Important Formulas: Number System

Law of Exponents

Basic statement: Exponents (indices) describe repeated multiplication of a base number. The following are the standard laws of exponents used throughout Number System topics.

• a^p · a^q = a^p+q - product of powers with same base adds the exponents.
• a^p / a^q = a^p-q (a = 0) - quotient of powers with same base subtracts the exponents.
• (a^p)^q = a^pq - power of a power multiplies the exponents.
• (ab)^p = a^p · b^p - power of a product equals product of powers.
• a^-p = 1 / a^p (a = 0) - negative exponent gives reciprocal.
• a⁰ = 1 (a = 0) - zero exponent equals one.
• a¹ = a - power one is the base itself.

Number System - Basic Definitions and Classification

• Natural numbers (N): 1, 2, 3, ...
• Whole numbers (W): 0, 1, 2, 3, ...
• Integers (Z): ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers (Q): numbers expressible as p/q where p and q are integers and q = 0. Decimal expansion is either terminating or repeating.
• Irrational numbers: numbers not expressible as p/q. Their decimal expansions are non‐terminating and non‐repeating (for example, √2, π).
• Real numbers (R): union of rational and irrational numbers; representable on the number line.

Closure and basic properties

• Closure under addition and multiplication: Natural numbers are closed under addition and multiplication. Integers are closed under addition, subtraction and multiplication. Rational numbers are closed under addition, subtraction, multiplication and division (except division by zero).
• Density: Between any two distinct real numbers there exists a rational number and also an irrational number.

Useful Formulas and Short Tricks

• Converting recurring decimals to fractions: Use multiplication to align repeating parts, subtract, solve for the variable.
• HCF × LCM = product of the two numbers when numbers are positive integers.
• To compare large numbers: Express in prime factor form or scientific notation before comparing.
• To check if a number is a perfect square: All prime exponents in its prime factorisation must be even.
• Negative bases with fractional exponents: Avoid non-integer fractional exponents of negative numbers in real numbers; consider complex numbers only when required.

Common Mistakes to Avoid

• Confusing (a^p)^q with a^p+q; remember exponents multiply for power of a power.
• Using (a + b)^p = a^p + b^p - this is false except for special cases (p = 1).
• For rationalising denominators with sums/differences of surds always use the conjugate.
• When converting repeating decimals, ensure you multiply by the correct power of 10 to align the repeating block.

Summary

This document collects the essential formulas and methods from the Number System topic useful for Class 8/9 level mathematics: definitions of number types, laws of exponents, prime factorisation, HCF and LCM (including Euclid's algorithm), divisibility tests, decimal ↔ fraction conversions, surds and rationalisation, fractional exponents, and scientific notation. Practise these methods through examples to build fluency and avoid the common mistakes listed above.