Journey from Natural Numbers to Complex Numbers

4Real Numbers

We have gone through the properties of rational numbers in the previous chapter. An important thing about rational numbers is that with the addition and multiplication operations, it is a field. We know that x ² = 2 does not have a rational solution for x.

We go for a broader shortcoming of rational numbers rather than extending it to have the solution of the equation given above and other equations similar to it.

Definition 4.1: For a subset B of set A with the total order  ≤  is bounded above if ∃ u ∈ A ∍ ∀ x ∈ B , x ≤ u and u is an upper bound of set B in A.

An important point to be noted is that if u is an upper bound and if u ≤ v then v is an upper bound.

Example 4.1: A = {1, 11, 21} as the subset of N with usual ≤ ordering ...