**Worksheet on properties of rational numbers :**

Properties of rational numbers worksheet is much useful to the students who would like to practice problems on rational numbers.

1. What will be the sum of two rational numbers ? Discuss your answer.

2. Are (2/9 + 4/9) and (4/9 + 2/9) equal ? What do you come to know from the result ?

3. Are {2/9 + (4/9 + 1/9)} and {(2/9 + 4/9) + 1/9} equal ? What do you come to know from the result ?

4. What will be the product of two rational numbers ? Discuss your answer.

5. What will be the difference of two rational numbers ? Discuss your answer.

6. Are (5/9 x 2/9) and (2/9 x 5/9) equal ? What do you come to know from the result ?

7. Are (5/9 - 2/9) and (2/9 - 5/9) equal ? what do you come to know from the result ?

8. Are {2/9 - (4/9 - 1/9)} and {(2/9 - 4/9) - 1/9} equal ? What do you come to know from the result ?

9. What will you get when you add zero to a rational number ? Explain your answer.

10. Are { 1/3 x (2/5 + 1/5) } and { (1/3x2/5) + (1/3x1/5) } equal ? What do you come to know from the result.

11. Are { 1/3 x (2/5 - 1/5) } and { (1/3x2/5) - (1/3x1/5) } equal ? What do you come to know from the result.

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

13. Find the additive inverse of "-2/3"

14. Find the multiplicative inverse of 2/3

15. Find the multiplicative inverse of 2 1/3

16. Find the multiplicative inverse of zero.

17. Find the additive inverse of zero.

18. Find the multiplicative inverse of 1

19. Find the additive inverse of 1

20. Find the multiplicative inverse of "∞"

**Problem 1 : **

What will be the sum of two rational numbers ? Discuss your answer.

**Solution :**

The sum of any two rational numbers is always a rational number. This is called ‘Closure property of addition’ of rational numbers. Thus, Q 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 = 2/3 is a rational number.

**Problem 2 : **

Are (2/9 + 4/9) and (4/9 + 2/9) equal ? What do you come to know from the result ?

**Solution :**

2/9 + 4/9 = 6/9 = 2/3

4/9 + 2/9 = 6/9 = 2/3

Hence, 2/9 + 4/9 = 4/9 + 2/9

From the above working, it is clear that the addition of two rational numbers is commutative.

**Problem 3 : **

Are {2/9 + (4/9 + 1/9)} and {(2/9 + 4/9) + 1/9} equal ? What do you come to know from the result ?

**Solution :**

2/9 + (4/9 + 1/9) = 2/9 + 5/9 = 7/9

(2/9 + 4/9) + 1/9 = 6/9 + 1/9 = 7/9

Hence, 2/9 + (4/9 + 1/9) = (2/9 + 4/9) + 1/9

From the above working, it is clear that the addition of rational numbers is associative.

**Problem 4 :**** **

What will be the product of two rational numbers ? Discuss your answer.

**Solution :**

The product of two rational numbers is always a rational number. Hence Q is closed under multiplication.

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

then (a/b)x (c/d) = ac/bd is also a rational number.

**Example : **

5/9 x 2/9 = 10/81 is a rational number.

**Problem 5 :**** **

What will be the difference of two rational numbers ? Discuss your answer.

**Solution :**

The difference between any two rational numbers is always a rational number.

Hence Q 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 = 1/3 is a rational number.

**Problem 6 :**** **

Are (5/9 x 2/9) and (2/9 x 5/9) equal ? What do you come to know from the result ?

**Solution :**

5/9 x 2/9 = 10/81

2/9 x 5/9 = 10/81

Hence, 5/9 x 2/9 = 2/9 x 5/9

From the above working, it is clear that the multiplication of two rational numbers is commutative.

**Problem 7 :**** **

Are (5/9 - 2/9) and (2/9 - 5/9) equal ? what do you come to know from the result ?

**Solution :**

5/9 - 2/9 = 3/9 = 1/3

2/9 - 5/9 = -3/9 = -1/3

Hence, 5/9 - 2/9 ≠ 2/9 - 5/9

From the above working, it is clear that the subtraction of two rational numbers is not commutative.

