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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.
With base e the natural logarithm behaves like the common logarithm in base 10, as ln(1 e) = 0, ln(10 e) = 1, ln(100 e) = 2 and ln(1000 e) = 3 (or more precisely the representation in base e of 3, which is of course a non-terminating number).
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 ]
Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating. Finite decimal representations can also be seen as a special case of infinite repeating decimal representations. For example, 36 ⁄ 25 = 1.44 = 1.4400000...; the endlessly repeated sequence is the one-digit sequence "0".
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 ...
The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872. [32] The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work A comprehensive textbook of classical mathematics: A contemporary interpretation. [37]
Fractions: A representation of a non-integer as a ratio of two integers. These include improper fractions as well as mixed numbers . Continued fraction : An expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of ...
Every non-negative rational number can be represented as a recurring base-φ expansion, as can any non-negative element of the field Q[√ 5] = Q + √ 5 Q, the field generated by the rational numbers and . Conversely any recurring (or terminating) base-φ expansion is a non-negative element of Q[√ 5]. For recurring decimals, the recurring ...