Simple continued fraction in the context of Numerators and denominators

In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.

The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number p/q is a "good" approximation of a real number α if the absolute value of the difference between p/q and α may not decrease if p/q is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of simple continued fractions.

Mathematics emerged independently in China by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system (binary and decimal), algebra, geometry, number theory and trigonometry.

Since the Han dynasty, as diophantine approximation being a prominent numerical method, the Chinese made substantial progress on polynomial evaluation. Algorithms like regula falsi and expressions like simple continued fractions are widely used and have been well-documented ever since. They deliberately find the principal nth root of positive numbers and the roots of equations. The major texts from the period, The Nine Chapters on the Mathematical Art and the Book on Numbers and Computation gave detailed processes for solving various mathematical problems in daily life. All procedures were computed using a counting board in both texts, and they included inverse elements as well as Euclidean divisions. The texts provide procedures similar to that of Gaussian elimination and Horner's method for linear algebra. The achievement of Chinese algebra reached a zenith in the 13th century during the Yuan dynasty with the development of tian yuan shu.

A continued fraction is a mathematical expression written as a fraction whose denominator contains a sum involving another fraction, which may itself be a simple or a continued fraction. If this iteration (repetitive process) terminates with a simple fraction, the result is a finite continued fraction; if it continues indefinitely, the result is an infinite continued fraction. The special case in which all numerators are equal to one is referred to as a simple (or regular) continued fraction. Any rational number can be expressed as a finite simple continued fraction, and any irrational number can be expressed as an infinite simple continued fraction.

Different areas of mathematics use different terminology and notation for continued fractions. In number theory, the unqualified term continued fraction usually refers to simple continued fractions, whereas the general case is referred to as generalized continued fractions. In complex analysis and numerical analysis, the general case is usually referred to by the unqualified term continued fraction.