0.3The Real Number System
Classify numbers as natural, whole, integer, rational, or irrational. Compare and order real numbers on the number line.
Why This Matters
Understanding real number types helps you know which tools to use when solving problems. You'll encounter integers in statistics, rationals in chemistry, and irrationals like π in geometry and physics — knowing the difference matters.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do mathematicians organize all numbers into categories, and why does it matter which category a number belongs to?
Lesson Overview
Every number you use in algebra belongs to the real number system. Real numbers are organized into nested subsets: Natural numbers are the counting numbers (1, 2, 3, …). Whole numbers add zero. Integers add negative whole numbers. Rational numbers include any number that can be written as a fraction p/q where q ≠ 0 — this includes all terminating and repeating decimals. Irrational numbers cannot be written as fractions; their decimals go on forever without repeating (e.g., √2, π). Together, rational and irrational numbers make up all real numbers.
Worked Examples
Classify −4 in all applicable sets.
Is it natural? No (negative).
Is it a whole number? No (negative).
Is it an integer? Yes (negative whole number).
Is it rational? Yes (−4 = −4/1).
Is it real? Yes.
Classify 0.75 in all applicable sets.
0.75 = 3/4 — can be written as a fraction.
Not an integer (not a whole number).
Rational? Yes.
Real? Yes.
Classify √9 in all applicable sets.
√9 = 3 — a perfect square.
Natural? Yes. Whole? Yes. Integer? Yes. Rational? Yes. Real? Yes.
Order from least to greatest: −2, 1.5, −3/2, 0, √4
Convert all to decimals: −2, 1.5, −1.5, 0, 2
Order on number line: −2, −1.5, 0, 1.5, 2
Evaluate |−8| + |3| − |−2|
|−8| = 8
|3| = 3
|−2| = 2
8 + 3 − 2 = 9
Guided Practice
Answers are in the Answer Key section.
Guided Practice Video: The Real Number System
Watch the guided practice walkthrough for the real number system, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Classify 7 in all applicable number sets.
Hint: Start with the smallest set (natural) and work outward.
Classify −2/3 in all applicable number sets.
Hint: Can it be written as a fraction? Is it an integer?
Is √5 rational or irrational? Explain.
Hint: Try to find a fraction equal to √5. Is 5 a perfect square?
Order from least to greatest: 1/2, −1, 0.3, −0.75, 2
Hint: Convert fractions to decimals, then place on a number line.
Evaluate |−6| − |4| + |−1|
Hint: Replace each absolute value with its positive value first.
Key Vocabulary
Natural Numbers (ℕ)
The counting numbers: 1, 2, 3, 4, 5, … Does not include 0 or negatives.
Example: 1, 2, 3, 100
Whole Numbers (𝕎)
Natural numbers plus zero: 0, 1, 2, 3, 4, …
Example: 0, 1, 2, 50
Integers (ℤ)
Whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …
Example: −5, −1, 0, 4
Rational Numbers (ℚ)
Numbers expressible as p/q where p and q are integers and q ≠ 0. Includes all terminating and repeating decimals.
Example: 1/2, 0.75, −3, 0.333…
Irrational Numbers
Real numbers that cannot be written as fractions. Non-terminating, non-repeating decimals.
Example: √2 ≈ 1.41421…, π ≈ 3.14159…
Absolute Value
The distance of a number from zero on the number line. Always non-negative.
Example: |−5| = 5, |5| = 5
Practice Questions
Interactive Practice — 5 Questions
Which sets does −4 belong to?
Classify √9 in all applicable number sets.
Evaluate: |−8| + |3| − |−2|
Order from least to greatest: −2, 1.5, −3/2, 0, √4
Is 0.333… (0.3̄) rational or irrational?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Classify −3 in all applicable number sets.
Classify 0.5 in all applicable number sets.
Is √16 rational or irrational? Explain.
Order from least to greatest: −1, 0.5, −2, 3/4, 0.
Evaluate: |−7| − |3| + |−1|
Common Mistakes
Thinking all negative numbers are integers — e.g., classifying −1.5 as an integer.
Integers are whole numbers and their negatives (…, −2, −1, 0, 1, 2, …). −1.5 is rational but not an integer.
Assuming all fractions are rational — forgetting that the denominator cannot be zero.
A rational number is any ratio a/b where b ≠ 0. Division by zero is undefined.
Confusing irrational numbers with negative numbers — thinking √4 is irrational.
√4 = 2, which is a whole number. Irrational numbers cannot be expressed as a fraction, like √2 or π.
Placing a number in only one set when it belongs to several — e.g., saying 5 is only a natural number.
5 is a natural number, a whole number, an integer, a rational number, and a real number all at once.
Math Tips
Think of the number sets as nested boxes: Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real. If a number is in a smaller box, it is also in every larger box.
Quick test for rational vs. irrational: if the decimal terminates (0.25) or repeats (0.333…), it is rational. If it goes on forever with no repeating block (√2, π), it is irrational.
SAT/ACT tip: questions about number classification often ask which set a number does NOT belong to. Always check the most restrictive set first (natural numbers).
Perfect squares (1, 4, 9, 16, 25, …) have rational square roots. Non-perfect squares (2, 3, 5, 6, 7, …) have irrational square roots.