Numbers and Operations

Introduction to Numbers and Operations

Numbers are the foundation of mathematics. Here's what you need to know, condensed but comprehensive:

• Natural Numbers: The counting numbers: 1, 2, 3, ...
• Whole Numbers: Natural numbers including zero: 0, 1, 2, 3, ...
• Integers: Whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational Numbers: Numbers that can be expressed as fractions $$\frac{p}{q}$$ where p and q are integers and q ≠ 0
• Irrational Numbers: Numbers that cannot be expressed as fractions, like $$\pi$$ and $$\sqrt{2}$$
• Real Numbers: All rational and irrational numbers

Basic Operations - Essential Concepts

Addition (+)

Addition combines two numbers to get their sum.

Example: $$5 + 3 = 8$$

Key insight: Addition is commutative (a + b = b + a) and associative ((a + b) + c = a + (b + c)).

Subtraction (-)

Subtraction finds the difference between two numbers.

Example: $$9 - 4 = 5$$

Key insight: Subtraction is not commutative (a - b ≠ b - a) and can be viewed as adding the negative (a - b = a + (-b)).

Multiplication (×)

Multiplication is repeated addition.

Example: $$6 × 7 = 42$$

Key insight: Multiplication is commutative (a × b = b × a) and distributive over addition (a × (b + c) = a × b + a × c).

Division (÷)

Division splits a number into equal parts.

Example: $$20 ÷ 4 = 5$$

Key insight: Division is not commutative (a ÷ b ≠ b ÷ a) and can be viewed as multiplying by the reciprocal (a ÷ b = a × (1/b)).

Order of Operations (PEMDAS) - Critical for Problem Solving

When evaluating expressions with multiple operations, follow this order:

1. Parentheses
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Example: $$2 + 3 × 4 = 2 + 12 = 14$$

Example: $$(2 + 3) × 4 = 5 × 4 = 20$$

Common pitfall: Forgetting that multiplication and division have the same precedence and are evaluated from left to right. Same for addition and subtraction.