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Dear Student :-

Hello student today we are discusses class 9th ch1 number system.
(1) Numbers :- Arithmetical value representing a particular quantity. The various types of numbers are Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers.etc

(2) Natural Numbers :- Natural numbers (N) are positive numbers i.e. 1, 2, 3 ..and so on.
(3) Whole number :- Whole numbers (W) are 0, 1, 2,..and so on. Whole numbers are all Natural Numbers including ‘0’
If ‘0’ is included in the collection of natural numbers, then the collection are known as whole numbers.
W = {0, 1, 2, 3, …}
(4) Integers :- Integers are the numbers that consists of entire numbers together with the negative numbers. The integers can be represented as:
Z = {……., -3, -2, -1, 0, 1, 2, 3, ……….}
(5) Rational Number:- A number r is called a rational number, if it can be written in the form pq, where p and q are integers and q ≠ 0. The collection of rational numbers is denoted by Q.
Q = 2/3, 3/5, etc. all are rational numbers.
(6) Irrational Number :- The number ‘a’ which cannot be written in the form of p/q is called irrational, where p and q are integers and q ≠ 0 or you can say that the numbers which are not rational are called Irrational Numbers.
Example - √7, √11 √2, √5, π, …etc.
(6) Number Line:

(7) Equivalent Rational Number: The rational number whose numerator and denominator both are equal or they are reducible to equal.
e.g.,

( Some Import Formulas
o √ab = √a√b
o √ab = √a√b
o (√a+√b) (√a-√b) = a – b
o (a+√b)(a−√b) = a²−b
o (√a+√b)(√c+√d) = √ac+√ad+√bc+√bd
o (√a+√b)(√c−√d) = √ac−√ad+√bc−√bd
o (√a+√b)2 = a+2√(ab)+b
(9) Rationalisation:-
Rationalisation is converting an irrational number into a rational number. Suppose if we have to rationalize
1/√a.
1/√a × 1/√a = 1/a
Rationalisation of 1/√a+b:
(1/√a+b) × (1/√a−b) = (1/a−b²)
(10) Laws of Exponents for Real Numbers
If a, b, m and n are real numbers then:
• am × an= am+n
• (am) n = amn
• am/an = am−n
• ambm=(ab)m
Here, a and b are the bases and m and n are exponents.
(11) Exponential representation of irrational numbers
If a > 0 and n is a positive integer, then: n√a=a1n Let a > 0 be a real number and p and q be rational numbers, then:
• ap × aq = ap + q
• (ap)q = apq
• ap/ aq= ap−q
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