Euclid Geometry:decimal expansion

# Decimal Expansion

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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 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/(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

From decimal expansion to Real numbers