FANDOM


repeating or recurring dozenal is dozenal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if and only if its dozenal representation is repeating or terminating (i.e. all except finitely many digits are zero). For example, the dozenal representation of 4/E becomes periodic just after the dozenal point, repeating the single digit "4" forever, i.e. 0.444... A more complicated example is 9453/222, whose dozenal becomes periodic at the second digit following the dozenal point and then repeats the sequence "920" forever, i.e. 43.692092092092...

The infinitely repeated digit sequence is called the repetend or reptend. If the repetend is a zero, this dozenal representation is called a terminating dozenal rather than a repeating dozenal, since the zeros can be omitted and the dozenal terminates before these zeros.[1] Every terminating dozenal representation can be written as a dozenal fraction, a fraction whose divisor is a power of 10 (e.g. 3.497 = 3497/1000); it may also be written as a ratio of the form k/2m3n (e.g. 3.497 = 3497/2633). However, every number with a terminating dozenal representation also trivially has a second, alternative representation as a repeating decimal whose repetend is the digit E. This is obtained by decreasing the final non-zero digit by one and appending a repetend of E. 1.000... = 0.EEE... and 3.497000... = 3.496EEE... are two examples of this. (This type of repeating dozenal can be obtained by long division if one uses a modified form of the usual division algorithm.[2])

Any number that cannot be expressed as a ratio of two integers is said to be irrational. Their dozenal representation neither terminates nor infinitely repeats but extends forever without regular repetition. Examples of such irrational numbers are the square root of 2 and π.