1.1 The Real Number System

Learning Objectives

After completing this section, you will be able to:

  1. Classify numbers into subsets of the real number system.
  2. Represent real numbers on a number line
  3.  Comparing Numbers with Inequalities
  4. Evaluate and interpret absolute value as distance from zero

Everyday Context

What kind of numbers do you use every day? Checking your bank balance, working with fractions in a recipe, or calculating a square root are all examples. Understanding what type of number you’re dealing with helps you know what rules apply and lets us communicate more effectively when talking about mathematics.

In this section, you’ll learn how mathematicians organize numbers into categories so we can better describe their properties and how they behave.

 

Classifying Numbers

The real number system is built in layers:

  • Natural Numbers (Counting Numbers): 1, 2, 3, …
  • Whole Numbers: Natural numbers plus 0.
  • Integers: Whole numbers and their negatives (…, –3, –2, –1, 0, 1, 2, 3, …).
  • Rational Numbers: A rational number is any number that can be written as a fraction of two integers, p/q, where q ≠ 0. This includes fractions, integers, terminating decimals (such as 0.45), and repeating decimals (such as 0.333…).
  • Irrational Numbers: Numbers that cannot be written as a fraction of integers. Their decimal expansions never end and never repeat (π, √2).
  • Real Numbers: All rational and irrational numbers combined.

All the numbers we use in this course are real numbers. Figure 1.1 illustrates how the number sets we’ve discussed in this section fit together.

A diagram showing the classification of real numbers. On the left side, nested rectangles illustrate subsets: Counting Numbers inside Whole Numbers, inside Integers, inside Rational Numbers. On the right side, a separate large region labeled Irrational Numbers is next to the Rational Numbers. Both Rational and Irrational Numbers are contained within the larger category labeled Real Numbers.
Figure 1.1A (Real Numbers by OpenStax licensed CC BY 4.0)

 

📌 Quick Facts

  • All integers are rational (for example, 3 = 3/1).
  • Terminating or repeating decimals are rational.
  • Non-terminating, non-repeating decimals are irrational.

Rational vs. Irrational Numbers

Rational Numbers:

  • Can be written as a ratio of integers.
  • Example: 7.3 = 73/10 → rational.
  • Example: –3 = –3/1 → rational.

Irrational Numbers:

  • Cannot be expressed as a ratio of integers.
  • Decimal form does not terminate and does not repeat.
  • Examples: π ≈ 3.14159…, √2 ≈ 1.41421…

📌 Note: Not every number is real. For example, √(–9) is not a real number because no real number squared equals –9. These belong to a different system called the complex numbers, which you’ll see in later courses.

Test:

  • If a decimal stops or repeats, it’s rational.
  • If it doesn’t stop and doesn’t repeat, it’s irrational. 

Let’s pause and look at a few examples before moving on.

 

Example 1.1.1

Classify the numbers: {0.47, 0.333…, √5, π}.

Solution
  • 0.47 → rational (terminates).
  • 0.333… → rational (repeats).
  • √5 → irrational (non-repeating, non-terminating).
  • π → irrational (non-repeating, non-terminating).

Example 1.1.2

Decide if each number is rational or irrational: 12, −4.5, 2/7.

Solution
  • 12 → an integer, so rational.
  • −4.5 → terminating decimal, so rational.
  • 2/7 → fraction of integers, so rational.

Example 1.1.3

Classify: √49 and √20.

Solution
  • √49 = 7, which is an integer → rational.
  • √20 ≈ 4.4721…, non-terminating and non-repeating → irrational.

Try it

Classify each number into the correct subset of the real number system.
Be sure to select all answers that apply for each question.
As you check your work, read the feedback to confirm your understanding of why each choice is correct or incorrect.

 

Numbers on the Number Line

The number line is a visual tool to compare and order numbers.

  • Integers are evenly spaced: … –2, –1, 0, 1, 2, …
  • Fractions and decimals fit between integers.
    • Example: 3/4 is between 0 and 1.
    • Example: 0.4 is between 0.3 and 0.5.
  • Irrational numbers are located by approximation.
    • Example: √2 ≈ 1.41 (between 1.4 and 1.5).

Rule: Moving right means larger numbers; moving left means smaller numbers.

 

Example 1.1.4

 Locate and label the following on a number line: 4,\frac{3}{4},-\;\frac{1}{4},-3,\frac{6}{5},-\;\frac{5}{2},\;\frac{7}{3}.

Solution

Locate and plot the integers, 4, -3

Locate the proper fraction 3/4 first. The fraction 3/4  is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot 3/4  Similarly plot -1/4

Now locate the improper fractions \frac{6}{5},-\;\frac{5}{2},\frac{7}{3}.  It is easier to plot them if we convert them to mixed numbers and then plot them as described above: \frac{6}{5}=1\frac{1}{5},-\;\frac{5}{2}=-2\frac{1}{2},\frac{7}{3}=2\frac{1}{3}.

