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Comment by dmurray

4 months ago

This seems like an incredible magic trick, but once you look closely, it is a magic trick - any series of numbers can be expressed as a constant with this formula.

In a different world, we settled on a system of notation based on continued fractions rather than decimals for writing non-integers. In this world, nobody marvels at the irregularity of pi or e, the fact that they seem to go on for ever without a pattern - both numbers have elegant and regular representations as infinite sums of fractions.

In this world, we all found it slightly harder to make change at the grocery store, but perhaps we made up for that by producing a million high-school Ramanujans.

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dmurray

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toth  4 months ago

> In a different world, we settled on a system of notation based on continued > fractions rather than decimals for writing non-integers. In this world, nobody > marvels at the irregularity of pi or e, the fact that they seem to go on for > ever without a pattern - both numbers have elegant and regular representations > as infinite sums of fractions.

There are some less widely-known topics in math that seem to make some of those that learn them want to "evangelize" about them and wish they had a more starring role. Continued fractions are one.

Now, don't get me wrong. Continued fractions are very cool, some of the associated results are very beautiful. More people should know about them. But they never will be a viable alternative to decimals. Computation is too hard with them for one.

Also, while e has a nice regular continued fraction expansion [1], that is not the case for pi [2]. There is no known formula for the terms, they are as irregular as the decimal digits. There are nice simple formulas for pi as infinite sums of fractions (simplest is probably [3]) but those are not continued fractions.

[1] https://oeis.org/A003417 [2] https://oeis.org/A001203 [3] https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

  • dmurray  4 months ago

    > But they never will be a viable alternative to decimals. Computation is too hard with them for one.

    I think this is too narrow-minded.

    You could make the same argument for ideograms vs alphabetic writing: one is clearly superior and you could never have a technological superpower that relies primarily on the other, but thanks to historical path dependency we actually have both.

    I could imagine a world where the SI system never took off in engineering, due to stubborn people at inopportune moments. Engineers and physicists would still get their jobs done in imperial units, just like American carpenters do today.

    Also I did elide the distinction between continued fractions and infinite sums of fractions, but again we can use our imagination and say that if continued fractions were commonplace, we'd all be a lot more familiar with the infinite sums too.

Someone  4 months ago

> any series of numbers can be expressed as a constant with this formula.

Not any series for this specific approach. As simple examples:

- series containing zeroes will lead to division by zero when computing the number

- if a series repeats a number, the term for it will be zero, and the number cannot be recovered

- if a series grows too fast, the next term ‘overflows’ into the previous one.

  • dmurray  4 months ago

    Good points! Showing what constraints have to be met and proving that the prime numbers meet the requirements does seem worthy of a paper. But once that paper is in, hopefully we don't need 300,000 papers, one for each other sequence in the OEIS, proving that it can or can't be expressed like this.

    • wslh  4 months ago

      I think another interesting avenue for future research is proving whether measures applied to prime numbers (e.g. their statistical distribution) have an equivalence in the Buenos Aires Constant. Intuitively, it seems likely, but I don't have a proof, just a hunch.

moralestapia  4 months ago

>it is a magic trick

John D. Cook is an expert at posing as a 5-6 figure consultant by making use of extremely trivial Math.

Not a complain, but the opposite, I wish I had it like that!

  • ttoinou  4 months ago

    I wish you were right but what we see as something simple is sometimes made simple by huge amount of work.

    Look at how we learn maths and physics right now, it is much simpler than what they had back then. Centuries of human work / knowledge / capital were compressed into tiny powerful equations / definitions / demonstrations, empowering new humans to look at more complex stuff more easily. And there is a good opportunity right now to simplify this even more