The Real Number System and Its Subsets

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Textbook Reference

This source is free and openly licensed. College Algebra 2e Section 1.1 opens Chapter 1.

Key idea

The real numbers are not one flat list. They are a set of nested boxes, each box sitting inside a larger one.

Counting numbers are inside the whole numbers, which are inside the integers, which are inside the rationals, which are inside the reals. Every time mathematics ran into a question the current box could not answer, a larger box was built. The whole numbers cannot answer “what is $3 - 5$”, so the integers were built to hold negatives. The integers cannot answer “what is $1 \div 3$”, so the rationals were built to hold fractions. The rationals cannot answer “what is the length of a diagonal of a unit square”, so the irrationals were added to complete the real line.

This means: when someone asks “what kind of number is this”, the honest answer is “the smallest box it fits in”. The number $5$ is a counting number, so it is also everything larger. The number $\sqrt{2}$ fits in no rational box, so it is irrational. Knowing the boxes tells you which rules are safe to use.

Prerequisite Check

This is a foundational skill, so the only thing assumed is grade-school arithmetic. Before this lesson, make sure you can do all of the following:

• Add, subtract, multiply, and divide whole numbers
• Recognize a fraction such as $\dfrac{3}{4}$ and a decimal such as $0.75$
• Read a number line from left to right
• Understand that a negative number such as $-3$ lies to the left of zero

If any of these is unfamiliar, a review of basic arithmetic will help before continuing.

Quick Reference

The nested sets of real numbers.

The deciding test for rational versus irrational. A real number is rational exactly when its decimal expansion either stops (terminates) or repeats a pattern forever. It is irrational when its decimal expansion goes on forever with no repeating pattern.

• $0.25$ terminates, so it is rational ($\frac{1}{4}$).
• $0.3333\dots$ repeats, so it is rational ($\frac{1}{3}$).
• $3.14159265\dots$ (the digits of $\pi$) never repeats, so it is irrational.

Key Concepts

1. Natural Numbers, Whole Numbers, and Integers

The natural numbers are the counting numbers $1, 2, 3, \dots$. Add the number $0$ and you have the whole numbers. Add the negatives of the natural numbers and you have the integers $\dots, -2, -1, 0, 1, 2, \dots$.

The pattern is one of inclusion. Every natural number is a whole number (move down one row and the answer stays Yes). Every whole number is an integer. The boxes nest.

Important: zero is a whole number but not a natural number. Some textbooks outside the United States include $0$ among the counting numbers. The OpenStax convention used here starts the natural numbers at $1$. When in doubt, state which convention you are using.

2. Rational Numbers

A rational number is any number that can be written as a ratio (a fraction) $\dfrac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. The word “rational” comes from “ratio”.

Every integer is rational, because an integer $n$ can be written as $\dfrac{n}{1}$. So the integer $-7$ is the rational number $\dfrac{-7}{1}$.

Rational numbers are exactly the numbers whose decimal form terminates or repeats.

Example 1. Show that $0.75$, $0.181818\dots$, and $-5$ are all rational.

• $0.75 = \dfrac{75}{100} = \dfrac{3}{4}$. A ratio of integers, so rational.
• $0.181818\dots$ repeats the block $18$ forever. Every repeating decimal is a ratio of integers; here it equals $\dfrac{18}{99} = \dfrac{2}{11}$.
• $-5 = \dfrac{-5}{1}$. A ratio of integers, so rational.

Important: division by zero is never allowed. In a fraction $\frac{a}{b}$, the denominator $b$ must not be zero. The expression $\frac{3}{0}$ is undefined; it is not a real number at all. Whenever a denominator could be zero, that value must be excluded.

3. Irrational Numbers

An irrational number is a real number that cannot be written as a ratio of two integers. Its decimal expansion runs forever with no repeating pattern.

The two irrational numbers a student meets first are $\sqrt{2}$ (the length of the diagonal of a square with side $1$) and $\pi$ (the ratio of a circle’s circumference to its diameter).

Example 2. Classify each number as rational or irrational: $\sqrt{9}$, $\sqrt{10}$, $\dfrac{22}{7}$.

• $\sqrt{9} = 3$, an integer, so it is rational. (A square root is irrational only when the number under the root is not a perfect square.)
• $\sqrt{10}$ is not a perfect square root; its decimal $3.16227\dots$ never repeats. Irrational.
• $\dfrac{22}{7}$ is a ratio of integers, so it is rational. (It is a common approximation of $\pi$, but it is not equal to $\pi$. The approximation is rational; the true value of $\pi$ is irrational.)

