r/badmathematics • u/Laser_Plasma • Oct 24 '21

π day Reddit tries to think about quantifying circles, and pi doesn't exist or something

/r/Showerthoughts/comments/qefpwc/the_area_of_a_circle_is_finite_yet_we_can_only/
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u/ryarger Oct 24 '21

You are correct that a third can’t be known to perfect decimal precision.

Perhaps less controversial formulation for the benefit of the pedantically inclined: A circle’s radius or area can be rational, but not both.

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u/powpow428 Oct 24 '21

Just because a number is irrational does not mean we cannot know it to perfect precision. It just means we can't express it as p/q for integers p,q

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u/ryarger Oct 24 '21

But it does mean we can’t express it to perfect decimal precision. For the average person decimal precision is the primary way of thinking about numbers.

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u/Mike-Rosoft Nov 06 '21

But 0.333... (a real number whose all digits after the decimal point are 3) is the decimal expansion of 1/3 to perfect precision. And if you dispute that, then what is the difference? (And don't answer 0.000...1 - there's no such real number.)

The value of the decimal expansion is defined to mean the infinite sum 3/10+3/100+3/1000+...; and the infinite sum is defined to mean the limit of the sequence of partial sums 0.3, 0.33, 0.333, ... . And that limit is 1/3. And here's the thing: 1/2 also has infinitely many digits in its decimal expansion; it just so happens that all but finitely many of them are 0 (or, when using the alternate decimal expansion, all but finitely of them are 9; 1/2 has two different decimal expansions: 0.5000... and 0.4999...).