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.
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?
I also never heard the expression, I think in English the correct expression is recurring decimal.