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Unit 1: Real And Complex Numbers
Explanation Of Exercise 1.2
Real Numbers:
The set of real numbers are the union rational and irrational number i.e. = QUQ'.
Properties of Real Numbers
In real numbers there exist properties with respect to addition and multiplication. For real number a, b the sum is a + b and product is written as a.b or a x b or simply ab.
Properties of Real Numbers with respect to Addition
1. Closure Property:
"The sum of any two real numbers is also a real number."
i.e. ∀ a, b ∈ R ⇒ a+b ∈ R is called closure property w.r.t addition.
Here (a, b are real number and there addition is also an other real number)
e.g.
(i) 5,7 ∈ R ⇒ 5+7 = 12 ∈ R
2. Commutative property:
"For any two Real Numbers (a and b) is same."
i.e. a+b = b+a ∀ a, b ∈ R
is called commutative property w.r.t addition.
3. Associative property:
"For any three real number (a, b and c) the addition follow by the method called associative property".
(a + b) + c = a + (b + c) ∀ a, b, c ∈ R
is called associative property w.r.t addition.
e.g.
(4 + 5) + 6 = 4 + (5 + 6)
4. Additive identity:
"For any existing number no change observed".or There exist a number 0 ∈ R such that
a+0 = a = 0+a, ∀ a ∈ R
'0' is called additive identity.
e.g.
(i) 3 + 0 = 3 = 0 + 3
(ii)
7 /
8 + 0 =
7 /
8 = 0 +
7 /
8, etc.
5. Additive inverse:
As, for each a ∈ R, there exist -a ∈ R Such that
a + (-a) = 0 = (-a) + a
so a and -a are the additive inverse of each other.
e.g.
6 + (-6) = 0 = -6 + 6
Here 6 and -6 are additive inverse of each other.
Properties of Real Number with respect to Multiplication
1. Closure property:
"The product of any two real numbers (a and b) is again a real number".
i.e. a, b ∈ R ⇒ ab ∈ R, is called closure property w.r.t (with respect to) multiplication.
e.g.
(i) 5, 7 ∈ R ⇒ (5)(7) = 35 ∈ R
2. Commutative property:
"For any two real number a and b".
ab = ba ∀ a, b ∈ R, is called commutative property w.r.t multiplication.
3. Associative property:
For any three real numbers a, b and c. (ab)c = a(bc) is called associative property with respect to multiplication.
4. Multiplicative identity:
"For any real number a ∈ R, multiply with 1 given the same real number so, "1" is called multiplicative identity.
i.e. a x 1 = a = l x a ∀ a ∈ R
e.g.
(i) 3 x 1 = 3 = 1 x 3
(ii)
3/
5 x 1 = 1 x
3/
5 =
3/
5, etc.
5. Multiplicative inverse:
For each real number a ∈ R (a ≠ 0) there exists an element
1/
a or a
-1 ∈ R gives the result 1 so, the
1/
a or a
-1 and a are called multiplicative inverse of each other.
i.e. a x (
1/
a) = (
1/
a) x a = 1
e.g.
(i) 3 x (
1/
3) = (
1/
3) x 3 = 1
Here 3 and
1/
3 are multiplicative inverse of each other.
Distributive property of Multiplication over addition
For any three real number a, b, c such that:
-
a(b + c) = ab + ac, it is called Distributive property of multiplication over addition. (Left Distributive Property)
-
(ii) (a + b)c = ac + bc, it is called distributive property of multiplication over addition. (Right Distributive Property)
e.g.
- 3(5 + 7) = 3 x 5 + 3 x 7, (Left Distributive Property)
-
(3 + 7)2 = 3 x 2 + 7 x 2, (Right Distributive Property)
Note: a(b - c) = ab - ac, it is called Distributive property of multiplication over subtraction.
Properties Of Equalities Of Real Numbers
(1) Reflexive Property:
If a ∈ R then a = a called reflexive property.
(2) Symmetric Property:
If a, b ∈ R then a = b ⇔ b = a is called symmetric property.
(3) Transitive Property:
If a, b, c ∈ R then a = b and b =c ⇔ a = c is called transitive property.
(4) Additive Property:
If a, b, c ∈ R then a = b ⇔ a+c =b+c is called additive property.
(5) Multiplicative Property:
If a, b and c ∈ R then a = b then ac=bc is called multiplicative property.
(6) Cancellation Property For Addition:
If a, b and c ∈ R, if a + c = b+ c then a = b is called cancellation property for addition.
(7) Cancellation Property For Multiplication:
If a, b, c ∈ R, and c ≠ 0 if ac = bc then, a = b is called cancellation property with multiplication.
Properties Of Inequalities Of Real Numbers:
Following are the properties of inequalities of real numbers.
(i) Trichotomy Property:
If a, b, c ∈ R, then a > b or a < b or a = b
(ii) Transitive Property:
If a, b, c ∈ R, then
(a) a < b and b < c ⇒ a < c.
(b) a > b and b > c ⇒ a > c.
(iii) Additive Property:
If a, b, c ∈ R, then
(a) a < b ⇒ a + c < b + c.
(b) a > b ⇒ a + c > b + c.
(iv) Multiplicative Property:
If a, b, c ∈ R and c > 0, then
(a) a > b ⇒ ac > bc.
(a) a < b ⇒ ac < bc.
Similarly, if c < 0, then
(a) a > b ⇒ ac < bc.
(a) a < b ⇒ ac > bc.
(v) Reciprocative Property
If a, b ∈ R and a, b are of same sign.
(vi) Cancellation Property:
If a, b, c ∈ R
(a) a + c > b + c ⇒ a > b
(b) a + c < b + c ⇒ a < b
similarly,
(c) ac > bc ⇒ a b, where c > 0
(d) ac < bc ⇒ a < b, where c > 0
