We can think of numbers as being points on a continuous number line (or plane, if we want to include lateral/imaginary numbers like “i”). There are so many of them that we can’t possibly think of all of them individually at once, or have a unique symbol for each and every one of them. So we choose a few numbers (0, 1, and the natural numbers all the way up to but not including the chosen base), and some symbols to show relationships, and now we have the tools to describe every number we can possibly think of. But we only have finitely many symbols. The continuum of real numbers is infinite, so there are always going to be some numbers (that do exist) that require an infinite number of symbols to represent.
Decimals are one such system that can technically describe any number with a finite selection of symbols used infinitely many times. If you think about what decimals and the place value system really mean, it’s the sum of an infinite power series that converges to the number we want to represent. The complexity of numbers in general is one of the things that makes mathematics beautiful, in my opinion.
We can think of numbers as being points on a continuous number line (or plane, if we want to include lateral/imaginary numbers like “i”). There are so many of them that we can’t possibly think of all of them at once, or have a unique symbol for each and every one of them. So we choose a few numbers (0, 1, and the natural numbers all the way up to but not including the chosen base), and some symbols to show relationships, and now we have the tools to describe every number we can possibly think of. But we only have finitely many symbols. The continuum of real numbers is infinite, so there are always going to be some numbers (that do exist) that require an infinite number of symbols to represent.
Decimals are one such system that can technically describe any number with a finite selection of symbols used infinitely many times. If you think about what decimals and the place value system really mean, it’s the sum of an infinite power series that converges to the number we want to represent. The complexity of numbers in general is one of the things that makes mathematics beautiful, in my opinion.