Math 8: Rational and Irrational Numbers

Overview

This lecture covers the distinction between rational and irrational numbers, decimal representations, and converting between fractions and decimals, including repeating and terminating decimals.

Note-Taking Setup

• Always write your class (Math 8), date, and name at the top of your notes.
• Title the notes with the module and lesson, e.g., "Module 10: Real Numbers, Lesson 1: Understand Rational and Irrational Numbers."

Rational and Irrational Numbers

• A rational number can be written as a ratio a/b, where a and b are integers, and b ≠ 0.
• Rational numbers can be written as fractions or decimals (either terminating or repeating).
• Terminating decimals have an end (e.g., 0.375).
• Repeating decimals go on forever with a repeating sequence (e.g., 0.111...).
• An irrational number cannot be written as a ratio a/b.
• Irrational numbers have non-ending, non-repeating decimals (e.g., π, √2).

Examples of Rational and Irrational Numbers

• 3/8 is rational because it equals 0.375, a terminating decimal.
• 0.2 is rational, equal to 1/5.
• 0.111... is rational, equal to 1/9, and can be written using ellipsis (...) or an overbar.
• π (pi) is irrational; its decimal never ends or repeats.
• √2 and √3 are irrational numbers; √4 is rational (equals 2).

Decimal Representations

• Repeating decimals can be shown with ellipsis (...) or an overbar (e.g., 0.235... or 0.235̅).
• Only decimals with a repeating pattern that goes on forever are rational.

Converting Decimals to Fractions

• To convert a terminating decimal to a fraction, use place value (e.g., 0.825 = 825/1000, then reduce).
• To reduce fractions, divide numerator and denominator by their greatest common factor (GCF).

Converting Repeating Decimals to Fractions

• Let x = the repeating decimal (e.g., x = 0.5̅).
• Multiply both sides by 10 (for one repeating digit) or 100 (for two), align the decimal parts.
• Subtract to eliminate repeating part, then solve for x.
• Example: 0.5̅ = 5/9; 0.18̅ = 18/99.

Key Terms & Definitions

• Rational Number — a number expressed as a fraction a/b, with integers a, b and b ≠ 0.
• Irrational Number — a number that cannot be written as a simple fraction.
• Terminating Decimal — a decimal that ends.
• Repeating Decimal — a decimal with a digit or group of digits that repeats forever.
• Ellipsis — three dots (...) indicating a repeating pattern.
• Overbar — a line over digits in a decimal to show repeating digits.
• Greatest Common Factor (GCF) — the largest integer that divides both numerator and denominator.

Action Items / Next Steps

• Add success criteria for the examples in your notes during class.
• Review vocabulary and worked examples before the next lesson.
• Prepare any questions about converting between decimals and fractions.