I think the problem is that people are often only taught base-10 so they confuse numerals with numbers. If you learn base 2 and then base 16 and then base pi, you start to realize that numbers are something more abstract than whichever numeral system we use to represent them. Rightly or wrongly, the way I imagine integers now is an infinite set of different numerals (base infinity?) such that there is only ever 1 digit (I don't actually have concrete pictorials assigned to those numerals).

Interesting. I find it funny to imagine what if you did have pictorial representations. They'd have more and more complex strokes and knots, and you would also need an infinitely large paper or infinitely precise pen in order to not end up repeating a number sooner or later as you enumerate them.

Come to think of it, this is one way of thinking about the relationship between symbols and geometry.

Who says pictorials have to be 2-dimensional? ;)

Base 12 is a better base overall. It is divisible by more numbers. That is why it is used in various monetary, time and measurement systems.

It's the one issue I have with the metric system... But that ship has sailed :) look up the dozenal society if you're curious how fervent some supporters might be.

The Babylonians though it was 60. (But imagine having to remember additional 50 numeric characters.)

> remember additional 50 numeric characters

The numerals are not distinctly varied like our Arabic numerals. Quite the opposite, they are repetitive and completely systematic and require 80% less effort to remember.

https://commons.wikimedia.org/wiki/File:Babylonian_numerals....

Any other argument for a different base aside, in base 12 you have the analogous “problem” that 0.BBB... = 1. None of the usual bases, equipped with this “...” power, avoid the “problem” (non-unique representation).

I think the confusion comes when mathematicians say "a sheet of paper can be 0.333... (0.3_, ie 0.3 recurring) units long" or something because in experienced reality we can always choose a measure that's rational (in the maths sense). That 1/3 of a meter can be measured in one-third-meter units and be precise and easily written.

Now sure, make a square using those measures and measure the diagonal, like an awkward mathematician - "see, see, we need irrationals!" - but then we can just cut another measure that's exactly that length ... stupid mathematicians!

Yeah representation is not reality.

Now, says the ratio of those two measuring sticks ...

Whatever base you use, it will only be useful for rationals. You can't use any base to significantly improve the representation of pi. In base 7, 3.1 would be a nice approximation, but you can't go beyond approximations.

It doesn't have a terrible representation in base pi ;)

;) As someone fond of integers, I can't advocate base pi.

You can't fix irrationals by changing base. But, there are other approaches.

Continued fractions give very approachable representations for common irrational numbers like e and sqrt(2). While π doesn't have a good continued fraction, it has some very well-behaved generalized continued fractions.

https://en.wikipedia.org/wiki/Continued_fraction