How can we be completely sure that a number is really infinite and does not end in millions of additional digits?
In: Mathematics
I presume you’re asking how we decide whether a number’s decimal expansion is infinitely long, not whether the number itself is infinite (which is impossible, since all real numbers are finitely large).
The general answer is that we can’t prove that the decimal digits will go on forever simply by computing those digits, because, as you note, they could always end at some point beyond our calculations. So we resort to using other techniques to prove that a number’s decimal expansion is infinite. One way is to prove that the number is irrational, which is usually done by contradiction: you assume it’s rational and show that this assumption leads to a contradiction with itself or other known facts, which then means that the assumption is false, i.e. the number is irrational. But there are also rational numbers with infinitely long decimal expansions, eg 1/3 = 0.3333…. To prove that the decimal expansion of 1/3 is infinitely long, it’s enough to show that the decimal expansion 0.333… equals 1/3 (which is done by considering the infinitely long decimal expansion as the limit of a sequence of finitely long decimal expansions and showing that the sequence converges to 1/3).
It’s also worth noting that decimal expansions aren’t necessarily unique, so some numbers can have both finite and infinitely long decimal expansions; for instance 1 = 0.999… (which can proven using the same technique mentioned for the case of 1/3 = 1, that is, considering the infinitely long decimal expansion as a limit and proving that the limit is 1). This non-uniqueness of decimal expansions is yet another reason mathematicians generally don’t find decimals to be particularly useful in theoretical work.
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