Unit fraction

From Free net encyclopedia

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.

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History

Unit fractions were used by the ancient Egyptians to facilitate mathematical computations. In modern notation unit fractions may be indicated as '2, '3, '4, '5 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 3 '7 especially when using a ruler divided into parts as calculator makes the process trivially simple.

For calculations involving multiplication or division by doubling 3 '7 may be taken as 3 '8 '64

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.

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