Krdytkyu

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.

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.

It is helpful to have vocabulary to distinguish between what an object fundamentally is and how it is represented.

A fraction $a/b$ is said to be irreducible or in simplest form if $a$ and $b$ are integers with no common factors, i.e., $gcd(a,b) = 1$. So, for example, $2/3$ is irreducible, while $200/300$ is not.

An integer can be written down in many ways. Not all of those ways are simple, but fundamentally the number is the same. Take $2$, for instance. We could write it as $10/5$: this is not the simplest form of the fraction, but the number is still an integer. Or we could express it as
$$1 + 1/2 + 1/4 + 1/8 + cdots;$$
again, this is not as straightforward as simply writing "$2$", but the expression still represents an integer.

$2$ 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.

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.