WORKSHEET ON PROPERTIES OF RATIONAL NUMBERS

Question 1 :

Is addition of two rational numbers closure under addition?

Question 2 :

Is subtraction of two rational numbers closure under addition?

Question 3 :

Is multiplication of two rational numbers closure under multiplication?

Question 4 :

Is division of two rational numbers closure under multiplication?

Question 5 :

Verify the commutative property for addition and multiplication of the rational numbers -⁵⁄₇ and ⁸⁄₉.

Question 6 :

Verify the commutative property for multiplication of the rational numbers ⁶⁄₇ and ⅘.

Question 7 :

Is subtraction of two rational numbers commutative?

Question 8 :

Is division of two rational numbers commutative?

Question 9 :

Verify the associative property for addition of the following rational numbers.

-¹⁰⁄₁₁, ⅚, -⁴⁄₃

Question 10 :

Verify the associative property for multiplication of the following rational numbers.

-¹⁰⁄₁₁, ⅚, -⁴⁄₃

Question 11 :

Is subtraction of rational numbers associative?

Question 12 :

Is division of rational numbers associative?

Question 13 :

Is multiplication of rational numbers distributive over addition?

Question 14 :

Is multiplication of rational numbers distributive over subtraction?

Question 15 :

What will you get when you multiply a rational number by 1 ? Explain your answer. 

Question 16 :

Find the additive inverse of -⅔.

Question 17 :

Find the multiplicative inverse of ⅔.

Question 18 :

Find the multiplicative inverse of 2⅓.

Question 19 :

Find the multiplicative inverse of zero. 

Question 20 :

Find the additive inverse of zero. 

Question 21 :

Find the multiplicative inverse of 1.

Question 22 :

Find the additive inverse of 1.

Question 23 :

Find the multiplicative inverse of ∞.

Answers

1. Answer :

Addition two rational numbers is always a rational number. Thus, set of rational numbers is closed under addition.

If ᵃ⁄b and ᶜ⁄d are any two rational numbers, then

(ᵃ⁄b + ᶜ⁄d) is also a rational number

Example : 

²⁄₉ + ⁴⁄₉ = ⁶⁄₉

= ⅔

⅔ is a rational number

2. Answer :

Subtraction of two rational numbers is always a rational number. Thus, set of rational numbers is closed under subtraction.

If ᵃ⁄b and ᶜ⁄d are any two rational numbers, then

(ᵃ⁄b - ᶜ⁄d) is also a rational number. 

Example : 

⁵⁄₉ - ²⁄₉ = ³⁄₉

= ⅓

⅓ is a rational number.

3. Answer :

Multiplication of two rational numbers is always a rational number. Thus, set of rational numbers is closed under multiplication.

If ᵃ⁄b and ᶜ⁄d are any two rational numbers, then

ᵃ⁄b x ᶜ⁄d  = ᵃᶜ⁄bd

ᵃᶜ⁄bd is a rational number.

Example : 

⁵⁄₉ x ²⁄₉ = ¹⁰⁄₈₁

¹⁰⁄₈₁ is a rational number.

4. Answer :

The collection of all non-zero rational numbers is closed under division.

If ᵃ⁄b and ᶜ⁄d are two rational numbers, such that b ≠ 0 and c/d ≠ 0, then (ᵃ⁄b ÷ ᶜ⁄d) is always a rational number. 

Example : 

⅔ ÷ ⅓ = ⅔ x ³⁄₁ 

= ²⁄₁

²⁄₁ is a rational number.

5. Answer :

Let a = -⁵⁄₇  and b = ⁸⁄₉.

Commutative Property for addition :

a + b = b + a

a + b = -⁵⁄₇ + ⁸⁄₉

Least common multiple of the denominators 7 and 9 is 63. So, multiply the fractions by appropriate numbers to make the denominator for both the fractions as 63. 

