Finite Sums of Unit Fractions
Any positive rational number can be written as the sum of unit fractions, in multiple ways. For example,
The ancient Egyptians used sums of distinct unit fractions in their notation for more general rational numbers, and so such sums are often called Egyptian fractions. There is still interest today in analyzing the methods used by the ancients to choose among the possible representations for a fractional number, and to calculate with such representations. The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős–Graham conjecture and the Erdős–Straus conjecture concern sums of unit fractions, as does the definition of Ore's harmonic numbers.
In geometric group theory, triangle groups are classified into Euclidean, spherical, and hyperbolic cases according to whether an associated sum of unit fractions is equal to one, greater than one, or less than one respectively.
Read more about this topic: Unit Fraction
Famous quotes containing the words finite, sums and/or unit:
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)
“If God lived on earth, people would break his windows.”
—Jewish proverb, quoted in Claud Cockburn, Cockburn Sums Up, epigraph (1981)
“During the Suffragette revolt of 1913 I ... [urged] that what was needed was not the vote, but a constitutional amendment enacting that all representative bodies shall consist of women and men in equal numbers, whether elected or nominated or coopted or registered or picked up in the street like a coroner’s jury. In the case of elected bodies the only way of effecting this is by the Coupled Vote. The representative unit must not be a man or a woman but a man and a woman.”
—George Bernard Shaw (1856–1950)