There are many possible representations of real numbers which could in theory be used in programming languages. For instance, you could represent them as funct

There are many possible representations of real numbers which could in theory be used in programming languages. For instance, you could represent them as functions which given a natural number produces a rational approximation – say f(n) should be a rational number closed than 1/2ⁿ to the number you represent (Cauchy sequences). Then addition of two numbers f and g, would be[0] the function (f+g)(n) ≔ f(n) + g(n), where + denotes rational summation on the left-hand side. Or you could have a function which given a rational number input produces a boolean which tells you if the real number is less than the input. (Dedekind cuts) No mainstream language has real numbers as a primitive[2]. In fact, not many languages have even arbitrary integers as a primitive type. But many mainstream languages have functions as first class objects, and therefore there is nothing stopping you from using them to represent real numbers. Not all of these representations are created equal. For instance if you wanted to represent real numbers as a function which gives you the n-th digit in the decimal expansions, you cannot implement addition. This because you would have to look arbitrarily far into the digits of each number to decide even the first digit.[1] Someone is bound say "Oh, but that is just computable numbers and there are only countably many of those". This is true if you accept classical logic. In constructive mathematics (which is most relevant to computer science), however, one can only prove that there is a surjection from the natural numbers (but there is no constructive bijection). In fact it is consistent to assume all real numbers are computable. [0]: Most likely, you want (f+g)(n) ≔ f(n+1) + g(n+1) to get a close enough approximation, but this is a technicality. [1]: Imagine 0.00000⋯ + 0.99999⋯. At any point later the first number could become non-zero and the first digit would be 1. Or the second number could become smaller than 9 and the result would have to have first digit 0. No way to tell what the result would be without looking at infinitely many digits. [2]: I know of at least one, non-mainstream, language which had built-in support for real numbers. It is called RealPCF – and I am not sure it was even implemented on a computer, or if it was just a theoretical construct.