1.1 Real Numbers: Algebra Essentials

Real numbers form the foundation of algebra, encompassing rational and irrational numbers. Understanding their subsets and representations on a number line is crucial for solving equations and graphing functions.

Algebraic operations and expressions build on real numbers, introducing variables and rules for manipulation. Mastering order of operations, properties of real numbers, and simplification techniques is essential for solving complex mathematical problems.

Real Number System

Subsets of real numbers

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• Real numbers encompass all rational and irrational numbers
  • Rational numbers can be expressed as a ratio of two integers $\frac{a}{b}$ where $b \neq 0$
    • Integers are whole numbers and their negatives ($..., -3, -2, -1, 0, 1, 2, 3, ...$)
      • Whole numbers include natural numbers and zero ($0, 1, 2, 3, ...$)
        • Natural numbers (counting numbers) start at 1 and continue infinitely ($1, 2, 3, ...$)
    • Rational numbers include terminating decimals ($0.5, 0.25, 0.125$) and repeating decimals ($0.\overline{3} = 0.333..., 0.\overline{16} = 0.161616...$)
  • Irrational numbers cannot be expressed as a ratio of two integers
    • Include non-terminating, non-repeating decimals ($\pi, \sqrt{2}, [e](https://www.fiveableKeyTerm:e)$)

Representation of Real Numbers

• Number line: A visual representation of real numbers on a horizontal line
• Interval notation: A way to represent a set of numbers using parentheses or brackets
• Set theory: A branch of mathematics that deals with the properties of collections of objects (including real numbers)

Algebraic Operations and Expressions

Order of operations application

• PEMDAS mnemonic device guides order of operations
  1. Parentheses: Perform operations within parentheses first
  2. Exponents: Evaluate exponents, powers, and roots
  3. Multiplication and Division: Multiply and divide from left to right
  4. Addition and Subtraction: Add and subtract from left to right

Properties of real numbers

• Commutative property allows changing order of operands
  • Addition: $a + b = b + a$
  • Multiplication: $a \times b = b \times a$
• Associative property allows grouping operands differently
  • Addition: $(a + b) + c = a + (b + c)$
  • Multiplication: $(a \times b) \times c = a \times (b \times c)$
• Distributive property distributes multiplication over addition
  • $a(b + c) = ab + ac$
• Identity property leaves value unchanged when operating with identity element
  • Addition: $a + 0 = a$
  • Multiplication: $a \times 1 = a$
• Inverse property results in identity element when operating with inverse
  • Addition: $a + (-a) = 0$
  • Multiplication: $a \times \frac{1}{a} = 1$ for $a \neq 0$

Evaluation of algebraic expressions

• Substitute given values for variables in expression
• Apply order of operations to simplify resulting expression
• Evaluate simplified expression to find final value

Simplification of complex expressions

• Combine like terms by adding or subtracting coefficients of terms with same variables and exponents
• Factor out common factors from terms
• Simplify fractions by reducing numerator and denominator by greatest common factor (GCF)
• Apply properties of exponents
  • Multiply powers with same base: $a^m \times a^n = a^{m+n}$
  • Divide powers with same base: $\frac{a^m}{a^n} = a^{m-n}$
  • Power of a power: $(a^m)^n = a^{mn}$
  • Power of a product: $(ab)^m = a^m b^m$
  • Power of a quotient: $(\frac{a}{b})^m = \frac{a^m}{b^m}$ for $b \neq 0$
• Rationalize denominators containing radicals by multiplying numerator and denominator by conjugate of denominator
• Use absolute value to represent the distance of a number from zero on the number line

Arithmetic Operations and Polynomials

• Arithmetic operations (addition, subtraction, multiplication, and division) are fundamental in algebraic manipulations
• Polynomials are expressions consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents