decimal

a fraction whose denominator is a whole power of the number 10. The decimal is written without a denominator, setting off in the numerator to the right of the decimal point as many digits as there are zeros in the denominator (for example, 485,634/1,000 = 485.634 and 3/100 = 0.03). In such notation, the part to the left of the decimal point designates the integer part of the fraction. The first digit after the decimal point designates the number of tenths; the second, the number of hundredths; and so forth.

The decimal notation of rational numbers whose denominator does not have other prime factors except 2 and 5 contains a finite number of digits (for example, 4/25 = 0.16). In general, the digits in the decimal notation of a rational number begin repeating at some position; such a number is an infinite repeating decimal (for example, 7/6 = 1.1666 …). Irrational numbers are nonrepeating infinite decimals (for example, Decimal = 1.41421 . … In all cases, the decimal of a_ka_k-1 … a₀b₁b₂ … can be written in the form

where a_k, a_k-1, … , a₀, b₁b₂, are the numerals 0, 1, 2, … , 9 (a_k ≠ 0) in the corresponding digit of the number. For example, 382.1274 = 3 x 10² + 8 × 10 + 2 + 1/10 + 2/10² + 7/10³ + 4/10⁴, that is, here a₂ = 3, a₁ = 8,a₀ = 2, b₁ = 1, b₂ = 2, b₃ = 7, and b₄ = 4. Decimals were already used in the 14th-15th centuries. The Samarkand mathematician Al Kashi described the decimal system in 1427. In Europe, the decimal was introduced by S. Stevin in 1584.