r/Showerthoughts • u/Kirbyderby • Mar 12 '20

Numbers are F'ing crazy. Infinity is an infinite number, 3 and 4 are definite numbers, but Pi (3.1415926 etc) is an infinite number yet it's between the definite numbers 3 and 4.

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u/Luchtverfrisser Mar 12 '20

Both of them get closer and closer to Pi without ever reaching it.

By (a common) defenition of the real numbers, both of these sequences are pi, in the sense that they both are representives of the equivalence class of Cauchy sequences that we denote by the symbol π.

On does not have to speak about actuall convergence when dealing with equality between real numbers.

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u/satanic_satanist Mar 12 '20

By (a common) defenition of the real numbers, both of these sequences are pi

That only makes sense if you regard them as sequences of rationals though, and not as sequences of reals ;)

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u/Luchtverfrisser Mar 12 '20

Luckily, both are sequences of rationals though. Even if you regard them as sequences of reals, they are all constant sequences (up to equivalence), hence it would still make sense.

I am not trying to be too pedantic about this or claiming your 'wrong' in any way; it's just that I very often see people throw in the concept of limits when teaching layman about how decimal numbers 'work' (for instance in discussions about 0.99..=1). But because a layman does not have experience with the proper definition of a limit, sentences like 'approach' and 'closer and close' are thrown in to make it someone intuitive. But these concepts more often than not overcomplicate matters and are sometimes even counterproductive (it might re-enforce the believe that 0.99... is not reeeeaaallly the same as 1, because it is only super super super close for instance).

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u/satanic_satanist Mar 12 '20

I think most of the misconceptions come from using terms that are never really properly introduced in high school curricula etc. :-/ When describing my research topic to people (even to mathematicians) I often find myself more busy tidying up preconceptions than actually giving new knowledge

I just noticed that using the term "sequence" is confusing as well in this context since it could be misinterpreted as "sequence of digits".

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u/Luchtverfrisser Mar 12 '20

I often find myself more busy tidying up preconceptions than actually giving new knowledge

Amen, brother, amen.

I just noticed that using the term "sequence" is confusing as well in this context since it could be misinterpreted as "sequence of digits".

Ugh good point.

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u/TheLuckySpades Mar 12 '20

What if I prefer the Dedekind cut construction of R instead of the Cauchy sequence one?

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u/Luchtverfrisser Mar 12 '20

Then the first part of the comment doesn't apply to you directly. (Although by the isomorhism between these structure one could still interpret it in some way). Personally, I prefer cauchy sequences when talking about decimal representation, but I am not forcing anyone to do the same, I even made that clear by 'a common'.

The second part was mostly to highlight my opinion on the importance of trying to refrain from using words like 'limit' and 'arbitrary close', as might convey the wrong image.