Foundations of Algebra

The Real Number System

Classifying numbers as natural, whole, integer, rational, or irrational.

Foundational20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

A Number Has More Than One Name

The number 5 is a natural number, a whole number, an integer, and a rational number, all at the same time. These aren't competing labels — they're nested categories, like how a single animal can be a poodle, a dog, and a mammal all at once. Knowing which categories a number belongs to tells you what kinds of operations and reasoning are safe to use with it, which matters as soon as equations start involving fractions, negatives, and roots.

Definition — Natural and Whole Numbers

Natural numbers are the counting numbers: 1, 2, 3, and so on. Whole numbers include everything in the natural numbers, plus 0. Neither includes negative numbers or fractions.

Definition — Integers

The whole numbers together with their negative counterparts: ..., −3, −2, −1, 0, 1, 2, 3, .... Integers have no fractional or decimal part.

Definition — Rational Numbers

Any number that can be written as a fraction of two integers, a/b, where b is not zero. This includes every integer (since 5 = 5/1), every terminating decimal (0.75 = 3/4), and every repeating decimal (0.333... = 1/3).

Definition — Irrational Numbers

A number that cannot be written as a fraction of two integers — its decimal expansion goes on forever without repeating. √2 and π are the two most common examples you'll encounter in this course.

Every rational number and every irrational number together make up the real numbers — essentially every number you can locate somewhere on a number line. Rational and irrational numbers never overlap: a number is one or the other, never both.

Number	Natural	Whole	Integer	Rational	Irrational
7	yes	yes	yes	yes	no
0	no	yes	yes	yes	no
−4	no	no	yes	yes	no
3/5	no	no	no	yes	no
√7	no	no	no	no	yes

Classifying five example numbers

Worked Example — Classifying a Number

Classify −8. It is not natural (natural numbers start at 1) and not whole (whole numbers aren't negative), but it is an integer, since it's a negative whole number. It's also rational, since it can be written as −8/1. It is not irrational, since it's rational.

Tip

A quick way to test whether a number is rational: try to write it as a fraction of two integers, or check whether its decimal form terminates or eventually repeats. If you can do either, it's rational.

Common Mistakes

Assuming every square root is irrational, such as calling √9 irrational.
√9 = 3, a whole number — it simplifies to a rational value. A square root is only irrational when the number under the root isn't a perfect square.
Thinking a negative number can't be rational.
Being negative and being rational are unrelated properties. −3/4 is negative and rational at the same time — rational only depends on whether the number can be written as a fraction of integers.

Key Takeaways

Natural ⊂ whole ⊂ integer ⊂ rational ⊂ real — each category is nested inside the next.
A rational number can always be written as a fraction of two integers; an irrational number never can.
Every real number is either rational or irrational, never both.

Summary

Classifying numbers gives you the vocabulary to describe exactly what kind of value you're working with. Next, you'll use the real numbers as the foundation for simplifying algebraic expressions.

Track your progress

Back to Foundations of Algebra Simplifying Algebraic Expressions→