Comment by toth
2 days 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
> 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.