How do we represent real numbers for calculation?

• This includes paper calculation and also electronic calculation

• The usual method is with a decimal expansion

• Theorem: every real number r has a unique representation as a₀, a₁, a₂...

• a₀ is any integer; 0 ≤ a_i ≤ 9 when i > 0

• For example:

  • 173 / 2 = 86 . 5 0 0 0 0 0 0 ....

  • 1 / 3 = 0 . 3 3 3 3 3 3 3 3 3 3 ....

  • 1 0 / 7 = 1 . 4 2 8 5 7 1 4 2 8 5 7 ....

  • = 1 . 4 1 4 2 1 3 5 ....

  • π = 3 . 1 4 1 5 9 2 6 5 3 5 ....

• (Actually, the representation is not quite unique)

  • (Since 86 . 5 0 0 0 0 0 0 .... = 86 . 4 9 9 9 9 9 9 ....)