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He also gave two other approximations of π: π ≈ 22 ⁄ 7 and π ≈ 355 ⁄ 113, which are not as accurate as his decimal result. The latter fraction is the best possible rational approximation of π using fewer than five decimal digits in the numerator and denominator. Zu Chongzhi's results surpass the accuracy reached in Hellenistic ...
Fractions such as 22 / 7 and 355 / 113 are commonly used to approximate π, but no common fraction (ratio of whole numbers) can be its exact value. [21] Because π is irrational, it has an infinite number of digits in its decimal representation, and does not settle into an infinitely repeating pattern of digits.
k is a decimal digit and R is a fraction that must be converted to decimal. It usually has only a single digit in the numerator, and one or two digits in the denominator, so the conversion to decimal can be done mentally. Example: find the square root of 75. 75 = 75 × 10 2 · 0, so a is 75 and n is 0.
An easy mnemonic helps memorize this fraction by writing down each of the first three odd numbers twice: 1 1 3 3 5 5, then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits: 1 1 3 分之(fēn zhī) 3 5 5. (In Eastern Asia, fractions are read by stating the denominator first ...
If N is chosen to be a power of ten, each term in the right sum becomes a finite decimal fraction. The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series.
In some contexts it is desirable to round a given number x to a "neat" fraction – that is, the nearest fraction y = m/n whose numerator m and denominator n do not exceed a given maximum. This problem is fairly distinct from that of rounding a value to a fixed number of decimal or binary digits, or to a multiple of a given unit m.
Approximation theory is a branch of mathematics, and a quantitative part of functional analysis. Diophantine approximation deals with approximations of real numbers by rational numbers. Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain.
Approximation theory. Theory of getting acceptably close inexact mathematical calculations. In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What is meant by best and simpler will depend on the application.