= -⁽⁵ˣ⁹⁾⁄₍₇ₓ₉₎ + ⁽⁸ˣ⁷⁾⁄₍₉ₓ₇₎

= ⁴⁵⁄₆₃ + ⁵⁶⁄₆₃

= ⁽⁻⁴⁵ ⁺ ⁵⁶⁾⁄₆₃

= ¹¹⁄₆₃ ----(1)

b + a = ⁸⁄₉ + (-⁵⁄₇)

= ⁸⁄₉ - ⁵⁄₇

= ⁵⁶⁄₆₃ - ⁴⁵⁄₆₃

= ⁽⁵⁶ ⁻ ⁴⁵⁾⁄₆₃

= ¹¹⁄₆₃ ----(2)

From (1) and (2)

a + b = b + a

-⁵⁄₇ + ⁸⁄₉ = ⁸⁄₉ + (-⁵⁄₇)

6. Answer :

Let a = ⁶⁄₇ and b = ⅘.

Commutative Property for multiplication :

a x b = b x a

a x b = ⁶⁄₇ x ⅘

= ⁽⁶ˣ⁴⁾⁄₍₇ₓ₅₎

= ²⁴⁄₃₅ ----(1)

b x a = ⅘ x ⁶⁄₇

= ⁽⁴ˣ⁶⁾⁄₍₅ₓ₇₎

= ²⁴⁄₃₅ ----(2)

From (1) and (2)

a x b = b x a

⁶⁄₇ x ⅘ = ⅘ x ⁶⁄₇

7. Answer :

Subtraction of two rational numbers is NOT commutative.

If ᵃ⁄b and ᶜ⁄d are any two rational numbers, then

ᵃ⁄b - ᶜ⁄d ≠ ᶜ⁄d - ᵃ⁄b

Example : 

⁵⁄₉ - ²⁄₉ = ³⁄₉ = ⅑

²⁄₉ - ⁵⁄₉ = ⁻³⁄₉ = ⁻¹⁄₉

And,

⁵⁄₉ - ²⁄₉ ≠ ²⁄₉ - ⁵⁄₉

Therefore, Commutative property is NOT true for subtraction of two rational numbers.

8. Answer :

Division of two rational numbers is NOT commutative.

If ᵃ⁄b and ᶜ⁄d are two rational numbers, then

ᵃ⁄b ÷ ᶜ⁄d ≠ ᶜ⁄d ÷ ᵃ⁄b

Example :

⅔ ÷ ⅓ = ⅔ x ³⁄₁ = ²⁄₁

⅓ ÷ ⅔ = ⅓ ÷ ³⁄₂ = ½

And, 

⅔ ÷ ⅓ ≠ ⅓ ÷ ⅔

Therefore, Commutative property is NOT true for division two rational numbers.

9. Answer :

Addition is associative for rational numbers.

That is, for any three rational numbers a, b and c,

(a + b) + c = a + (b + c)

Let a = -¹⁰⁄₁₁, b = ⅚ and c = -⁴⁄₃.

a + b = -¹⁰⁄₁₁ + ⅚

Least common multiple of the denominators 11 and 6 is 66. So, multiply the fractions by appropriate numbers to make the denominator for both the fractions as 66. 

= -⁶⁰⁄₆₆ + ⁵⁵⁄₆₆

= ⁽⁻⁶⁰ ⁺ ⁵⁵⁾⁄₆₆

= -⁵⁄₆₆

(a + b) + c = -⁵⁄₆₆ + (-⁴⁄₃)

= -⁵⁄₆₆ - ⁽⁴ˣ²²⁾⁄₍₃ₓ₂₂₎

= -⁵⁄₆₆ - ⁸⁸⁄₆₆

= ⁽⁻⁵ ⁻ ⁸⁸⁾⁄₆₆

= -⁹³⁄₆₆

= -³¹⁄₂₂ ----(1)

b + c = ⅚ + (-⁴⁄₃)

= ⅚ - ⁴⁄₃

= ⅚ - ⁽⁴ˣ²⁾⁄₍₃ₓ₂₎

= ⅚ - ⁸⁄₆

= ⁽⁵ ⁻ ⁸⁾⁄₆

= -³⁄₆

= -½

a + (b + c) = -¹⁰⁄₁₁ + (-½)

= -¹⁰⁄₁₁ - ½

= -⁽¹⁰ˣ²⁾⁄₍₁₁ₓ₂₎ - ⁽¹ˣ¹¹⁾⁄₍₂ₓ₁₁₎

= -²⁰⁄₂₂ - ¹¹⁄₂₂

= ⁽⁻²⁰ ⁻ ¹¹⁾⁄₆₆

= -³¹⁄₆₆ ----(2)

From (1) and (2),

(a + b) + c = a + (b + c)

Hence, Associative Property is true for addition of rational numbers.

