Arithmetic with Rational Numbers

May 14, 2024

Take quiz

Review flashcards

Arithmetic with Rational Numbers

Overview

  • Focus on arithmetic for the TEAS (Test of Essential Academic Skills).
  • Main topic: Order of operations with rational numbers.

Order of Operations (PEMDAS)

  • **Parentheses: ** Perform operations inside parentheses first.
  • **Exponents: ** Calculate powers and square roots next.
  • **Multiplication and Division: ** Perform these from left to right. Neither takes precedence.
  • **Addition and Subtraction: ** Perform these from left to right. Neither takes precedence.

Example 1

Equation: 4 + (3 * 2) - 8 / 2

  1. Parentheses: 3 * 2 = 6
  2. Exponents: None
  3. Multiplication/Division: 8 / 2 = 4
  4. Addition/Subtraction: 4 + 6 - 4 = 6

Correct Answer: 6

Example 2

Equation: 15 + (3 + 2)^2 - 9 * 6 + 2^3

  1. Parentheses: 3 + 2 = 5
  2. Exponents: 5^2 = 25 and 2^3 = 8
  3. Multiplication/Division: 9 * 6 = 54
  4. Addition/Subtraction: 15 + 25 - 54 + 8 = -6

Correct Answer: -6

Practice Question

Equation: 5 * (2 + 3) - 4^2 + 6

  1. Parentheses: 2 + 3 = 5
  2. Exponents: 4^2 = 16
  3. Multiplication/Division: 5 * 5 = 25
  4. Addition/Subtraction: 25 - 16 + 6 = 15

Correct Answer: 15

Rational and Irrational Numbers

  • Rational Numbers: Numbers expressible as a ratio of two integers (denominator ≠ 0).
    • Example: 1 = 1/1, -7 = -7/1
  • Non-Whole Rational Numbers: Express fractions as decimal form.
    • Example: 3.75 = 375/100 or 15/4
  • Repeating Decimals: Can be expressed as fractions.
    • Example: 0.333... = 1/3

Irrational Numbers

  • Definition: Cannot be expressed as a simple ratio of two integers.
    • Examples: Pi (π), e, sqrt(2)
  • Properties: Non-terminating and non-repeating.
  • Common Misconception: Irrational numbers are rare.
    • Truth: Any non-perfect square root and combining irrational numbers with rational numbers usually results in irrational numbers.

Example

Which of the following is irrational?

  • 8/2 = 4 (rational)
  • sqrt(16) = 4 (rational)
  • sqrt(15) = 3.87298... (irrational)
  • 2.75 = 2.75 (rational)

Correct Answer: sqrt(15)

Ordering Rational Numbers

  • Method 1: Number Line: Plot numbers on a number line.
  • Method 2: Stacking: Stack numbers vertically by align decimal points.

Practice

Arrange from least to greatest: -3/4, 0.5, -1.5, 2/3, 1.75

  1. Plot on number line
  2. Convert to decimals for easier comparison

Result: -1.5, -3/4, 0.5, 2/3, 1.75

Comparing Rational Numbers

  • Use: Greater than (>), Less than (<), and Equal to (=).
  • Tip: Think of the symbols as a hungry alligator wanting to eat the larger number.

Practice

  1. 3/4 is not greater than 0.75 (False)
  2. -1/2 < 0.6 (True)
  3. 0.5 > 1/3 (True)
  4. 2.5 > 5/3 (True)

Correct Answer: 3/4 > 0.75 is not true