A number line labeled from -6 to 6 with blue dots marking specific rational numbers: -3, -5/2, -1/4, 3/4, 6/5, 7/3, and 4. Each point is plotted in its correct position relative to the whole numbers.

Try It 

Locate each number on the number line by selecting the interval where it belongs.
Use your understanding of fractions and mixed numbers to estimate their positions.

 

Try it (Draggable)

Drag each number to the correct position on the number line.
Use your understanding of negative numbers, fractions, and mixed numbers to estimate where each value belongs.
Numbers to the left are smaller, and numbers to the right are larger.

 

Comparing Numbers with Inequalities

We use inequality symbols to compare numbers:

  • a < b: “a is less than b” (a is to the left of b).
  • a > b: “a is greater than b” (a is to the right of b).

 

Example 1.1.5A

Order each pair using < or >.

  1. –5 __ –2
  2. 3/4 __ 0.8
  3. √3 __ 1.7
Solution
  1. –5 < –2 (–5 is less than –2; placing these on a number line can help).
  2. 3/4 = 0.75, and 0.75 < 0.8, so 3/4 < 0.8.
  3. √3 ≈ 1.732, and 1.732 > 1.7, so √3 > 1.7.

Example 1.1.5B

Order each pair using < or >.

  1. –7 __ –12
  2. –4 __ 2
  3. –15 __ –9
Solution
  1. –7 is to the right of –12 on the number line, so –7 > –12.
  2. –4 is to the left of 2 on the number line, so –4 < 2.
  3. –15 is to the left of –9 on the number line, so –15 < –9.

Example 1.1.6

Compare each pair of numbers using < or >.

  1. 2/3 __ 0.7
  2. 5/8 __ 0.62
  3. 1.25 __ 6/5
Solution
  1. 2/3 ≈ 0.666…, and 0.666… < 0.7, so 2/3 < 0.7.
  2. 5/8 = 0.625, and 0.625 > 0.62, so 5/8 > 0.62.
  3. 1.25 = 1.25 and 6/5 = 1.2, so 1.25 > 6/5.

Example 1.1.7

Order each pair using < , >, or =.

  1. √50 __ 7
  2. √20 __ 4.5
  3. √36 __ 6
Solution
  1. √50 ≈ 7.07, and 7.07 > 7, so √50 > 7.
  2. √20 ≈ 4.47, and 4.47 < 4.5, so √20 < 4.5.
  3. √36 = 6, so √36 = 6.

Try It

 

 

 

Absolute Value

The absolute value of a number is its distance from 0 on the number line.

The absolute value of a number n is written as | n |.

For example,

  • -5 is 5 units away from 0 so | – 5| = 5
  • 5 is 5 units away from 0 so | 5 | = 5

Figure 1.2 illustrates this idea.

 

A number line showing points at -5, 0, and 5. Arrows and brackets indicate that both -5 and 5 are 5 units from 0. Labels read: “-5 is 5 units from 0, so |-5| = 5” and “5 is 5 units from 0, so |5| = 5,” illustrating the concept of absolute value.
Figure 1.1B The integers 5 and -5 are 5 units away from 0. (Image by OpenStax licensed CC BY 4.0)

The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from  on the number line is zero units.

 

Example 1.1.8

Evaluate.

  1. Evaluate |–25|
  2. Evaluate |0|
Solution
  1. |–25| = 25 (distance from 0 is 25).
  2. |0| = 0 (0 is at the origin).

Example 1.1.9

Simplify.

  1. Simplify: |7 – 12|
  2. Simplify: |–4 + 9|
Solution
  1. |7 – 12| = |–5| = 5. (Evaluate what is inside the absolute value first.)
  2. |–4 + 9| = |5| = 5.

Example 1.1.10

Compare the two expressions: |–18| and –|18|. Use an inequality symbol to indicate which is larger.

Solution

Evaluate each expression:

  • |–18| = 18
  • –|18| = –(18) = –18

So, |–18| > –|18|.

Try It

 

Key Takeaways

  • The real number system is organized in layers: Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real, with Irrational numbers also part of the real numbers.
  • Rational numbers can be written as a fraction of two integers. This includes integers, fractions, terminating decimals, and repeating decimals.
  • Irrational numbers cannot be written as fractions and have decimals that never end and never repeat (e.g., π, √2).
  • All integers are rational numbers, but not all rational numbers are integers.
  • The number line helps us visualize and compare numbers:
    • Numbers increase as you move to the right.
    • Numbers decrease as you move to the left.
  • Inequality symbols compare values:
    • < means “less than”
    • > means “greater than”
  • Absolute value represents distance from 0 on the number line and is always nonnegative.
  • Key test:
    • If a decimal terminates or repeats → rational
    • If a decimal does not terminate and does not repeat → irrational

Content from this section was adapted from “1.3 Add and Subtract Integers” and “1.8. The Real Numbers” in Elementary Algebra 2e by OpenStax and is used under a Creative Commons Attribution 4.0 International License. Access for free at https://openstax.org/details/books/elementary-algebra-2e.

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Essentials of Mathematics Copyright © 2026 by Scott McDaniel is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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