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:
“All great amusements are dangerous to the Christian life; but among all those which the world has invented there is none more to be feared than the theater. It is a representation of the passions so natural and so delicate that it excites them and gives birth to them in our hearts, and, above all, to that of love.”
—Blaise Pascal (1623–1662)
“There is a quality even meaner than outright ugliness or disorder, and this meaner quality is the dishonest mask of pretended order, achieved by ignoring or suppressing the real order that is struggling to exist and to be served.”
—Jane Jacobs (b. 1916)
“And when all bodies meet
In Lethe to be drowned,
Then only numbers sweet
With endless life are crowned.”
—Robert Herrick (1591–1674)