3

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.

Asan Ramzan
  • 131

3 Answers3

4

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.

====

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.

fleablood
  • 124,253
1

$2$ 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.

Théophile
  • 24,627
Nuclear Hoagie
  • 1,976
-4

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.

marty cohen
  • 107,799
  • I disagree. The number either is or is not an integer whether we know whether it is or is not. So it IS an integer before we simplify it. It's just that before we simplify it we don't KNOW if it is an integer. But that doesn't mean it isn't an integer. (because ... it is.) – fleablood Nov 27 '18 at 21:22
  • Suppose the expression encodes something we don't know the answer for - for example, 1 if $\pi$ is a normal number, 1/2 if it is not. Is the value of the expression an integer? – marty cohen Nov 27 '18 at 23:38
  • 1
    " Is the value of the expression an integer?" If $\pi$ is normal then it is an integer. If $\pi$ is not normal it is not an integer. The fact that we don't know if it's an integer or not doesn't prevent it from being an integer or a non-integer. – fleablood Nov 28 '18 at 00:04
  • No. A normal number is a real number: https://en.wikipedia.org/wiki/Normal_number – marty cohen Nov 28 '18 at 03:47
  • 1
    Um.... What are you talking about??? Of course normal numbers are real numbers. Everything that isnt complex is a real number. You asked if given the function $f(x) = 1$ if $x$ is normal; $f(x)$ if $x$ is not normal. You asked whether $f(\pi)$ is an integer (which it is if it is $1$) or if it was not an integer (which it will not be if it is $\frac 12$). And the answer is: We don't know. But EITHER $f(\pi)$ is an integer OR it is not. We can NOT say $f(\pi)$ is a non-integer, just because we don't know that it is an integer. We don't know it is not. – fleablood Nov 28 '18 at 05:38
  • "Suppose the expression encodes something we don't know the answer for" Then we don't know the answer for it. That's all. That doesn't mean the answer is "No". It means the answer is either "Yes" or "No" but not both and not neither, but we do not and probably never will know which. All numbers are either an integer, or a non-integer. And it doesn't matter how we express them. The fact that we don't KNOW which some numbers are doesn't mean they aren't either. – fleablood Nov 28 '18 at 05:42