Real Numbers
Real numbers are represented by a limiting series of rational numbers given by a function from the natural numbers to the rationals. A real number can have many series representations just like the rational numbers can. The series \(f(n)\) is such that it must satisfy: \( \forall_{t:\mathbb{N}} \exists_{n:\mathbb{N}} \forall_{m:\mathbb{N}} (m>n) \rightarrow |f(m)-f(n)|\lt\frac{1}{t} \). We can add, multiply, subtract and divide reals by doing those operations on the Cauchy sequences (subject to some limitations.)Note that reals are the "completion" of the rationals and the complex numbers are the "completion" of the complex rationals (using the standard metric).
Depending on our purposes, we can either work with the real numbers as symbols: Or we can work with their definitions: To create a natural number we can write: We can multply two real numbers together. We can either keep them in their symbolic form. or we can expand out the definitions: To get an approximate rational value we can take the 100th term in the series: