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Number Toolkit

Number

Rational numbers

  • Any value expressible as one integer over another, with a non-zero denominator, is a rational number
  • Every integer is rational, because it can be written over a denominator of 1
  • Every terminating decimal is rational
    • 0.35 = 7/20
  • Every recurring decimal is rational
    • 0.333… = 1/3

Irrational numbers

  • An irrational number admits no such integer-over-integer form
  • Its decimal form never terminates and never settles into a repeating block
  • The square root of a number that is not a square number is irrational, and is called a surd
    • √7 is irrational, but √36 is not, because it equals 6
  • π is irrational
  • A non-zero rational number multiplied by an irrational number is still irrational
    • 5√2 and π/4 are both irrational
Exam tip

Leave the answer as a surd. This is marked as a final answer, so writing √17 and then "= 4.12" beside it loses the mark, because a terminating decimal is rational.

Worked example

Naming an irrational number in a range

Write down an irrational number between 3 and 4. Explain how you know it is irrational.

Solution:

  • Square both ends of the range: 3² = 9 and 4² = 16
  • Any square root of a number between 9 and 16 therefore lies between 3 and 4
  • Choose a value in that range that is not a square number, such as 10
  • √10 is one possible answer, and √11, √12, √13, √14 and √15 all work equally well
  • It is irrational because 10 is not a square number, so its square root neither terminates nor recurs
  • √16 would not do, because 16 is a square number and √16 = 4, which is rational