Number System Overview

Overview

This lecture introduces the number system, covering the origins, types, and properties of numbers, along with key terms like prime, composite, and perfect numbers—essential for competitive exams.

Introduction to the Number System

• Mathematics relies fundamentally on numbers and calculation.
• Numbers originated from the practical need to count and keep track of items.
• The concept of numbers evolved from basic counting methods like using stones.

Types of Numbers

• Natural Numbers: Counting numbers starting from 1 (1, 2, 3, ...).
• Whole Numbers: Natural numbers including zero (0, 1, 2, 3, ...).
• Integers: Whole numbers plus their negatives (... -3, -2, -1, 0, 1, 2, ...).
• Rational Numbers: Numbers that can be written as p/q where q ≠ 0 (fractions, terminating, and repeating decimals).
• Irrational Numbers: Numbers that cannot be written as p/q; their decimal expansions are non-terminating and non-repeating (e.g., √2, π).
• Real Numbers: The set of all rational and irrational numbers.
• Imaginary Numbers: Numbers involving the square root of negative numbers (not included in real numbers).
• Complex Numbers: Numbers of the form a + bi where a and b are real and i is the imaginary unit.

Decimal Representation

• Terminating Decimals: Decimals that end (e.g., 2.3).
• Non-Terminating, Repeating Decimals: Decimals that repeat a pattern (e.g., 0.333...).
• Non-Terminating, Non-Repeating Decimals: Decimals that neither end nor repeat (e.g., π, √2).

Special Types of Numbers

• Even Numbers: Divisible by 2 (e.g., 0, 2, 4, ...).
• Odd Numbers: Not divisible by 2 (e.g., 1, 3, 5, ...).
• Prime Numbers: Numbers with exactly two distinct positive divisors: 1 and itself (e.g., 2, 3, 5, 7, ...).
• Composite Numbers: Numbers with more than two divisors (e.g., 4, 6, 8, ...).
• Coprime Numbers: Two numbers with only 1 as their common factor (e.g., 4 and 5).
• Twin Primes: Pairs of primes differing by 2 (e.g., 3 and 5, 11 and 13).
• Perfect Numbers: Numbers equal to the sum of their proper divisors (e.g., 6, 28).

Key Terms & Definitions

• Natural Numbers — Counting numbers starting from 1.
• Whole Numbers — Natural numbers plus zero.
• Integers — All positive, negative whole numbers, and zero.
• Rational Numbers — Can be expressed as p/q, q ≠ 0.
• Irrational Numbers — Cannot be expressed as p/q; non-terminating, non-repeating decimals.
• Real Numbers — Set including all rational and irrational numbers.
• Imaginary Numbers — Numbers involving square roots of negatives.
• Complex Numbers — Combination of real and imaginary numbers (a + bi).
• Even Numbers — Divisible by 2.
• Odd Numbers — Not divisible by 2.
• Prime Numbers — Numbers with only two positive divisors: 1 and itself.
• Composite Numbers — Numbers with more than two divisors.
• Coprime Numbers — Pairs of numbers with only 1 as common factor.
• Twin Primes — Pair of primes with difference of 2.
• Perfect Numbers — Equal to the sum of their proper divisors.

Action Items / Next Steps

• Make detailed notes on different types of numbers and their properties.
• Memorize examples of rational, irrational, prime, composite, coprime, and perfect numbers.
• Prepare for upcoming topics: factors, LCM, HCF, and unit digits.
• Review and practice identifying types of numbers for exams.