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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.
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 ...
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 ...
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).
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".
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 ]
Enjoy a classic game of Hearts and watch out for the Queen of Spades!
By their nature, all numbers expressed in floating-point format are rational numbers with a terminating expansion in the relevant base (for example, a terminating decimal expansion in base-10, or a terminating binary expansion in base-2). Irrational numbers, such as π or √2, or non-terminating rational numbers, must be approximated. The ...