# Decimal Expansion

The decimal expansion of a rational number is either terminating or non-terminating repeating (or recurring) without knowing when it is terminating and when it is non-terminating repeating. In this section, we will explore exactly when the expansion of decimal of a rational number is terminating and when it is non-terminating repeating.1) 0.125 = 125/1000 = 125/10

^{3}= 5

^{3}/(2

^{3}x 5

^{3}) = 1/2

^{3}2) 0.00134 = 134/100000 = 134/10

^{5}= (2 x 67)/(2

^{5}x 5

^{5}) = 67/(2

^{4}x 5

^{5})

3) 0.7 = 7/10 = 7/(2 x 5)

**Note : As we know that 2 and 5 are the only prime factors of 10.**

**1) Theorem 1: Let x be a rational number whose expansion of decimal terminates. Then x can be expressed in the form p/q, where p and q are co-prime, and the prime factorization of q is of the form 2**

2) Theorem 2 : Let x = p/q be a rational number, such that the prime factorization of q is of the form 2

3) Let x =p/q be a rational number, such that the prime factorization of q is not of the form 2

Examples :

^{n}5^{m}, where n, m are non-negative integers.2) Theorem 2 : Let x = p/q be a rational number, such that the prime factorization of q is of the form 2

^{n}5^{m}, where n, m are non-negative integers. Then x has a expansion of decimal which terminates.3) Let x =p/q be a rational number, such that the prime factorization of q is not of the form 2

^{n}5^{m}, where n, m are non-negative integers. Then, x has a expansion of decimal which is non-terminating repeating (recurring).Examples :

1) State whether the following rational numbers will have terminating decimal-expansion or a non-terminating repeating expansion of decimal.

a) 17/8

**Solution :**

17/8 = 17/(2

^{3}x 5

^{0})

As the denominator is of the form 2

^{n}5

^{m}so the expansion of decimal of 17/8 is terminating.

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b) 64/255

**solution :**

64/255 = 64/ (5 x 3 x 17)

Clearly, 255 is not of the form 2

^{n}5

^{m}. So, the expansion of decimal of 64/255 is non-terminating repeating.

**Euclid's Geometry**

• Euclid Geometry

• Euclids division lemma

• Euclids division Algorithm

• Fundamental Theorem of Arithmetic

• Finding HCF LCM of positive integers

• Proving Irrationality of Numbers

• Decimal expansion of Rational numbers

• Euclid Geometry

• Euclids division lemma

• Euclids division Algorithm

• Fundamental Theorem of Arithmetic

• Finding HCF LCM of positive integers

• Proving Irrationality of Numbers

• Decimal expansion of Rational numbers

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