How can we be completely sure that a number is really infinite and does not end in millions of additional digits?
In: Mathematics
Quick explanation:
If a number’s irrational, then it _must_ have an unending decimal expansion.
Proof: if it didn’t, and the expansion ended at some point, you could write that as (the digits up to the end) / 10^n , where n is the decimal place it ended at. But this is a rational number, oops.
If a number’s rational, but has ANY prime factors other than 2 or 5 in its denominator? It also has an unending decimal expansion.
Proof: Close to the same one. But now we look at what the factors of that denominator, 10^n , are: n factors of 2, and n factors of 5. No others.
So: if all you have are factors of 2 and 5, you can multiply by 2/2 or 5/5 (which both = 1) until the denominator has equal amounts of them … and then you have a terminating expansion, just convert that number back into a decimal that ends at the m-th place, where the denominator with balanced numbers of 2s and 5s had m of each.
And: if you have ANY other factors on the bottom, then you can never get the fraction into a form where 10^n is on the bottom – there’s always gonna be other factors multiplying it, and so you can’t turn it back into a terminating decimal.
–Dave, hope that’s relatively clear
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