Lesson 1.7

Classifying Real Numbers

Not all numbers are created equal. The real number system is a family tree — from the counting numbers you learned as a child all the way to the mysterious irrationals like .

Introduction

Every number you encounter in algebra is a real number. Real numbers can be organized into subsets: Natural, Whole, Integers, Rational, and Irrational. Knowing which category a number belongs to helps you understand its properties.

Past Knowledge

You know how to work with whole numbers, fractions, and decimals.

Today's Goal

Classify any number as Natural, Whole, Integer, Rational, or Irrational.

Future Success

Properties of these number sets determine what operations are valid in equations.

Key Concepts

1. The Number Family Tree

Set	Symbol	Description
Natural		Counting numbers
Whole		Natural + zero
Integer		Whole + negatives
Rational		Can be written as
Irrational	—	Cannot be a fraction; non-repeating decimals

2. The Nesting Pattern

Every natural number is also whole, integer, rational, AND real.

3. Rational vs. Irrational — The Test

Rational ✓

Decimals that terminate (end) or repeat. Example: ,

Irrational ✗

Decimals that go on forever without repeating. Example:

Worked Examples

Example 1: Classify a Whole Number

Basic

Classify .

Check Each Set

Natural? Yes — it's a counting number.
Whole? Yes — all naturals are whole.
Integer? Yes — all whole numbers are integers.
Rational? Yes — .

is Natural, Whole, Integer, Rational, and Real.

Example 2: Classify a Fraction

Intermediate

Classify .

Check Each Set

Natural? No — it's negative and not a whole number.
Whole? No.
Integer? No — it's not a whole value.
Rational? Yes — it's already a fraction .

is Rational and Real.

Example 3: Classify a Square Root

Advanced

Classify and .

, a perfect square. So it's Natural, Whole, Integer, Rational, Real.

is NOT a perfect square. (non-repeating).

is Irrational and Real.

Common Pitfalls

All Square Roots Are Irrational

is rational. Only roots of non-perfect squares are irrational.

Forgetting That Integers Are Rational

is rational because . Every integer can be written as a fraction.

Real-Life Applications

In computer science, choosing the right number type (integer vs. floating point) affects both speed and accuracy. Financial software uses exact rational arithmetic (fractions) to avoid rounding errors — because a rounding error of on millions of transactions adds up fast.