# Rationals…

Standard

A few days ago I noticed some fascinating properties of so called rational numbers.

Natural Bias:

Our definition of a number being rational or irrational is very much biased. We implicitly assume our numbers to be in decimals (base 10), and then define rational numbers as those numbers which have terminating or recurring decimal representation.

But it is interesting to note that, for example, √5 is irrational in base-10 (non-terminating, non-repeating decimal representation) but if we consider “golden-ratio base“, √5 = 10.1, has terminated representation, just like rational number!!

Ability to complete themselves:

When we construct numbers following Peano’s Axioms we can “easily” create (set of) natural numbers ($\mathbb{N}$), and from them integers ($\mathbb{Z}$) and rational numbers ($\mathbb{Q}$). Notice that $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$.

But it is comparatively difficult to create real numbers ($\mathbb{R}$) from rational numbers ($\mathbb{Q}$) although still we want to create a set from its subset. Notice that unlike previous cases, to create $\mathbb{R}$ we will first have to create so-called irrational numbers ($\overline{\mathbb{Q}}$) from $\mathbb{Q}$. The challenge of creating the complementary set ($\overline{\mathbb{Q}}$) of a given set ($\mathbb{Q}$) using the given set ($\mathbb{Q}$) itself makes it difficult to create $\mathbb{R}$ from $\mathbb{Q}$ . We overcome this difficulty by using specialized techniques like Dedekind cut or Cauchy sequences (the process is called “completion of rational numbers”).