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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 loge(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.
History
Unit fractions were used by the ancient Egyptians to facilitate mathematical computations. In modern notation Egyptian fractions may be indicated as and so forth.
Unlike systems with a fixed base such as the decimal system which are forced to a long series of ever closer approximations, the variable base allowed the Egyptians to get to an exact equivalent relatively quickly.
Even irrational numbers such as π can be closely approximated using for example especially when using a ruler divided into parts as calculator makes the process trivially simple.
For calculations involving multiplication or division by doubling, may be taken as .
Using fractions 2/3 or 3/4 is adequate in terms of pure mathematics but breaks down in human terms as for example in distributing rations of grain and beer where the number of people is not really a variable.
See also
External links
Fractions
Fractions | Elementary arithmetic
Fracción unitaria | Fraction unitaire | Enotski ulomek | 單位分數
"Unit fraction"