Number system

Awais Bakshy
MATHEMATICS.
Chapter: Sets.

Mathematics.
Chapter: Number Systems.
Number System.
Number system is the way to represent a number in different
forms.
Real Numbers.
The sum of rationaland irrationalnumbers is calledreal
numbers.
Real number = RationalNumbers+ IrrationalNumbers.
Denoted By Capital R
Rational Numbers.
A rationalnumber is a number which can be put in the form
𝑃/q) where p, q ϵ Z ∧ q ≠ 0. The number√16, 3, 7, 4 etc. are
rational numbers. √16 Can be reduced to form
𝑝
𝑞⁄ where p,
q ϵ Z and q ≠ 0 because √16 =4=4
1⁄ .
Denoted by Capital Q.

IrrationalNumbers.
Irrational Numbers are those numbers which cannot be put
into the form
𝑝
𝑞⁄ where p, q ϵ Z and q≠0. The numbers√2,
√3, √5,
√5
16
.
Denoted By Capital 𝑄,
(Q Dash)
Fractions.
Numbers that can be written as ratios of 2 integers
E.g.
2
3
,
6
8
,
2
7
,
3
5
.
Integers.
Positive Integers + 0 (Zero) + Negative Integers.
Denoted By CapitalZ.
Z= {0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, …}
Negative Integers.
Denoted by 𝑍′
(Z Dash)
𝑍′
= {-1,-2, -3, -4, -5, -6, …}
Natural Numbers.
N ={1, 2, 3, 4, 5, 6, 7, …}
Whole Numbers.

W = {0, 1, 2, 3, 4, 5, 6, 7,8, …}
Even Numbers.
E= {0, 2, 4, 6, 8, 10, 12, 14, …}
Odd Number.
O ={1, 3, 5, 7, 9, 11, 13, …}
Properties of Real Numbers.
Addition Laws:
1. Closure Property w.r.t addition.
∀ a, b ϵ R,a+b=R (∀ Stands for “for all”)
2. Associative propertyw.r.t addition.
∀ a, b, c ϵ R, a+(b+c)=(a+b)+c
3. Additive identity.
∀ a ϵ R , ∃ 0 ϵ R such that a+0=0+a=a
(∃ stand for “there exist”)
0 (read as zero) is calledthe identity element of
addition.
4. Additive inverse.
∀ a ϵ R, ∃ (-a) ϵ R such that
a+(-a)=(-a)+a=0
5. Commutative Property w.r.t addition.
∀ a, b ϵ R, a+b=b+a

Multiplicative Laws:
i. Closure Property w.r.t Multiplication.
∀ a, b ϵ R,a.b=R (a.b is usually written as ab)
ii. Associative propertyw.r.t multiplication.
∀ a, b, c ϵ R, a(bc)=(ab)c
iii. Multiplicative identity.
∀ a ϵ R , ∃ 1 ϵ R such that a.1=1.a=a 1 is called the
multiplicativeidentity of real numbers.
iv. Multiplicative inverse.
∀ a(≠0) ϵ R, ∃ (𝑎−1
) ϵ R such that a. 𝑎−1
= 𝑎−1
.a=1
(𝑎−1
is also written as
1
𝑎
)
v. Commutative Property w.r.t addition.
∀ a, b ϵ R, ab=ba
DistributivePropertyof Multiplicationoveraddition.
∀ a, b, c ϵR such that;
a (b+c)= ab+ac ( left D.Law)
(a+b) c= ac+bc ( Right D.Law)

Exercise.
Q No 1.
Name of the properties of real numbers given below.
a. 25+48=48+25
b. √11 ×√3=√3×√11
c. √5(3+7)=√5
3
+√5
7
d. (
11
2
+
9
5
) +
7
10
=
11
2
+(
9
5
+
7
10
)
e. 0 +
4
5
=
4
5
+ 0
f. (
√5
2
× 1) =
√5
2
Q No 2.
Write the additive and multiplicativeinverse of the
following.
a.
1
𝑎
b. 1
c. √57
d. 0
e.
√3
7
f.
√5
−
2
g. −7

Q No 3.
Which of the followingset have the closure property w.r.t
addition and multiplication.
a. 0
b. {0,-1}
c. {0, 1}
d. {-1, -2, -3, …}
e. Set of real numbers.
f. Set of rational numbers.
g. Set of whole numbers.
Q No 4.
Which of the followingset have additive and multiplicative
identity.
a. Set of real numbers.
b. Set of whole numbers.
Q No 5.
Which of the followingsets have the additive and
multiplicativeinverse.
a. Set or rational numbers.
b. Set of prime numbers.
c. Set of real numbers.
d. Set of integer numbers.

Number system

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