*by Jeff Secker* *| Originally posted on *

I mention rational and irrational numbers in The Kairos Codex. A *rational number* is one which can be expressed as a ratio (the ‘ratio’ part of *ratio*nal) of two integers, as illustrated by the following examples:

- 2.25 = 9/4
- 3.3333333… = 10/3
- 2.0 = 2/1
- 0.15 = 15/100
- 18.181818… = 200/11
- 8.02439024 = 987/123

Integers, including zero, are rational numbers:

- 3 = 3/1
- 176 = 176/1
- 0 = 0/1

On the other hand, *irrational numbers* cannot be expressed as a ratio of two integers. Four famous examples (the rock stars of irrational numbers!) are provided below. Note that the three dots indicates that the digits go on forever without repeating.

- π = 3.1415926535… (pi)
- √2 = 1.4142135623… (the square root of 2)
*e*= 2.7182818284… (Euler’s number, a mathematical constant)- ϕ= 2.6180339887… (the golden ratio)

There are an infinite number of irrational numbers; five more examples are √3, √5, √11, √21 and π^{2}. There are rational-number approximations to these irrational numbers. For example, the first three digits of 22/7 are 3.14, which match the first two decimals of pi. An even better rational-number approximation to pi is 355/113, which match the first six decimals. But there are no fractions which would evaluate to these irrational numbers.

Rational numbers, irrational numbers and infinities:

The set of rational numbers (represented by **Q**) contains all integers (represented by **Z**) and some of the real numbers (represented by **R**). The set of real numbers, **R,** contains all rational numbers, **Q**, and all irrational numbers (represented by **P**). There are an infinite number of both rational numbers and irrational numbers; however, there are a *countable infinity* of rational numbers whereas there are an *uncountable infinity* (i.e., a larger infinity) of irrational numbers. (See Maths in a Minute, University of Cambridge.)