When Pi is calculated in decimal you get a never ending remainder. Is this simply a quirk of the decimal system that wouldn’t exist if we were born with 12 fingers instead of 10?

What would happen if your used hexadecimal or unidecimal to find the value of pi. Is there a way to find a theorical base system which would produce a whole number for pi. Like base 42, for example?

In: 15

In short, no. Changing your base will never get you different results beyond the result being represented in a different base. It’s like if you had a 2 foot board and you wanted it longer, so you measured it in metric. Different representation, same board.

It can be calculated in any base you want. Irrationality implies that the decimal is non-terminating in *every* base, whereas to be a rational number, there must be some base where the decimal terminates (trivially, when the base is the denominator, though other bases may also terminate).

This goes both ways. If there is a base where a number terminates, that number *must* be rational. If there were a base where pi terminates, pi would necessarily be rational. Since we know pi is irrational, there can be no base where it terminates.

No. Pi is irrational. It cannot be represented by the ratio of two integers. That holds regardless of what base number system you are using. Changing the base would determine whether or not a rational number would terminate or repeat, but it does not have any special effect on irrational numbers.

Pi is also “transcendental” which means that it’s not the root of a rational number either (it is not like the square root of two).

Although the numerical base does not matter, you did bring it up, so I figured it would be cool to point out that one of the [best ways of calculating large numbers of digits of pi](https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula) is actually designed to work best in base-16.

If pi could be expressed in another number base (let’s call that base b). I.e. pi=x_b/y_b (ratio of two integers in base b), then you could just convert x_b and y_b to base 10 and now have pi as a ratio of two integers in base 10, which we know isn’t possible.

So no, you cannot express pi as a rational number by simply changing bases. Pi is irrational as irrational can be.

This is true regardless of your decimal system, because pi is an irrational number. It cannot be produced by dividing one integer by another. This property remains true regardless of your base.