Real Numbers

Class 9 | Maths | Number Systems

1. Real Numbers:

Real numbers include all rational and irrational numbers.
Rational numbers can be expressed as fractions, and irrational numbers cannot be expressed as fractions.

2. Rational Numbers:

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers.
A rational number can be represented as a/b, where ‘a’ and ‘b’ are integers, and ‘b’ is not equal to zero (b ≠ 0).

3. Irrational Numbers:

Irrational numbers cannot be expressed as a simple fraction or ratio of two integers.
Examples of irrational numbers include √2, π (pi), and e (Euler’s number).

4. Decimal Expansion:

Rational numbers have either a finite or a repeating decimal expansion.
For example, 1/2 = 0.5 (finite) and 1/3 = 0.333… (repeating).

5. Euclid’s Division Lemma:

Given two positive integers ‘a’ and ‘b’, there exist unique integers ‘q’ and ‘r’ such that:
a = bq + r, where 0 ≤ r < b.
‘q’ is the quotient, and ‘r’ is the remainder.

6. HCF (Highest Common Factor) and LCM (Least Common Multiple):

HCF of two numbers is the largest number that divides both of them.
LCM of two numbers is the smallest multiple that is divisible by both of them.

7. Fundamental Theorem of Arithmetic:

Every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way (up to the order of factors).

8. Prime Numbers:

Prime numbers are natural numbers greater than 1 that have exactly two distinct positive divisors: 1 and the number itself.

9. Prime Factorization:

Expressing a number as a product of its prime factors is called prime factorization.
For example, prime factorization of 12 = 2^2 * 3.

10. Properties of Real Numbers:
– Closure property: The sum or product of two real numbers is also a real number.
– Commutative property: a + b = b + a and ab = ba.
– Associative property: (a + b) + c = a + (b + c) and (ab)c = a(bc).
– Distributive property: a(b + c) = ab + ac.

11. Laws of Exponents:
– Product of Powers: a^m * a^n = a^(m + n).
– Quotient of Powers: a^m / a^n = a^(m – n).
– Power of a Power: (a^m)^n = a^(m * n).
– Power of a Product: (ab)^n = a^n * b^n.