>>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?