Types of Numbers

Numbers are organized into categories, the same way animals are sorted into species and families. Each type of number has specific traits, and knowing the categories helps you tell which rules apply when you're working with a given number.

Think of the number system as a series of nested boxes. The smallest box holds the counting numbers you learned as a child. Each larger box contains everything in the smaller boxes, plus some new numbers. The largest box — the real numbers — contains almost every number you'll meet in algebra.

Natural Numbers

Natural numbers are the counting numbers: 1, 2, 3, 4, 5, and so on forever. These are the first numbers humans ever used, probably for counting sheep or tracking days. Some mathematicians call these "counting numbers" because that's exactly what they're for.

Natural Numbers: ${1, 2, 3, 4, 5, ...}$

Notice that natural numbers don't include zero. Why? Because you don't count "zero sheep" when you're counting your flock. Natural numbers are strictly positive whole numbers used for counting discrete objects.

Examples of natural numbers: 7, 42, 1,000,000

Not natural numbers: 0, -5, 3.7, $\frac{1}{2}$

Whole Numbers

Whole numbers are the natural numbers plus zero. That single addition makes a difference.

Whole Numbers: ${0, 1, 2, 3, 4, 5, ...}$

Zero is useful for representing "nothing" or "none." If you have zero apples, that's a whole number but not a natural number. Whole numbers still don't include negatives or fractions.

Examples of whole numbers: 0, 15, 238

Not whole numbers: -3, 4.5, $\frac{2}{3}$

Integers

Integers expand the number system to include negative whole numbers, giving you positives, negatives, and zero all in one set.

Integers: ${..., -3, -2, -1, 0, 1, 2, 3, ...}$

The word "integer" comes from Latin and means "whole" or "untouched." Integers are whole numbers that haven't been broken into pieces — no fractions or decimals allowed.

Integers are incredibly useful for representing real-world situations. Temperature can be negative (20 degrees below zero is -20). Your bank account can be negative (owing $50 is -50). Elevation can be negative (below sea level).

Examples of integers: -100, -7, 0, 8, 452

Not integers: 3.14, $\frac{1}{2}$, -0.5

Every whole number is an integer, and every natural number is an integer. But not every integer is a natural number (because of negatives and zero) or a whole number (because of negatives).

Rational Numbers

Here's where things get interesting. Rational numbers are any numbers that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.

Rational Numbers: Any number of the form $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$

The word "rational" comes from "ratio" — these are numbers that can be written as a ratio of two integers.

At first glance, you might think rational numbers are just fractions like $\frac{1}{2}$ or $\frac{3}{4}$. But integers are also rational! Why? Because any integer can be written as a fraction with a denominator of 1.

Examples:

$5 = \frac{5}{1}$ (integer, therefore rational)
$\frac{3}{4}$ (obviously rational)
$-2 = \frac{-2}{1}$ (integer, therefore rational)
$0.75 = \frac{3}{4}$ (can be written as a fraction)
$0.333... = \frac{1}{3}$ (repeating decimal, still rational)
$-\frac{7}{8}$ (rational)

Even repeating decimals like $0.666...$ are rational because they can be expressed as fractions (in this case, $\frac{2}{3}$). Terminating decimals like 0.25 are also rational (that's $\frac{1}{4}$).

The takeaway: every integer is rational, every fraction (positive or negative) is rational, and any decimal that either terminates or repeats is rational too.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a fraction of two integers. Their decimal representations go on forever without repeating any pattern.

The most famous irrational number is $\pi$ (pi), which starts as 3.14159... and continues forever without ever repeating. No matter how many digits you calculate, you'll never find a pattern that repeats.

Examples of irrational numbers:

$\pi = 3.14159...$
$\sqrt{2} = 1.41421...$
$\sqrt{3} = 1.73205...$
$e = 2.71828...$ (Euler's number)
$\sqrt{5}, \sqrt{7}, \sqrt{11}$ (most square roots)

Notice that not all square roots are irrational. $\sqrt{4} = 2$, which is rational. $\sqrt{9} = 3$, also rational. But if you take the square root of any number that isn't a perfect square, you get an irrational number.

How can you tell if a square root is irrational?

If the number under the square root sign is a perfect square (1, 4, 9, 16, 25, 36, ...), the square root is rational. Otherwise, it's irrational.

$\sqrt{16} = 4$ → rational
$\sqrt{15}$ → irrational (15 is not a perfect square)

Why are these numbers called "irrational"? Not because they don't make sense, but because they're not ratios of integers. No fraction of whole numbers can exactly equal $\pi$ or $\sqrt{2}$.

Real Numbers

Real numbers include both rational and irrational numbers. Essentially, any number you can place on a number line is a real number.

Real Numbers: All rational and irrational numbers together

The number line is completely filled with real numbers. Between any two real numbers, there are infinitely many others. That's why the real number line is drawn as a solid line: it's continuous, with no gaps.

Examples of real numbers:

All integers: -3, 0, 7
All fractions: $\frac{1}{2}, \frac{-5}{8}$
All decimals: 0.25, -3.7, 2.8181...
All irrational numbers: $\pi, \sqrt{2}, e$

If you've worked with a number anywhere in this course, it's almost certainly a real number.

What's NOT a real number?

Imaginary numbers. The square root of a negative number, like $\sqrt{-1}$, can't be plotted anywhere on the real number line. Those are imaginary numbers (denoted by $i$) and are covered in a separate lesson on imaginary numbers.

Classifying Numbers

Walk through a few numbers and identify every category each one belongs to.

Example 1: Classify the number 5. Think about it — which categories does 5 belong to? Show answerNatural (it's a counting number), whole, integer, rational (equals $\frac{5}{1}$), and real.

Example 2: Classify the number -3. Show answerNot natural (negative) and not whole (negative). Integer: yes. Rational: yes (equals $\frac{-3}{1}$). Real: yes.

Example 3: Classify the number $\frac{2}{3}$. Show answerNot natural, not whole, not integer. Rational: yes (already a fraction). Real: yes.

Example 4: Classify $\sqrt{7}$. Show answerNot natural, not whole, not an integer. Not rational (7 isn't a perfect square). Irrational: yes. Real: yes.

Example 5: Classify 0. Show answerNot natural (natural numbers start at 1). Whole: yes. Integer: yes. Rational: yes (equals $\frac{0}{1}$). Real: yes.

Why Does This Matter?

Understanding number types helps you know what operations you can perform and what properties apply. For instance:

When you divide two integers, you might not get an integer ($5 \div 2 = 2.5$), but you'll always get a rational number.
When you take the square root of a positive integer, you might get an irrational number.
When you see the word "integer" in a problem, you know you're working only with whole numbers (positive, negative, or zero), not fractions.

Different types of numbers behave differently on number lines, in equations, and in real-world applications. Knowing the categories is like knowing the difference between tools in a toolbox: each has the right job.

Practice Problems

Classify each number as: natural, whole, integer, rational, irrational, and/or real.

8 Show answerNatural, whole, integer, rational, real
-12 Show answerInteger, rational, real
$\frac{5}{6}$ Show answerRational, real
$\sqrt{25}$ Show answer$\sqrt{25} = 5$: Natural, whole, integer, rational, real
$\sqrt{10}$ Show answerIrrational, real (10 is not a perfect square)
0 Show answerWhole, integer, rational, real
$\pi$ Show answerIrrational, real
-0.75 Show answerRational, real (equals $\frac{-3}{4}$)

What's Next?

With the categories in hand, you can name what kind of number you're working with at any moment, and that often suggests what operations are safe. Next up: absolute value, which measures how far any number sits from zero on the number line — independent of whether it's positive or negative.