I am very new to this site (great site, btw), and have looked around for a similar Q, but found none. Much online talk about sqrt(2) being irrational. A quick Excel spreadsheet inquiry shows a LOT of square roots that look like they're irrational. It seems like only perfect squares and multiples of perfect squares (4*49) have obviously rational square roots (they're integers). Is there a way to tell the difference between a true irrational number and a huge rational number? (1/499 repeats for 500 digits) And finally, of all those irrational-looking decimal numbers, which is the first one (not a perfect square) to be deceptively rational?
I have my money on 18^.5, but I do not know how to tell what is what.

Thanks in advance
-Jim Huddle

First, the subject line: there is no smallest rational square root as there is no largest integer square root. They would be reciprocal. Just remember that any natural number - any positive number, in fact - is the root of its square.

>It seems like only perfect squares
>and multiples of perfect squares (4*49) have obviously
>rational square roots

That's correct. A square root of a non-perfect square (integer or rational) is irrational.

>Is there a way to
>tell the difference between a true irrational number and a
>huge rational number?

By looking at the decimal expansion? The only way to claim that a decimal expansion is that of a rational number is to observe a period, i.e., a repeated pattern. This, even if short, may start arbtrarily far from the decimal point.

>(1/499 repeats for 500 digits)

It also may be arbitrarily long, as this example shows.

To understand this imagine a rational number and its decimal expansion which is periodic. Now, you start changing some digits randomly. If you change a random digit in the recurring period it will no longer will be recurring and the resulting number will be irrational.

> And
>finally, of all those irrational-looking decimal numbers,
>which is the first one (not a perfect square) to be
>deceptively rational?

I do not understand that question but suspect that the answer is there is no first in whatever sense.

>I have my money on 18^.5, but I do not know how to tell what
>is what.

Why? What do you see in 18^.5?

>>It seems like only perfect squares and multiples of perfect
>>squares (4*49) have obviously rational square roots

>That's correct. A square root of a non-perfect square
>(integer or rational) is irrational.

At least it seems that way. It also seems that any number of
perfect squares multiplied together will create a perfect square.
Seems like a direct corollary to the prime permutations of the
Fundamental Theorem of Arithmetic. They are, in effect, perfect
square permutations, and no matter how many you multiply together,
you *seem* to get a perfect square.

I wonder if there are subtle distinctions between prime perfect
squares and compound perfect squares...

>>Is there a way to tell the difference between a true irrational
>>number and a huge rational number?

>By looking at the decimal expansion?...

I know it's difficult to judge by expansions. If you get enough
digits, OK. 1/7 = 0.142857 142857 142857 etc... And you can *assume*
the repetition means the number is rational. I was just wondering
if there was some mathematical (vs. manual) method of working out
whether a given decimal expansion was rational, irrespective of
the length of its period.

Has the "a square root of a non-perfect square is irrational"
statement been proven, somewhere?

>>And finally, of all those irrational-looking decimal numbers,
>>which is the first one (not a perfect square) to be
>>deceptively rational?

>I do not understand that question but suspect that the
>answer is there is no first in whatever sense.

If I am looking at a bunch of non-integer results from processing
square roots, and I do not know of the "rule" that says the non-
perfect-square roots are irrational, I might wonder, which, if
any of them, are rational. Especially after all the "hype" about
2^.5 (square root of 2) being ***irrational***!!! Well, it looks
like there's a LOT of irrational sqrt()s, so what could the big
deal be about sqrt(2)?

I'm wondering if, among all the *seemingly* irrational results,
there might be a rational result (like 7.142857142857), and what
(among the irrational masses) the first (smallest) one might be.
If *all non-perfect-square roots* are irrational by definition
or proof, I'm out of luck. Otherwise...

>>I have my money on 18^.5, but I do not know how to tell what
>>is what.

>Why? What do you see in 18^.5?

It's prime factors are 2,3,3. It has a perfect square (3x3) as a
factor, plus, it scared Theodorus :-)

http://www.cut-the-knot.org/proofs/Why17.shtml

That, and Alexander Bogomolny didn't come right out and say what
you did, that they're *all* irrational (non-perf-sq's). In fact, I
haven't found that *anywhere* online. I wonder why?

Thanks for the feedback!
-JH

waitaminute-- *you're* Alexander Bogomolny! right?
I didn't pick that up from alexb! Super site! Great manifesto.

Why didn't you mention the "all non-perf-sq's are irrational" thing
in Why17? Did I miss it? Has someone proved this? Do you have a
page?

>>>It seems like only perfect squares and multiples of perfect
>>>squares (4*49) have obviously rational square roots
>
>>That's correct. A square root of a non-perfect square
>>(integer or rational) is irrational.
>
>At least it seems that way. It also seems that any number
>of perfect squares multiplied together will create a perfect
>square.

Of course.

>Seems like a direct corollary to the prime permutations of
>the Fundamental Theorem of Arithmetic.

It is simpler

a²×b² = (ab)²

>I wonder if there are subtle distinctions between prime
>perfect squares and compound perfect squares...

Do not know about subtle. But "prime perfect squares" have only one factor different from 1 and the number itself; others have more than 1 such factors.

>>>Is there a way to tell the difference between a true irrational
>>>number and a huge rational number?

No. There is no way to tell one from another by looking at a finite part of there decimal expansion.

>I know it's difficult to judge by expansions.

It is not just difficult, it is simply impossible.

>If you get enough digits, OK.

No, there is never enough digits. It's like asking for the largest number. Give me as long an expansion as you wish and I will modify something beyond what you see to spoil possible rationality of the number.

>I was just wondering
>if there was some mathematical (vs. manual) method of
>working out whether a given decimal expansion was
>rational, irrespective of the length of its period.

Again, no. You need the whole thing.

>Has the "a square root of a non-perfect square is
>irrational" statement been proven, somewhere?

Yes. See Proof 5 at

http://www.cut-the-knot.org/proofs/sq_root.shtml

>
>>>And finally, of all those irrational-looking decimal numbers,
>>>which is the first one (not a perfect square) to be
>>>deceptively rational?
>
>>I do not understand that question but suspect that the
>>answer is there is no first in whatever sense.
>
>If I am looking at a bunch of non-integer results from
>processing square roots, and I do not know of the "rule"
>that says the non-perfect-square roots are irrational,
>I might wonder, which, if any of them, are rational.

But now that you know the rule the mystery is gone, right?

>Especially after all the "hype" about
>2^.5 (square root of 2) being ***irrational***!!! Well, it
>looks like there's a LOT of irrational sqrt()s, so what
>could the big deal be about sqrt(2)?

sqrt(2) is the diagonal of the square with side 1.

>I'm wondering if, among all the *seemingly* irrational
>results,
>there might be a rational result (like 7.142857142857), and
>what
>(among the irrational masses) the first (smallest) one might
>be.

There is no first as there is no last.

Thank you. It will take some quality time to decipher Proof 5, but I look forward to the result. You have a fascinating and well-managed site. My hat is off to you; you are doing the world a great service. Keep up the great work!

-JH