Real Numbers and Order of Operations: Foundations for Algebra

Real Numbers and Order of Operations

Introduction

This module introduces foundational concepts essential for success in algebra, focusing on the properties and operations of real numbers and the systematic approach to simplifying mathematical expressions using the order of operations.

Real Numbers

Real numbers include all the numbers that can be found on the number line. They encompass rational and irrational numbers, and are used in nearly all algebraic calculations.

• Definition: Real numbers are the set of numbers that include natural numbers (1, 2, 3, ...), whole numbers (0, 1, 2, ...), integers (..., -2, -1, 0, 1, 2, ...), rational numbers (fractions, decimals that terminate or repeat), and irrational numbers (numbers that cannot be written as fractions, such as $\sqrt{2}$ or $\pi$).

• Examples: $5$,$-3$, $0$, $\frac{1}{2}$, $\sqrt{2}$, $\pi$

• Key Properties:

  • Closure: Real numbers are closed under addition, subtraction, multiplication, and division (except division by zero).

  • Commutativity: $a + b = b + a$ and $a \times b = b \times a$

  • Associativity: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$

  • Identity: $a + 0 = a$ and $a \times 1 = a$

  • Inverse: $a + (-a) = 0$ and $a \times \frac{1}{a} = 1$ (for $a \neq 0$)

Operations with Real Numbers

Performing operations with real numbers is fundamental in algebra. The four basic operations are addition, subtraction, multiplication, and division.

• Addition: Combining two or more numbers to get their sum. Example: $3 + 5 = 8$

• Subtraction: Finding the difference between two numbers. Example: $7 - 2 = 5$

• Multiplication: Repeated addition of a number. Example: $4 \times 3 = 12$

• Division: Splitting a number into equal parts. Example: $10 \div 2 = 5$

Order of Operations

The order of operations is a set of rules that determines the sequence in which calculations are performed in a mathematical expression. This ensures consistency and accuracy in simplifying expressions.

• PEMDAS/BODMAS: The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). BODMAS is an alternative acronym used in some regions.

• Steps:

  1. Parentheses (or Brackets): Simplify expressions inside parentheses first.

  2. Exponents: Evaluate powers and roots.

  3. Multiplication and Division: Perform these operations from left to right.

  4. Addition and Subtraction: Perform these operations from left to right.

• Example: Simplify $2 + 3 \times (4^2 - 1)$

  • Step 1: Parentheses: $4^2 - 1 = 16 - 1 = 15$

  • Step 2: Multiplication: $3 \times 15 = 45$

  • Step 3: Addition: $2 + 45 = 47$

Summary Table: Real Number Operations

Operation	Example	Result
Addition	$7 + 2$	$9$
Subtraction	$5 - 3$	$2$
Multiplication	$4 \times 6$	$24$
Division	$12 \div 4$	$3$