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 = 125/103 = 53/(23 x 53) = 1/23 2) 0.00134 = 134/100000 = 134/105 = (2 x 67)/(25 x 55 ) = 67/(24 x 55 )
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 2n5m, 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 2n5m, 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 2n5m, 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/(23 x 50)
As the denominator is of the form 2n5m 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 2n5m. 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

From decimal expansion to Real numbers

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