That said (and regardless of whether there is 100% historical conformance to that stereotype), there was an intended meaning to the “perfect” as distinct from the “square”.

25.5025 is a square of 5.05 (obviously). But the “perfect” was in this historical community of textbook-authoring educators referring to the square being an integer without any nonzero fractional/etc portion of the real (or any other, e.g., complex, quaternion, octonion) number. The “perfect” in the context of “perfect square” is in effect synonymous with the term ‘square positive integer’.