I think a lot of people are mistaking "Real numbers are useless." For "irrational numbers are useless." The square root of two is useful. e is useful. π is useful. Every solution to every equation that anyone has ever written is potentially useful. 1/5 - ThreadSky

drwhen.bsky.social • 50 days ago

I think a lot of people are mistaking "Real numbers are useless." For "irrational numbers are useless."

The square root of two is useful. e is useful. π is useful. Every solution to every equation that anyone has ever written is potentially useful.

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hexygen.bsky.social•50 days ago

I think the point is that the square root of two approximated to 20 digits is good enough for every practical application.

Evidently, nobody ever *used* an irrational number in any computation or they would still be writing it on their endless piece of paper (or magnetic tape).

badibulgator.bsky.social•50 days ago

If you only count as a computation the manipulation of the decimal (or binary) representations of some numbers, yes.

But when I use SymCalc to do calculations such as this⬇️, I’d say that it *does* use irrational numbers in a computation — no infinite tape needed.

drwhen.bsky.social•49 days ago

If you are measuring a physical object, then yeah, 20 significant digits is plenty. But if you want to prove, for example, that some particular formula is correct, you definitely want exact values.

drwhen.bsky.social•50 days ago

But if you collected together every possible number that anyone could possibly define using any combination of mathematical symbols and sentences in any language that have ever existed or will ever exist, it would not even begin to scratch the surface of the "real numbers".

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hughgnu.bsky.social•50 days ago

No. The real numbers stretch out to an infinite abyss our imaginations would be scandalized to behold. And that's not counting the negative reals that go really negative with shameless extravangance.

drwhen.bsky.social•50 days ago

The vast majority of real numbers cannot be defined or described ever.

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drwhen.bsky.social•50 days ago

But there is a use for talking about the set of real numbers because we can't actually define "every number that can ever be defined" in a meaningful way (if you think you have such a definition, I can define one that you missed, Cantor's diagonalization style).

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drwhen.bsky.social•50 days ago

So we prove claims about "every real number" including the useless ones we can't even describe and we know that those claims will be true even when someone finds a new way to define a number.

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