10. Answer :

Multiplication is associative for rational numbers.

That is, for any three rational numbers a, b and c,

(a x b) x c = a x (b x c)

Let a = -¹⁰⁄₁₁, b = ⅚ and c = -⁴⁄₃.

a x b = -¹⁰⁄₁₁ x ⅚

= ⁽⁻¹⁰ ˣ ⁵⁾⁄₍₁₁ ₓ ₆₎

= -⁵⁰⁄₆₆

= -²⁵⁄₃₃

(a x b) x c = (-²⁵⁄₃₃) x (-⁴⁄₃)

= ⁽⁻²⁵ˣ⁻⁴⁾⁄₍₃₃ₓ₃₎

= ¹⁰⁰⁄₉₉ ----(1)

b x c = ⅚ x (-⁴⁄₃)

= ⁽⁵ˣ⁻⁴⁾⁄₍₆ₓ₃₎

= -²⁰⁄₁₈

= -¹⁰⁄₉

a x (b x c) = -¹⁰⁄₁₁ x -¹⁰⁄₉

= ⁽⁻¹⁰ ˣ ⁻¹⁰⁾⁄₍₁₁ ₓ ₉₎

= ¹⁰⁰⁄₉₉ ----(2)

From (1) and (2),

(a x b) x c = a x (b x c)

Hence, Associative Property is true for multiplication of rational numbers.

11. Answer :

Subtraction of rational numbers is NOT associative.

If ᵃ⁄b, ᶜ⁄d and ᵉ⁄f are any three rational numbers, then

ᵃ⁄b - (ᶜ⁄d - ᵉ⁄f) ≠ (ᵃ⁄b - ᶜ⁄d) - ᵉ⁄f

Example :

²⁄₉ - (⁴⁄₉ - ⅑) = ²⁄₉ - ³⁄₉ = ⁻¹⁄₉

(²⁄₉ - ⁴⁄₉) - ⅑ = ⁻²⁄₉ - ¹⁄₉ = ⁻³⁄₉ = ⁻¹⁄₃

And,

²⁄₉ - (⁴⁄₉ - ⅑) ≠ (²⁄₉ - ⁴⁄₉) - ⅑

Therefore, Associative property is NOT true for subtraction of rational numbers.

12. Answer :

Division of rational numbers is not associative.

If ᵃ⁄b, ᶜ⁄d and ᵉ⁄f are any three rational numbers, then

ᵃ⁄b ÷ (ᶜ⁄d ÷ ᵉ⁄f) ≠ (ᵃ⁄b ÷ ᶜ⁄d) ÷ ᵉ⁄f

Example :

²⁄₉ ÷ (⁴⁄₉ ÷ ⅑) = ²⁄₉ ÷ 4 = ¹⁄₁₈

(²⁄₉ ÷ ⁴⁄₉) ÷ ⅑ = ½ ÷ ⅑ = ⁹⁄₂

And,

²⁄₉ ÷ (⁴⁄₉ ÷ ⅑) ≠ (²⁄₉ ÷ ⁴⁄₉) ÷ ⅑

Therefore, Associative property is NOT true for division of rational numbers.

13. Answer :

Multiplication of rational numbers is distributive over addition.

If ᵃ⁄b, ᶜ⁄d and ᵉ⁄f  are any three rational numbers, then

ᵃ⁄b x (ᶜ⁄d + ᵉ⁄f) = (ᵃ⁄b x ᶜ⁄d) + (ᵃ⁄b x ᵉ⁄f)

Example :

⅓ x (⅖ + ⅕) :

= ⅓ x ⁽² ⁺ ¹⁾⁄₅

= ⅓ x ³⁄₅

= ³⁄₁₅

= ⅕ ----(1)

⅓ x ⅖ + ⅓ x ⅕ :

= ²⁄₁₅ + ¹⁄₁₅

  = ⁽² ⁺ ¹⁾⁄₁₅

= ³⁄₁₅

= ⅕ ----(2)

From (1) and (2), 

⅓ x (⅖ + ⅕) = ⅓ x ⅖ + ⅓ x ⅕

Therefore, Multiplication is distributive over addition.

