Decimal-expansion:Terminating and Non-terminating

Let x be a rational number whose decimal expansion terminates. Then we can express x in the form p/q , where p and q are coprime.On division of p by q two main things happen-either the remainder becomes 0 or never becomes zero and we get a repeating string of remainders.

 

Terminating decimal expansion:

 Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2n5m,where n, m are non-negative integers. Then x has a decimal expansion which terminates.

(i) ³⁶/₁₀₀

On dividing 36 by 100, we get

 

Therefore, ³⁶/₁₀₀= 0.36, which is a terminating decimal.

 

  (ii)  ³²⁹/₄₀₀

 

On dividing 329 by 400, we get

While dividing 329 by 400, the remainder is 0.

 

Therefore,³²⁹/₄₀₀=0.8225, which is a terminating decimal.

Non Terminating decimal expansion:

Let x = p/q be a rational number, such that the prime factorisation of q is not of the form 2n5m,where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring)..

(iii)   ³/₁₃

On dividing 3 by 13, we get 0.230769

while dividing 3 by 13 the remainder is 3, which will continue to be 3 after carrying out continuous divisions.

Therefore,0.23076913..... or 0.23076913, which is a non-terminating and recurring decimal.

Therefore, ³/₁₃= 0.23076913, which is a non-terminating decimal.

(iv) ²/₁₁

On dividing 2 by 11, we get

We can observe that while dividing 2 by 11, first the remainder is 2 then 9, which will continue to be 2 and 9 alternately.

Therefore, ²/₁₁=0.1818 .....

or 20.1818 , which is a non-terminating and recurring decimal.

(v) ¹/₁₁

 

On dividing 1 by 11, we get

 

We observe that while dividing 1 by 11, the quotient = 0.09 is repeated.

Therefore, ¹/₁₁= 0.0909..... or ¹/₁₁=0.09, which is a non-terminating and 

 

recurring decimal.