Search results
Results from the WOW.Com Content Network
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.
Conventionally, the decimal representation without trailing 9's is preferred. Moreover, in the standard decimal representation of , an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself if is an integer.
In this context, the usual decimals, with a finite number of non-zero digits after the decimal separator, are sometimes called terminating decimals. A repeating decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., 5.123144144144144... = 5.123 144 ). [ 4 ]
Every nonzero terminating decimal has two equal representations (for example, 8.32000... and 8.31999...). Having values with multiple representations is a feature of all positional numeral systems that represent the real numbers.
Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics. Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic), and in many ...
For example, to convert 146 in decimal to negaternary: ... terminating representations correspond to fractions where the denominator is a power of the base; repeating ...
A decimal separator is a symbol that separates the integer part from the fractional part of a number written in decimal form. Different countries officially designate ...
Conversely any recurring (or terminating) base-φ expansion is a non-negative element of Q[√ 5]. For recurring decimals, the recurring part has been overlined: 1 / 2 = 0. 010 φ 1 / 3 = 0. 00101000 φ 1 / 4 = 0. 001000 φ 1 / 5 = 0. 001001010100100100 φ 1 / 10 = 0 ...