Is a number written in the square root/fraction form called a non-integer even if it can be simplified to an integer

Question

This is very simple question, but I cannot get the ansewer from the internet.

Is a number written in the square root/fraction form called a non-integer even if it can be simplified to an integer.

For example 4/2, 12/4, sqr4, sqr64 etc... do these need to be simplified before we can call them integers.

Too make this easer to understand are sqr64 and 12/4 non-integers while 8 and 3 are integers.

Answer 1

If it can be simplified to an integer, it can be called an integer after the simplification.

Until the simplification is done, I would just call the expression "an expression" when it is not clear if it could be simplified to an integer.

Considering how expressions involving nested radicals can be sometimes amazingly simplified, I think that there would be cases where the fact that an expression simplifies to an integer is a surprise.

Answer 2

$2$ is an integer. $4/2$ is equal to $2$, and therefore has all the properties that the number $2$ has, including being an integer. The square root of $4$ is also equal to $2$, so it's an integer as well. In some cases, you'll probably need to simplify to recognize that it is indeed an integer, but that doesn't change its properties no matter how you write it.

For example, is $\sqrt{14883}$ an integer? How about $\sqrt{14884}$? It might be tough to tell unless you do the simplification, but one is an integer and one isn't.

Answer 3

No. Numbers are what they are. It doesn't matter how they are represented.

$7$ is an integer. Period.

It doesn't matter if is written as $5 + 2$ or $\sqrt{49}$ or $\sqrt{25} + \frac{\sqrt[3]{16}}{2^{\frac 13}}$ or $\ln (e^7)$.

Those are all equal to $7$ and $7$ is an integer. Period.

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That said, it might not be easy (or even possible) to tell if a number is or is not an integer. It's obvious that $7$ is an integer and $7.0000012142650469991421281354411.....$ isn't. But it isn't clear whether $\sqrt[7]{823543}$ or $\sqrt[7]{823544}$ are integers. (It turns out that those are the same numbers.)

But it doesn't matter whether we know if a number is an integer or not. It either is or isn't.