Important: the square root of a perfect square is rational. Students sometimes mark every square root as irrational. Check first whether the radicand (the number under the root) is a perfect square. $\sqrt{16} = 4$ is rational; $\sqrt{17}$ is irrational.

4. The Real Numbers as One Line

The real numbers $\mathbb{R}$ are all the rationals together with all the irrationals. Picture them as a single number line with no gaps: every point on the line is a real number, and every real number is a point on the line. The rationals fill the line densely, and the irrationals fill in the points the rationals miss.

Example 3. Place the smallest box label on each number: $-4$, $\dfrac{3}{8}$, $\sqrt{2}$, $0$, $\sqrt{25}$.

The smallest box is the most specific true name. Each number also belongs to every larger box: $-4$ is an integer and therefore also rational and also real.

5. Properties of Real Number Operations

Three properties of addition and multiplication are used constantly, often without naming them.

• Commutative: order does not matter. $a + b = b + a$ and $a \cdot b = b \cdot a$.
• Associative: grouping does not matter. $(a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
• Distributive: multiplication spreads over addition. $a(b + c) = ab + ac$.

Example 4. Use the distributive property to expand $3(x + 5)$.

\[ 3(x + 5) = 3 \cdot x + 3 \cdot 5 = 3x + 15 \]

The distributive property is the single most-used rule in all of algebra. Almost every “multiply it out” step is the distributive property at work.

Important: subtraction and division are NOT commutative. $5 - 3 = 2$ but $3 - 5 = -2$. $\frac{6}{2} = 3$ but $\frac{2}{6} = \frac{1}{3}$. Only addition and multiplication may be reordered freely.

Common Errors Summary

Common Misconceptions

Leveled Practice

Level 1: Direct Application

Problem 1. Classify each number using its smallest box: $12$, $-9$, $\dfrac{5}{6}$, $0$.

Problem 2. Is $0.1212121212\dots$ (the block $12$ repeating) rational or irrational?

Problem 3. Use the distributive property to expand $-2(a - 4)$.

Level 2: Reasoning

Problem 4. True or false: every integer is a rational number. Explain.

Problem 5. Decide whether $\sqrt{2} + 3$ is rational or irrational.

Level 3: Deeper Reasoning

Problem 6. A classmate says “$\sqrt{2}$ is just $1.41421356$, and that is a terminating decimal, so $\sqrt{2}$ is rational.” What is wrong with this argument?

Mastery Checklist

You have mastered this skill when you can do all of the following without referring to notes:

• Name the smallest set (natural, whole, integer, rational, or irrational) that a given number belongs to
• State why every integer is rational
• Decide whether a square root is rational or irrational by checking for a perfect square
• Recognize that a terminating or repeating decimal is rational and a non-repeating decimal is irrational
• Apply the commutative, associative, and distributive properties
• Explain why division by zero is undefined

Mental Model

Think of the real numbers as a set of nested boxes, smallest inside largest:

\[ \mathbb{N} \subset \text{whole} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \]

The irrational numbers sit beside $\mathbb{Q}$ inside $\mathbb{R}$ but outside every smaller box. To classify a number, find the smallest box it fits in; that is its most specific name. It automatically belongs to every box that contains that one.

Each box was built to answer a question the previous box could not: negatives gave us subtraction without limits, fractions gave us division without limits, and irrationals gave us lengths and ratios that fractions could not name. The real line is the finished picture, with no gaps anywhere.

Connections

Within MATH 114

• Order of operations: to evaluate an expression you must first know what kind of number each piece is and that division by zero is forbidden.
• Integer exponents: the exponent rules apply to all real bases; knowing the number system tells you when a result stays rational.
• Simplifying expressions: the distributive, commutative, and associative properties introduced here are the legal moves for rewriting any algebraic expression.

Toward Functions

• Domain and range: describing the set of allowed inputs of a function uses exactly this vocabulary of real-number subsets and the rule that denominators cannot be zero.

Audience Notes

For students who find math intimidating: You already know these numbers. This lesson only gives names to groups you have used since grade school. The one new idea is that the groups nest, like measuring cups that stack inside one another.

For students interested in proof: The classic proof that $\sqrt{2}$ is irrational assumes it equals a fully reduced fraction $\frac{a}{b}$ and derives a contradiction (both $a$ and $b$ would have to be even). It is a short, beautiful argument and a first taste of proof by contradiction.

For students interested in careers: Computers store rational numbers exactly but can only approximate irrational numbers, which is the source of rounding error in scientific software. Knowing which numbers are exact and which are approximate is a daily concern in engineering and data work.

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