**Problem 8 : **

Are {2/9 - (4/9 - 1/9)} and {(2/9 - 4/9) - 1/9} equal ? What do you come to know from the result ?

**Solution :**

2/9 - (4/9 - 1/9) = 2/9 - 3/9 = -1/9

(2/9 - 4/9) - 1/9 = -2/9 - 1/9 = -3/9

Hence, 2/9 - (4/9 - 1/9) ≠ (2/9 - 4/9) - 1/9

From the above working, it is clear that the subtraction of rational numbers is not associative.

**Problem 9 : **

What will you get when you add zero to a rational number ? Explain your answer.

**Solution :**

The sum of any rational number and zero is the rational number itself.

If a/b is any rational number,

then a/b + 0 = 0 + a/b = a/b

Zero is the additive identity for rational numbers.

**Example : **

2/7 + 0 = 0 + 2/7 = 27

**Problem 10 : **

Are { 1/3 x (2/5 + 1/5) } and { (1/3x2/5) + (1/3x1/5) } equal ? What do you come to know from the result.

**Solution :**

1/3 x (2/5 + 1/5) = 1/3 x 3/5 = 1/5 -----(1)

1/3 x (2/5 + 1/5) = 1/3 x 2/5 + 1/3 x 1/5 = (2 + 1) / 15 = 1/5 -----(2)

From (1) and (2), the given numerical expressions are equal.

From the above working, it is clear that the multiplication is distributive over addition.

**Problem 11 : **

Are { 1/3 x (2/5 - 1/5) } and { (1/3x2/5) - (1/3x1/5) } equal ? What do you come to know from the result.

**Solution :**

1/3 x (2/5 - 1/5) = 1/3 x 1/5 = 1/15 ------ (1)

1/3 x (2/5 - 1/5) = 1/3 x 2/5 - 1/3 x 1/5 = (2 - 1) / 15 = 1/15 -------(2)

From (1) and (2), the given numerical expressions are equal.

From the above working, it is clear that the multiplication is distributive over subtraction.

Let us look at the next problem on "Worksheet on properties of rational numbers"

**Problem 12 : **

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

**Solution :**

The product of any rational number and 1 is the rational number itself. ‘One’ 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 = 1x 5/7 = 5/7

**Problem 13 : **

Find the additive inverse of "-2/3"

**Solution :**

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 "-2/3" is 2/3

Let us look at the next problem on "Worksheet on properties of rational numbers"

**Problem 14 : **

Find the multiplicative inverse of 2/3.

**Solution :**

Multiplicative inverse of a fraction is nothing but its reciprocal.

Therefore, the multiplicative inverse of 2/3 is 3/2.

Let us look at the next problem on "Worksheet on properties of rational numbers"

**Problem 15 : **

Find the multiplicative inverse of 2 1/3.

**Solution :**

First let us convert the given mixed number into improper fraction.

That is,

2 1/3 = 7/3

Multiplicative inverse of 7/3 is nothing but its reciprocal 3/7

Therefore, the multiplicative inverse of 2 1/3 is 3/7.

Let us look at the next problem on "Worksheet on properties of rational numbers"

**Problem 16 : **

Find the multiplicative inverse of zero.

**Solution :**

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, the reciprocal of zero is undefined.

Let us look at the next problem on "Worksheet on properties of rational numbers"

**Problem 17 : **

Find the additive inverse of zero.

**Solution :**

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.

Let us look at the next problem on "Worksheet on properties of rational numbers"

**Problem 18 : **

Find the multiplicative inverse of 1.

**Solution :**

Multiplicative inverse of a fraction is nothing but its reciprocal.

Therefore, the multiplicative inverse of 1 is 1 itself.

(Because, the reciprocal of 1 is 1 itself)

Let us look at the next problem on "Worksheet on properties of rational numbers"

**Problem 19 : **

Find the additive inverse of 1

**Solution :**

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"

Let us look at the next problem on "Worksheet on properties of rational numbers"

**Problem 20 : **

Find the multiplicative inverse of "∞"

**Solution :**

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 zero.

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