Exercise 1.4 Class 10 Maths

Real Numbers

Class 10 Maths Exercise 1.4 Question 1

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

(i) \cfrac {13} {3125}     (ii) \cfrac {17} {8}    (iii) \cfrac {64} {455}    (iv) \cfrac {15} {1600}

(v) \cfrac {29} {343}    (vi) \cfrac {23} {2³×5²}  

 (vii) \cfrac {129} {2²× 5^7 ×7^5}

(viii) \cfrac {6} {15}   (ix) \cfrac {35} {50}   (x) \cfrac {77} {210}

Solutions: (i) \cfrac {13} {3125}  

Here, Denominator = 3125 = 5 × 5 × 5 × 5 × 5

So, The denominator consists prime factor = 5 , which is in the form 2^n × 5^m , where n = 0 , m = 5

So, \cfrac {13} {3125} has terminating decimal expansion.

(ii) \cfrac {17} {8}  

Here, Denominator = 8 = 2 × 2  × 2

Which is in the form 2^n × 5^m , where n = 3 , m = 0

So, \cfrac {17} {8}   has a terminating decimal expansion.

(iii) \cfrac {64} {455}

Here, Denominator = 455 = 5 × 7 × 13

Means, In addition to 5, 7 & 13 are also two factor of the denominator of the given fraction. So, The given fraction will have a non-terminating decimal fraction.

(iv) \cfrac {15} {1600}

Here Denominator = 1600 = 2 × 2 × 2  × 2 × 2 × 2 × 5 × 5

So, The denominator consists prime factor = 5 & 2 only , which is in the form 2^n × 5^m , where n = 6 , m = 2 . Decimal expansion is ternimating.

(v) \cfrac {29} {343}

Denominator = 343 = 7 × 7 × 7

So, The denominator consists prime factor = 7 only , which is’t in the form 2^n × 5^m . The decimal expansion of the given fraction will be non terminating.

SEE ALL THE HINTS IN BELLOW PORTION

(vi) \cfrac {23} {2³×5²}

Here clearly, The denominator consists prime factor = 5 & 2 only , which is in the form 2^n × 5^m , where n = 3 , m = 2 . Decimal expansion is ternimating.

(vii) \cfrac {129} {2²× 5^7 ×7^5}

Here clearly, The denominator consists prime factor = 2, 5 and also 7 , which is’t in the form 2^n × 5^m only. So, The decimal expansion of the given fraction will be non terminating.

(viii) \cfrac {6} {15} = \frac 25

So, The denominator consists prime factor = 5 , which is in the form 2^n × 5^m , where n = 0 , m = 1 . So, the given fraction has a terminating decimal expansion.

(ix) \cfrac {35} {50}

Denominator = 50 = 2 × 5 × 5 

So, The denominator consists prime factor = 2 & 5 , which is in the form 2^n × 5^m , where n = 1 , m = 2 . So, the given fraction has a terminating decimal expansion.

(x) \cfrac {77} {210} = \cfrac {11} {30}

Denominator = 30 = 2 × 3 × 5

So, The denominator consists prime factor = 2, 5 but 3 also , which is’t in the form 2^n × 5^m . The decimal expansion of the given fraction will be non terminating.

Class 10 Maths Exercise 1.4 Question 3

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If, they are rational, and of the form \frac p q , what can you say about the prime factors of q?

(i) 43.123456789

(ii) 0.1201120012000120000…

(iii) 43.\overline{123456789}

Solutions: (i) For this number, we can count the total number of digits after the decimal which is = 9 . That means, this number can be expressed in the form \frac p q easily, where q ≠ 0 . So, Clearly, this decimal number is a rational number.

(ii) For this number–

1. we can’t count the total number of digits after the decimal.
2. digits after the decimal are not repeating uniformly.

That means, this number can’t be expressed in the form \frac p q easily, where q ≠ 0 . So, Clearly, this decimal number is a irrational number.

(iii) For this number–

1. we can’t count the total number of digits after the decimal.
2. But, digits after the decimal are repeating uniformly ( uperline means repeating).

That means, this number can be expressed in the form \frac p q easily, where q ≠ 0 . So, Clearly, this decimal number is a rational number.