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The reciprocal of 3 / 7 , for instance, is 7 / 3 . The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. The reciprocal of a proper fraction is improper, and the reciprocal of an improper fraction not equal to 1 (that is, numerator and denominator are not equal) is a ...
From the relation to unit fractions, it can be shown that cyclic numbers are of the form of the Fermat quotient b p − 1 − 1 p {\displaystyle {\frac {b^{p-1}-1}{p}}} where b is the number base (10 for decimal ), and p is a prime that does not divide b .
1 ⁄ 7: 0.142... Vulgar Fraction One Seventh 2150 8528 ⅑ 1 ⁄ 9: 0.111... Vulgar Fraction One Ninth 2151 8529 ⅒ 1 ⁄ 10: 0.1 Vulgar Fraction One Tenth 2152 8530 ⅓ 1 ⁄ 3: 0.333... Vulgar Fraction One Third 2153 8531 ⅔ 2 ⁄ 3: 0.666... Vulgar Fraction Two Thirds 2154 8532 ⅕ 1 ⁄ 5: 0.2 Vulgar Fraction One Fifth 2155 8533 ⅖ 2 ...
Finally, the fractional part, 1 / 7 , is the reciprocal of 7, so its approximation in this scheme, 7, is exact ( 7 / 1 = 7 + 0 / 1 ) and produces the exact expression 4 + 1 / 2 + 1 / 6 + 1 / 7 for 415 / 93 .
The sum of the first k terms of the infinite series provides the closest possible underestimate of 1 by any k-term Egyptian fraction. [2] For example, the first four terms add to 1805/1806, and therefore any Egyptian fraction for a number in the open interval (1805/1806, 1) requires at least five terms.
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, [a] which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.
For instance, the primary pseudoperfect number 1806 is the product of the prime numbers 2, 3, 7, and 43, and gives rise to the Egyptian fraction 1 = 1 / 2 + 1 / 3 + 1 / 7 + 1 / 43 + 1 / 1806 .
Truncating the continued fraction at any point yields a rational approximation for π; the first four of these are 3, 22 / 7 , 333 / 106 , and 355 / 113 . These numbers are among the best-known and most widely used historical approximations of the constant.
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