Chapter 1: Real Numbers

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Real Numbers Solutions

Question:

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i)   frac{13}{3125}           (ii)   frac{17}{8}               (iii)  frac{64}{455}          (iv)   frac{15}{1600}        (v)  frac{29}{343}   

 

(vi)   frac{23}{2^35^2}        (vii)   frac{129}{2^25^77^5}       (viii)  frac{6}{15}          (ix)   frac{35}{50}             (x)  frac{77}{210}

Answer:

Solutions: We know that   x = \frac{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 decimal expansion which terminates.

 

And   x= \frac{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 decimal expansion which non-terminating repeating.

 

 

(i)   \frac{13}{3125}

 

If we factorize the denominator, we get
3125 = 5 × 5 × 5 × 5 × 5 = 5^{5}

So, the denominator is in the form of  5^{5} so, โ€‹ \frac{13}{3125}   is terminating decimal expansion.

 

(ii)   \frac{17}{8}  

 

If we factorize the denominator, we get
8 = 2 × 2 × 2 = 2^{3}
So, the denominator is in the form of   2^{3}  so, โ€‹ \frac{17}{8}   is terminating decimal expansion.     

 

 (iii)  \frac{64}{455} 

 

If we factorize the denominator, we get

455 = 5 × 7 × 13  

So, the denominator is not in the form of  2^{n}5^{m} so,โ€‹  \frac{64}{455}    

 is non-terminating repeating decimal expansion.  

 

(iv)   \frac{15}{1600}   

 

If we factorize the denominator, we get

1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 =  2^{6}5^{2}

So, the denominator is in the form of  2^{n}5^{m} . So, \frac{15}{1600}   is terminating decimal expansion.  

 

(v)  frac{29}{343}

 

If we factorize the denominator, we get

343 = 7 × 7 × 7 =  7^{3}   

So, the denominator is not in the form of  2^{n}5^{m} . So,   is non-terminating repeating decimal expansion.  

 

(vi)   frac{23}{2^35^2}

Here, the denominator is in the form of    2^{n}5^{m} . So, frac{23}{2^35^2}   is terminating decimal expansion.

 

(vii)   frac{129}{2^25^77^5}

Here, the denominator is not in the form of   2^{n}5^{m} only. So,  frac{129}{2^25^77^5} is non-terminating repeating decimal expansion.

 

(viii)  frac{6}{15} 

If we divide nominator and denominator both by 3 we get \frac{3}{5}
So, the denominator is in the form of 5^{m}.   So, frac{6}{15}  is terminating decimal expansion.    

 

(ix)    frac{35}{50}    

If we divide nominator and denominator both by 5, we get \frac{7}{10} 
If we factorize the denominator, we get

10 = 2 × 5

So, the denominator is in the form of  2^{n}5^{m} .  So,    frac{35}{50}     is terminating decimal expansion.

 

 (x)    frac{77}{210}

 

If we divide nominator and denominator both by 7, we get 
If we factorize the denominator, we get

30 = 2 × 3 × 5

So, the denominator is not in the form of 2^{n}5^{m} .  So,  frac{77}{210}   is non-terminating repeating decimal expansion.

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