14. Answer :

Multiplication of rational numbers is distributive over subtraction.

If ᵃ⁄b, ᶜ⁄d and ᵉ⁄f  are any three rational numbers, then

ᵃ⁄b x (ᶜ⁄d + ᵉ⁄f) = (ᵃ⁄b x ᶜ⁄d) + (ᵃ⁄b x ᵉ⁄f)

Example :

⅓ x (⅖ - ⅕) :

= ⅓ x ⁽² ⁻ ¹⁾⁄₁₅

= ²⁄₁₅ - ¹⁄₁₅

  = ⁽² ⁻ ¹⁾⁄₁₅

 = ¹⁄₁₅ ----(1)

⅓ x ⅖ - ⅓ x ⅕ :

= ²⁄₁₅ - ¹⁄₁₅

= ⁽² ⁻ ¹⁾⁄₁₅

= ¹⁄₁₅ ----(2)

From (1) and (2), 

⅓ x (⅖ - ⅕) = ⅓ x ⅖ - ⅓ x ⅕

Therefore, Multiplication is distributive over subtraction.

15. Answer :

The product of any rational number and 1 is the rational number itself. ‘1’ is the multiplicative identity for rational numbers.

If a/b is any rational number, then

ᵃ⁄b x 1 = 1 x ᵃ⁄b = ᵃ⁄b

Example : 

⁵⁄₇ x 1 = 1 x ⁵⁄₇ = ⁵⁄₇

16. Answer :

The additive inverse of a rational is the same rational number with opposite sign.

That is, if the given rational number is positive, then its additive inverse will be negative.

If the given rational number is negative, then its additive inverse will be positive.

Therefore, the additive inverse of -⅔ is ⅔.

17. Answer :

Multiplicative inverse of a fraction is nothing but its reciprocal.

Therefore, the multiplicative inverse of ⅔ is ³⁄₂.

18. Answer :

Convert the given mixed number to an improper fraction.

2⅓ = ⁷⁄₃

Multiplicative inverse of ⁷⁄₃ is nothing but its reciprocal ³⁄₇.

Therefore, the multiplicative inverse of 2⅓ is ³⁄₇.

19. Answer :

Multiplicative inverse of 0 is its reciprocal.

To get the reciprocal of 0, let us write 0 as a fraction with denominator 1.

That is ⁰⁄₁.

To get reciprocal of ⁰⁄₁, we have to write the denominator 1 as numerator and numerator 0 as denominator.

Then, we have ¹⁄₀.

Here, 1 is divided by 0.

In Math, any non zero number divided by zero is undefined.

Therefore, multiplicative inverse of 0 is undefined.

In other words, multiplicative inverse of zero does not exist.

20. Answer :

The additive inverse of a rational is the same rational number with opposite sign.

That is, if the given rational number is positive, then its additive inverse will be negative.

If the given rational number is negative, then its additive inverse will be positive.

But. zero is neither positive nor negative.

Hence, the additive inverse of zero is the same zero.

21. Answer :

Multiplicative inverse of a rational number is its reciprocal.

Therefore, the multiplicative inverse of 1 is 1 itself.

(Because, the reciprocal of 1 is 1 itself)

22. Answer :

The additive inverse of a rational is the same rational number with opposite sign.

That is, if the given rational number is positive, then its additive inverse will be negative.

If the given rational number is negative, then its additive inverse will be positive.

Therefore, the additive inverse of 1 is -1.

23. Answer :

The multiplicative inverse of ∞ is zero.

Let us see, how the multiplicative inverse of infinity is zero. 

In math infinity ∞ is the symbol that we use for the term 'Undefined'.

To get the reciprocal of ∞, let us write ∞ as a fraction ¹⁄₀.

Because, any non zero number divided by zero is undefined or infinity.

To get reciprocal of ¹⁄₀, we have write the denominator 0 as numerator and numerator 1 as denominator.

Then, we have ⁰⁄₁.

Here, 0 is divided by 1.

In Math, zero divided by any non zero number is zero.

Therefore, the multiplicative inverse of infinity ∞ is 0. 

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