# Rational Numbers

## NCERT Exercise 1.1 (Part 2)

Question 3: Verify that `-(-x)=x` for:

(i) `x=(11)/(15)`

**Solution:** Given, `x=(11)/(15)`

The additive inverse of `x=(11)/(15)` is `-x=(-11)/(15)`

Similarly, the additive inverse of `(-11)/(15)` is `(11)/(15)`

Or, `-((-11)/(15))=(11)/(15)`

Or, `-(-x)=x` proved

(ii) `x=-(13)/(17)`

**Solution:** Given, `x=-(13)/(17)`

The additive inverse of `x=-(13)/(17)` is `-x=(13)/(17)`

Similarly, the additive inverse of `(13)/(17)` is `-(13)/(17)`

Or, `-(13)/(17)+(13)/(17)=0`

Or, `-(-x)=x` proved

Question 4: Find the multiplicative inverse of the following

(i) `-13`

**Solution:** We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of `-13` is equal to `(1)/(-13)`

(ii) `(-13)/(19)`

**Solution:** We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of `(-13)/(19)` is equal to `(19)/(-13)`

(iii) `1/5`

**Solution:** We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of `1/5` is equal to 5

(iv) `-5/8xx(-3)/(7)`

**Solution:** Given, `-5/8xx(-3)/(7)`

`=((-5)xx(-3))/(8xx7)=(15)/(56)`

We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of `(15)/(56)` is equal to `(56)/(15)`

(v) `-1xx(-2)/(5)`

**Solution:** Given, `-1xx(-2)/(5)=2/5`

We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of `2/5` is equal to `5/2`

(vi) `-1`

**Solution:** We know that multiplicative inverse of a number is reciprocal of the number.

Thus, multiplicative inverse of `-1` is equal to `(1)/(-1)` or `-1`

**Alternate Method:**

The product of a number and its multiplicative inverse is equal to 1

Here, `-1xx-1=1`

Thus, multiplicative inverse of `-1` is `-1`

Question 5: Name the property under multiplication used in each of the following.

(i) `(-4)/(5)xx1=1xx(-4)/(5)=-4/5`

**Solution:** Here, 1 is the multiplicative identity.

Thus, property of multiplicative identity is used.

(ii) `-(13)/(17)xx(-2)/(7)=(-2)/(7)xx(-13)/(17)`

**Solution:** Here, multiplicative commutativity is used.

(iii) `(-19)/(29)xx(29)/(-19)=1`

**Solution:** Since, the product of given numbers is 1, so `(29)/(-19)` is the multiplicative inverse of `(-19)/(29)`

Thus, property of multiplicative inverse is used.

Question 6: Multiply `(6)/(13)` by the reciprocal of (-7)/(16)`

**Solution:** Reciprocal of `(-7)/(16)` is `(16)/(-7)`

So, `(6)/(13)xx(16)/(-7)`

`=(6xx16)/(13xx(-7))=(96)/(-91)`

Question 7: Tell what property allows you to compute `1/3xx(6xx4/3)` as `(1/3xx6)xx4/3`

**Solution:** The property of associativity

Question 8: Is `8/9` the multiplicative inverse of `-1\1/8`? Why or why not?

**Solution:** `-1\1/8=-7/8`

Since, `8/9xx(-7)/(8)=-7/9≠1`

So, `-1\1/8` is not the multiplicative inverse of `8/9`

Question 9: Is 0.3 the multiplicative inverse of `3\1/3` Why or why not?

**Solution:** `0.3=(3)/(10)`

The multiplicative inverse of `(3)/(10)` is `(10)/(3)=3\1/3`

Thus, `3\1/3` is the multiplicative inverse of 0.3.

Question 10: Write

(i) The rational number that does not have a reciprocal.

**Solution:** 0 (zero) is the rational number which does not have a reciprocal.

(ii) The rational numbers that are equal to their reciprocals.

**Solution:** 1 and – 1 are the rational numbers which are equal to their reciprocals.

(iii) The rational number that is equal to its negative.

**Solution:** 0 (zero) is the rational number which is equal to its negative.

Question 11: Fill in the blanks:

(i) Zero has __________ reciprocal.

**Solution:** no

(ii) The numbers ________ and ________ are their own reciprocals.

**Solution:** 1 and – 1

(iii) The reciprocal of – 5 is _____________.

**Solution:** `(1)/(-5)`

(iv) Reciprocal of `1/x` where `x≠0` is ______________.

**Solution:** `x`

(v) The product of two rational numbers is always a _____________.

**Solution:** rational number

(vi) The reciprocal of a positive rational number is ____________

**Solution:** positive