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Decimal Expansion of Rational Numbers By Arogya Singh Seema KC Prashant Rajbhandari

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Rational Number A rational number is any number that can be expressed as the ratio of two integers p/q, where q≠0. Some examples of rational numbers are: Rational numbers that can be expressed in a decimal form either terminates or repeats. For Example: 3/5 = 0.6 (Terminate) 2/3 = 0.6666 (Repeat)

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Conditions (Terminate) The conditions for a rational numbers to terminate is: A fraction (in simplest form/lowest terms) terminates in its decimal form, if the prime factors of the denominator are only 2’s and 5’s or a product of prime factors of 2’s and 5’s. The product of prime factors can also be expressed as (Q = P1n1*P2n2………..……………Pknk). The decimal expansion of an irrational number never terminates.

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Conditions (Cont…..) TerminateRepeat, Where 2 in the denominator is the prime factor of (2*1). Where 3 in the denominator is not the prime factor of 2’s & 5’s., Where 5 in the denominator is the prime factor of (5*1). Where 15 in the denominator is not the prime factor of 2’s & 5’s., Where 5 in the denominator is the prime factor of (2*5). Where 21 in the denominator is not the prime factor of 2’s & 5’s., Where 5 in the denominator is the prime factor of (2*2*5). Where 30 in the denominator is not the prime factor of 2’s & 5’s., Where 5 in the denominator is the prime factor of (5*5). Where 66 in the denominator is not the prime factor of 2’s & 5’s.

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Condition (Repeat) The conditions for a rational numbers to repeat is: A fraction (in simplest form/lowest terms) repeats in its decimal form, if the prime factors of the denominator are not 2’s and 5’s or a product of prime factors, 2’s and 5’s. The decimal expansion of an irrational number does not repeat. Some real numbers cannot be expressed by fractions. These numbers are called irrational numbers. For example: = 1.414213562 = 3.141592653 r = 0.10110111011110

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Conjecture Since p/q = p(1/q), it is sufficient to investigate the decimal expansions of 1/q. Our conjecture or assumption is that the decimal expansion of 1/q for enough positive integers can either be terminates or repeats. As it has already been describe above that a fraction (in simplest form/lowest terms) terminates in its decimal form if the prime factors of the denominator are only 2’s and 5’s or a product of primes factors, 2’s and 5’s. Otherwise it repeats. The product of prime factors can also be expressed in the equation of; Q = P1n1*P2n2………..……………Pknk

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Conjecture (Cont….) 1/2 = 0.5Terminate 1/3 = 0.Repeat 1/4 = 0.25Terminate 1/5 = 0.20Terminate 1/6 = 0.1Repeat 1/7 = 0.Repeat 1/8 = 0.125Terminate 1/9 = 0.Repeat 1/10 = 0.1Terminate

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Conjecture ( Cont…..) If we take 100 as a denominator it terminates because 100 is the product of prime factors of 2*2*5*5. Similarly, if we take 66 it repeats because 66 is the product of prime factors of 2*3*11. Therefore any number that is in the denominator and is the combination of 2’s and 5’s is terminated. The denominators of the first few unit fractions having repeating decimals are 3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, and 29.

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That’s all folks

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