Rational and irrational numbers

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 rational) 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 π2. 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.)