non-turnacating rationals

Any rational number can be expressed as a turnacating decimal, or a non-turnacating decimal which eventually turns in to an infinitely repeating sequence of numbers. Does anyone know why that last fact is true? Just something I'm curious to know.

Furthermore, any decimal number which ends with an infinitely repeating sequence of numbers is rational. I would also like to know why this is.

Also, is this a peculiarity of the decimal number system, or is this true in any base?

The second part is fairly straightforward. Here's a demonstration, but the principle would apply to any such number.

Let's say we've got the number 0.abcdefdefdef...(def repeats infinitely)

= 0.abc + def x 10^-6 + def x 10^-9 + def x 10^-12...

= 0.abc + def x 10^-6 x (1 + 10-3 + 10-6 + ...)

But that series in the brackets is a geometric series. We know what that adds up to:

1 + 10^-3 + 10^-6 + ... = 1/(1 - 10^-3), which is rational (= 1000/999)

So we've now got an expression that's just a (finite) sum of things we know is rational. Which must be rational. Not only do we know any such number is rational, it's not all that hard to work out what it is as a fraction.

There's also nothing in there that would work differently in other number bases - your geometric series would be different, that's all.

Incidentally: "turnacating"? I can see what you mean from context, but I've never heard that word before. It's not in the OED, and Googling doesn't have anything relevant. You sure that's right?

For the first part, think about how you calculate the decimal expansion of a fraction. Let's say:

X/Y = 0.abcdefghijklm....... (where X & Y integers, and X < Y)

We can obtain each decimal by multiplying by 10 and taking the integer part. Then subtract that:

(10X - aY)/Y = 0.bcdefghijklm........

and repeat.

Each time, the left-hand side is a fraction < 1 with Y as the denominator. But either:

1. It hits zero. In which case we stop, and we've got a finite-length ('turnacating'?) decimal, or

2. Since there are only Y-1 possible such non-zero fractions, eventually we have to hit one we've had before. And the pattern repeats from this point on.

You can also use that nifty trick to show that 0.999... = 1.

Originally posted by mtthw
Incidentally: "turnacating"? I can see what you mean from context, but I've never heard that word before. It's not in the OED, and Googling doesn't have anything relevant. You sure that's right?

Wow, I made up a word. That's new. I think the word I was looking for was truncate, or something. Thanks for the posts.

I think you meant "non-truncated".

Yea.

Can I use my upcoming quantum test as an excuse for this rather glaring mistake?