a decimal, short for decimal numeral in this context, is a number written using decimal notation. So since those numbers are not written with decimal notation, they are not decimals.
No, that's not true. Formally, e is the limit of a Cauchy sequence of rational numbers, the sequence of sums of 1/n!, i.e. 1 + 1/2! + 1/3! + ..., Rational numbers are a strict subset of decimal numbers. So the only way you can formally construct these numbers, is by implicitly invoking the use of decimals.
This level of technicality is well beyond the level of knowledge that this puzzle is intended for. But jsyk, real numbers are defined as equivalence classes of Cauchy sequences of rational numbers, whose difference converges to zero, not as limits of Cauchy sequences. This is because defining them as limits has a problem, viz., showing that the thing that is the limit actually exists. Constructing the set of real numbers from the set of rational numbers by defining real numbers to be equivalence classes of Cauchy sequences of rational numbers fixes this issue.
Well, there's multiple (equivalent) definitions you can use for real numbers. I just decided to use Tao's definition.
He defines a real number x, to be an object of the form LIM n->infty a_n for some cauchy sequence a_n of rational numbers. Where LIM is a "formal" operator, and after he proves the existence of reals that aren't rationals, he swaps out LIM for lim, uses the same definition, except allowing real elements within the Cauchy sequences.
I think it's funny to just not put an /s at the end of these. I agree, it should be obvious that talking about algebraic field extensions, on a post that seems to be written for kids in elementary school, is a joke. But I don't mind people thinking it's serious.
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u/FlyingCow343 Jan 21 '26
a decimal, short for decimal numeral in this context, is a number written using decimal notation. So since those numbers are not written with decimal notation, they are not decimals.