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.
Number System – an Introduction
Rational Numbers between two numbers
Number System – problem
Identities based on roots
