Unit fraction

A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n. Examples are 1/1, 1/2, 1/3, 1/42 etc.

The partial sum

1/1+1/2+1/3+...+1/n

gives the harmonic series, and is close to log_e(n)+γ as n increases. So the sum of all unit fractions is infinite.

The product of two unit fractions is again a unit fraction; the sum and difference may be unit fractions, though are often not.

• 1/m × 1/n = 1/(mn)
  • 1/2 × 1/5 = 1/10
  • 1/3 × 1/6 = 1/18
• 1/m + 1/n = (n+m)/(mn)
  • 1/2 + 1/5 = 7/10
  • 1/3 + 1/6 = 1/2
• 1/m – 1/n = (n-m)/(mn)
  • 1/2 – 1/5 = 3/10
  • 1/3 – 1/6 = 1/6

Any positive rational number can be written as the sum of distinct unit fractions. The result is an Egyptian fraction, but the expression is not unique. For example 0.8 = 1/2+1/4+1/20 = 1/3+1/5+1/6+1/10.