Representation of Real Numbers
Every real number can be expressed as a regular continued fraction in canonical form. Each convergent of that continued fraction is in a sense the best possible rational approximation to that real number, for a given number of digits. Such a convergent is usually about as accurate as a finite decimal expansion having as many digits as the total number of digits in the nth numerator and nth denominator. For example, the third convergent 333/106 for π (Pi) is roughly 3.1415094, which is not quite as accurate as the 6-digit 3.14159; the fourth convergent 355/113 = 3.14159292 is more accurate than the 6-digit decimal.
By the determinant formula it appears that the successive convergents Ak/Bk of a regular continued fraction are connected by the formula
This implies, in particular, that the greatest common divisor (Ak, Bk) = 1; in other words, each convergent of a regular continued fraction, as given by the fundamental recurrence formulas, is automatically expressed in lowest terms.
More detailed properties of best rational approximations and convergents of π are discussed in the continued fraction article.
Read more about this topic: Convergent (continued Fraction)
Famous quotes containing the words representation of, real and/or numbers:
“America loves the representation of its heroes to be not just larger than life, but stupendously, awesomely bigger than anything else. If blue whales built statues to each other they’d be smaller then these.”
—Simon Hoggart (b. 1946)
“It was a real treat when he’d read me Daisy Miller out loud. But we’d reached the point in our relationship when, in a straight choice between him and Henry James, I’d have taken Henry James any day even if Henry James were dead and not much of a one for the girls when living, either.”
—Angela Carter (1940–1992)
“The principle of majority rule is the mildest form in which the force of numbers can be exercised. It is a pacific substitute for civil war in which the opposing armies are counted and the victory is awarded to the larger before any blood is shed. Except in the sacred tests of democracy and in the incantations of the orators, we hardly take the trouble to pretend that the rule of the majority is not at bottom a rule of force.”
—Walter Lippmann (1889–1974)