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• Number sets are collections of numbers with specific properties

Natural Numbers (N)

• Natural numbers are positive integers starting from 1 and extending infinitely: {1, 2, 3, ...}
• They are used for counting and ordering

Whole Numbers (W)

• Whole numbers include all natural numbers and zero: {0, 1, 2, 3, ...}
• They are non-negative integers

Integers (Z)

• Integers consist of all whole numbers and their negative counterparts: {..., -3, -2, -1, 0, 1, 2, 3, ...}
• They include positive and negative whole numbers and zero

Rational Numbers (Q)

• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0
• They include all integers, fractions, and terminating or repeating decimals (e.g., 0.5, 0.333...)

Irrational Numbers

• Irrational numbers cannot be expressed as a fraction p/q
• Their decimal representations are non-terminating and non-repeating (e.g., √2, π)

Real Numbers (R)

• Real numbers encompass all rational and irrational numbers
• They can be represented on a number line

Complex Numbers (C)

• Complex numbers are expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (i² = -1)
• They extend the real number system to include solutions to equations that have no real solutions

Prime Numbers

• Prime numbers are natural numbers greater than 1 that have exactly two distinct positive divisors: 1 and the number itself
• Examples include 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29
• 2 is the only even prime number
• A prime number is only divisible by 1 and itself

Composite Numbers

• Composite numbers are natural numbers greater than 1 that have more than two distinct positive divisors
• Examples include 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18
• They can be expressed as a product of prime numbers
• A composite number is divisible by 1, itself and at least one other number

Identifying Prime and Composite Numbers

• To identify prime numbers, check if the number is divisible only by 1 and itself
• To identify composite numbers, check if the number has more than two factors

Factors

• Factors are numbers that divide evenly into a given number
• For example, the factors of 12 are 1, 2, 3, 4, 6, and 12

Multiples

• Multiples are numbers obtained by multiplying a given number by an integer
• For example, the multiples of 7 are 7, 14, 21, 28, 35, and so on

Highest Common Factor (HCF)

• The highest common factor (HCF) of two or more numbers is the largest number that divides evenly into each of them
• Also known as the greatest common divisor (GCD)
• For example, the HCF of 24 and 36 is 12
• The HCF can be found using methods like prime factorization or the Euclidean algorithm

Lowest Common Multiple (LCM)

• The lowest common multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of them
• For example, the LCM of 6 and 8 is 24
• The LCM can be found using methods like prime factorization or listing multiples

Prime Factorization

• Prime factorization is the process of expressing a composite number as a product of its prime factors
• Every composite number has a unique prime factorization
• For example, the prime factorization of 28 is 2 × 2 × 7, or 2² × 7

Method

• Divide the number by the smallest prime number that divides it evenly
• Continue dividing the quotient by prime numbers until the quotient is 1
• Express the original number as the product of these prime factors

Exponential Notation

• Exponential notation is a way of expressing repeated multiplication of the same number
• It consists of a base and an exponent
• Base: The number being multiplied
• Exponent: The number of times the base is multiplied by itself
• For example, in 2³, 2 is the base, and 3 is the exponent, so 2³ = 2 × 2 × 2 = 8

Laws of Exponents

• Product of Powers: aᵐ × aⁿ = aᵐ⁺ⁿ
• Quotient of Powers: aᵐ / aⁿ = aᵐ⁻ⁿ
• Power of a Power: (aᵐ)ⁿ = aᵐⁿ
• Power of a Product: (ab)ⁿ = aⁿbⁿ
• Power of a Quotient: (a/b)ⁿ = aⁿ/bⁿ
• Zero Exponent: a⁰ = 1 (if a ≠ 0)
• Negative Exponent: a⁻ⁿ = 1/aⁿ

Ratio

• A ratio is a comparison of two or more quantities
• It can be expressed in several ways: as a fraction, using a colon, or in words
• Ratios are used to describe relative sizes or amounts

Expressing Ratios

• Fraction: a/b (e.g., 1/2)
• Colon: a:b (e.g., 1:2)
• Words: "a to b" (e.g., 1 to 2)

Simplifying Ratios

• Simplify ratios by dividing each term by their greatest common factor (GCF)
• For example, the ratio 6:8 can be simplified to 3:4 by dividing both terms by 2

Proportions

• A proportion is an equality between two ratios
• If a/b = c/d, then a, b, c, and d are in proportion

Solving Proportions

• Use cross-multiplication to solve proportions
• If a/b = c/d, then ad = bc
• This method is useful for finding an unknown value in a proportion