8th Standard Maths - Chapter 1 Detailed Concepts

8th Standard Samacheer Kalvi Mathematics

Chapter 1: Numbers (எண்கள்) - Complete Detailed Concept Sheet

1. Rational Numbers (விகிதமுறு எண்கள்)

Definition: A number that can be expressed in the form ab, where a and b are integers and b ≠ 0, is called a rational number.
Inclusions: All natural numbers (1, 2, 3...), whole numbers (0, 1, 2...), integers (..., -2, -1, 0, 1, 2...), and fractions are rational numbers.
Example: 5 can be written as 51, so it is a rational number.
Number Line: Every rational number can be uniquely represented on a number line.
Zero Property: 0 is neither a positive nor a negative rational number.
Standard Form: A rational number ab is in standard form if denominator b is a positive integer, and HCF(a, b) = 1.
Example: Standard form of 12-18 is -23.
Density Property: There are infinitely many (unlimited) rational numbers between any two given rational numbers.

Subtraction: Subtracting two rational numbers is equivalent to adding the additive inverse of the second number to the first.
ab - cd = ab + (-cd)
Multiplication: Multiply numerators together and denominators together, then simplify to standard form.
ab × cd = a × cb × d
Division: Divide one rational number by another by multiplying the first number by the reciprocal (multiplicative inverse) of the second.
ab ÷ cd = ab × dc

Operation	Closure	Commutative	Associative	Distributive Property
Addition (+)	✓	✓	✓	Multiplication is distributive over Addition & Subtraction: a × (b ± c) = (a × b) ± (a × c)
Subtraction (-)	✓	✗	✗
Multiplication (×)	✓	✓	✓
Division (÷)	✗ (if ÷0)	✗	✗

Identities:
- 0 is the Additive Identity (a + 0 = a).
- 1 is the Multiplicative Identity (a × 1 = a).
Inverses:
- Additive inverse of ab is -ab.
- Multiplicative inverse (Reciprocal) of ab is ba (since ab × ba = 1).

Square Number: A natural number n is a square number if n = m² for some natural number m. (e.g., 4 × 4 = 16)
Square Root: Written as √n or n^½, is the value that gives n when multiplied by itself.
Prime Factorization Rule: The number of times a prime factor occurs in a square number is always twice the number of times it occurs in its square root.
Properties of Square Roots:
1. √(a × b) = √a × √b
2. √(a / b) = √a / √b (where b ≠ 0)

Cube Number: Obtained by multiplying a number by itself three times (n × n × n = n³). E.g., 3³ = 27.
Cube Root: Written as ∛n or n^⅓, is the value that when cubed gives the original number.

An expression representing repeated multiplication of the same factor is a power. Exponential format: aⁿ (a = base, n = exponent).
Laws of Exponents:
1. Product Rule: a^m × aⁿ = a^m+n
2. Quotient Rule: a^m ÷ aⁿ = a^m-n
3. Power Rule: (a^m)ⁿ = a^m×n
Other Important Results:
• a⁰ = 1 (where a ≠ 0)
• Negative exponent: a^-m = 1 / a^m
• (a × b)^m = a^m × b^m
• (a / b)^m = a^m / b^m

Scientific notation is used to express very large or very small numbers conveniently in the standard format:

S × 10ⁿ

Where 1 ≤ S < 10 (S is a number with decimals between 1 and 10).
n is an integer (positive for large numbers, negative for numbers smaller than 1).

Example 1: 456,000 = 4.56 × 10⁵
Example 2: 0.00078 = 7.8 × 10^-4