# class10 NCRT Real Numbers Exercise – 1.4 download pdf || UP Board

__Real Numbers__

__Exercise – 1.4__

__Real Numbers__

__Exercise – 1.4__

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

**(****i****)13/3125**

**Solution****:**

3125 = 5×5×5×5×5 = 5^{5}

We know that in a rational number p/q, q is written as
a factor of 2^{m}×5^{n} (where m and n are non-negative
integers) then its decimal expansion is terminating.

The prime factor of 3125 = 5^{5}

So this factor can be written as 2^{m}×5^{n} (2^{0}×5^{5}) so the decimal expansion will be terminating.

**(****ii****)17/8**

**Solution****:**

8 = 2×2×2 = 2^{3}

We know that in a rational number p/q, q is written as
a factor of 2^{m}×5^{n} (where m and n are non-negative
integers) then its decimal expansion is terminating.

The prime factor of 8 = 2^{3}

So this factor can be written as 2^{m}×5^{n} (2^{3}×5^{0}) so the decimal expansion will be terminating.

**(****iii****)****64/455**

**Solution****:**

455 = 4×7×13

We know that in a rational number p/q, q is written as
a factor of 2^{m}×5^{n} (where m and n are non-negative
integers) then its decimal expansion is terminating.

The prime factor of 455 = 4×7×13

So this factor cannot be written as 2^{m}×5^{n} so the decimal expansion will be
of non-terminating repeating.

**(****iv****)****15/1600**

**Solution****:**

1600 = 2×2×2×2×2×5×5 = 2^{6}×5^{2}

^{m}×5^{n} (where m and n are non-negative
integers) then its decimal expansion is terminating.

The prime factor of 1600 = 2^{6}×5^{2}

So this factor can be written as 2^{m}×5^{n} (2^{6}×5^{2}) so the decimal expansion will be terminating.

**(****v****)29/343**

**Solution****:**

343 = 7×7×7

^{m}×5^{n} (where m and n are non-negative
integers) then its decimal expansion is terminating.

The prime factor of 343 = 7^{3}

So this factor cannot be written as 2^{m}×5^{n} so the decimal expansion will be
of non-terminating repeating.

**Solution****:**

Denominator = 2^{3 }×5^{2}

^{m}×5^{n} (where m and n are non-negative
integers) then its decimal expansion is terminating.

The prime factor of denominator= 2^{3 }×5^{2}

So this factor can be written as 2^{m}×5^{n} (2^{3 }×5^{2}) so the decimal expansion
will be terminating.

**Solution****:**

Denominator = 2^{2}×5^{7}×7^{5}

We know that in a rational number p/q, q is written as
a factor of 2^{m}×5^{n} (where m and n are non-negative
integers) then its decimal expansion is terminating.^{}

The prime factor of denominator = 2^{2}×5^{7}×7^{5}

So this factor cannot be written as 2^{m}×5^{n} so the decimal expansion will be
of non-terminating repeating.

**(****viii****)6/15**

**Solution****:**

Denominator = 5^{}

^{m}×5^{n} (where m and n are non-negative
integers) then its decimal expansion is terminating.

The prime factor of denominator = 5

So this factor can be written as 2^{m}×5^{n} (2^{0 }×5^{1}) so the decimal expansion will be terminating.

**(****ix****)35/50**

**Solution****:**

Denominator = 10 = 2×5^{}

^{m}×5^{n} (where m and n are non-negative
integers) then its decimal expansion is terminating.

The prime factor of denominator = 2×5

So this factor can be written as 2^{m}×5^{n} (2^{1 }×5^{1}) so the decimal expansion will be terminating.

**(****x****)77/210**

**Solution****:**

Denominator = 2×3×5^{}

^{m}×5^{n} (where m and n are non-negative
integers) then its decimal expansion is terminating.

The prime factor of denominator = 2×3×5^{}

^{m}×5^{n} so the decimal expansion will be
of non-terminating repeating.

**2. Write
down the decimal expansions of those rational numbers in Question 1 above which
have terminating decimal expansions.**

** **

**(****i****)13/3125**

**Solution****:**

= 0.00416

**(****ii****)17/8**

**Solution****:**

= 2.125

**(****iii****)64/455**

**Solution****:**

Decimal expansion is non-termination repeating.

**(****iv****)15/1600**

**Solution****:**

= 0.009375

**(****v****)29/343**

**Solution****:**

Decimal
expansion is non-termination repeating.

**Solution****:**

= 0.115

**Solution****:**

Decimal expansion is non-termination repeating.

**(****viii****)6/15**

**Solution****:**

= 0.4

**(****ix****)35/50**

**Solution****:**

= 0.7

**(****x****)77/210**

**Solution****:**

Decimal
expansion is non-termination repeating.

**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 P/q, what can you say about the prime factor of q?**

**(i) 43.123456789**

**Solution****:**

The decimal expansion is terminating so it is a
rational number and it is of the form p/q. The prime factorization of q is of
the form 2^{m}×5^{n}

**(ii) 0.120120012000120000…**

**Solution****:**

The decimal expansion is non-terminating and non-recurring so it is an irrational number.

**Solution****:**

The decimal expansion is non-terminating and repeating,
so it is a rational number and it is of the form p/q. The prime factorization
of q is any prime number other than 2^{m}×5^{n}

^{class10 NCRT Real Numbers Exercise – 1.4 download pdf}

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