Belongs to the unit Real Numbers

1. Give an example to each of the property of real numbers associativity and commutativity of addition and multiplication; distributuvity of multiplication over addition.

**Solution:**

Associative property

(a) Let a = 2, b = 5 and c = 8

Then a + (b + c) = 2 + (5 + 8)

= 2 + 13

= 15

a + (b + c) = (2 + 5) + 8

= 7 + 8

= 15

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

This is associative property of addition

(b) Now a. (b. c) = 2. (5 . 8)

= 2.40

= 80

(a. b). c = (2.5).8

= 10.8

= 80

This is associative property of multiplication.

Commutative property:

Let a = 3 and b = 7

(a) a + b = 3 + 7 = 10

b + a = 7 + 3 = 10

a + b = b + a

This is commutative property of addition.

(b) a. b = 3.7 = 21

b. a = 7.3 = 21

∴ a. b = b .a

This is commutative property of multiplication.

Distribution of multiplication over addition.

Let a = 4, b = 6 and c = 9

a (b + c) = 4(6 + 9) = 4 x 15 = 60 …… (1)

ab + bc = 4.6 + 4.9

= 24 + 36

= 60 …… (2)

From (1) and (2) we see that

∴ a (b + c) = ab + bc

This is left distribution law.

Also

(b + c) a = (6 + 9). 4 = 15.4 = 60 ……. (3)

ba + ca = 6.4 + 9.4 = 24 + 36 = 60 …….. (4)

From (3) and (4)

(b + c) a = ba + ca

This is right distributive law.

2. What are the properties of R used in the following?

(i) 8 x 7 = 7 x 8

**Solution:**

Commutative property of multiplication.

(ii) n + (𝛑 + c) = (n + 𝛑) + c

**Solution:**

Associative property of addition

(iii) 0 + 0 = 0

**Solution:**

Identity element property with respect to addition. 0 is the identity element

(iv) 𝛑 x 1 = 𝛑

**Solution:**

Identity element property with respect to multiplication.

(v) √2 (1 + √2 ) = 2 + √2

**Solution:**

Distribution property of multiplication of over addition.

3. Find the additive inverse of each of the following:

(a) √𝟓

(b) 1 + 𝛑

(c) 7 + 2^{(1/4)}

(d) –𝟑/√𝝅

(e) (–3 + √𝟑)²

**Solution:**

(a) Additive inverse of √5 is – √5

(b) Additive inverse of (1 + 𝛑) is –(1 + 𝛑)

(c) Additive inverse of 7 + 2^{(1/4) }is -(7+ 2^{(1/4)})

(d) Additive inverse of (−3/√𝜋) is (3/√𝜋)

(e) Additive inverse of (–3 + √𝟑)² is -(–3 + √𝟑)²

4. Find the multiplicative inverse of each of the following: