> 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