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