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# Real numbers exercise 1.2.4

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.
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:
(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: