What is decimal expansion

When a rational number with is expanded, the period begins after terms and has length , where and are the smallest numbers satisfying 3. If is a Prime and 10 is a Primitive Root of , then the period of the repeating decimal is given by 5. The first few numbers with maximal decimal expansions, called Full Reptend Primes , are 7, 17, 19, 23, 29, 47, 59, 61, 97, , , , , , Sloane's A The decimals corresponding to these are called Cyclic Numbers.

No general method is known for finding Full Reptend Primes. Such numbers are called full reptend primes. The decimal period of a fraction with denominator equal to a prime factor above is therefore the power of 10 in which the factor first appears.

For example, 37 appears in the factorization of and , so its period is 3. Multiplication of any factor by a still gives the same period as the factor alone. A denominator obtained by a multiplication of two factors has a period equal to the first power of 10 in which both factors appear.

A table of the periods of small primes other than the special , for which the decimal expansion is not periodic, follows OEIS A Shanks ab computed the periods for all primes up to and published those up to. Portions of this entry contributed by Christopher Stover. Conway, J.

New York: Springer-Verlag, pp. Das, R. Monthly 56 , , Dickson, L. History of the Theory of Numbers, Vol. New York: Dover, pp.

Glaisher, J. Cambridge Philos. Havil, J. Gamma: Exploring Euler's Constant. Lehmer, D. Ogilvy, C. Excursions in Number Theory. New York: Dover, p. Rademacher, H. Rao, K. Monthly 62 , , Shanks, W. London 22 , , a.

All those who say programming isn't for kids, just haven't met the right mentors yet. The students will get to learn more about the world of programming in these free classes which will definitely help them in making a wise career choice in the future. Decimal Expansion Before going into a representation of the decimal expansion of rational numbers, let us understand what rational numbers are.

Rational numbers are denoted by Q. That means the number comes to end after a decimal point after a certain number of repetitions. Example: 0. Non-Terminating Decimals Non-Terminating decimals are those decimal numbers that have an infinite number of digits.

Example: 1. Repeating Decimals: Repeating decimals are those numbers in which a specific number repeats uniformly after a decimal point. Note: Non terminating and repeating decimals are rational numbers. Example: 4. Note: Non terminating and nonrepeating decimals are irrational numbers. Solution: Here 45 is nonrepeating and 23 is repeating. So in the numerator, we subtract the nonrepeating term 45 from the number and as we have two terms as repeating 2, 3.