Lesson 1.8

Ordering Real Numbers

Is bigger than ? To answer this, we place every number on the infinite ruler we call the Number Line.

Introduction

Ordering numbers is about location. Any two numbers can be compared: one is either to the left (smaller), to the right (larger), or at the exact same spot (equal).

Past Knowledge

You can classify numbers as integers, fractions, or decimals.

Today's Goal

Locate fractions and decimals on a number line and use inequality symbols (<, >, =).

Future Success

Inequalities (like ) are a major topic in algebra. Mastering order now makes that topic easy.

Key Concepts

1. The Inequality Symbols

Less Than

Points to the left (smaller).

Greater Than

Points to the right (larger).

Equal To

Same exact location.

2. Decimal Conversion Strategy

To compare fractions, convert them to decimals. It is much easier to see that than .

3. Negative Numbers Flip Logic

For positive numbers, bigger magnitude = larger ().
For negative numbers, bigger magnitude = smaller ().

Think of temperature: -10° is colder (lower) than -2°.

Worked Examples

Example 1: Compare Fractions

Basic

Compare and .

Convert to Decimals

Compare

0.666... is larger than 0.625.

Example 2: Negative Numbers

Intermediate

Compare and .

Visualize the Number Line

is further to the left (more negative) than .

Example 3: Ordering a List

Advanced

Order from least to greatest: .

Convert & Align

(Decimal)

Sort

Most negative first: . Then . Then . Then . Largest is .

Common Pitfalls

"Bigger" Negative Numbers

Thinking because 10 is bigger than 5. Remember: on the number line, left is always smaller.

Comparing Without Converting

Guessing that without checking. Convert to decimal () to prove it's actually less.

Real-Life Applications

Sorting algorithms (how Netflix ranks movies or Google ranks search results) rely entirely on these comparison rules. If a computer couldn't reliably say if A < B, it couldn't sort anything.