Types of Numbers

Mathematics has organized numbers into different categories, kind of like how we classify animals into species and families. Each type of number has specific characteristics, and understanding these categories helps you know what rules apply when you're working with them.

Think of the number system as a series of nested boxes. The smallest box contains the counting numbers you learned as a child. Each larger box includes all the numbers from the smaller boxes plus some new ones. By the time we get to the biggest box—the real numbers—we have almost every number you'll encounter 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 natural numbers plus zero. That's it—just one additional number, but it 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. Now we have positives, negatives, and zero.

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 $\frac{1}{4}$).

Key point: All integers are rational numbers. All fractions (positive and negative) are rational numbers. Many decimals are rational numbers.

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 other real numbers. This is why the real number line is often 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: $\pi, \sqrt{2}, e$

Basically, if you've thought of a number in this course so far, it's almost certainly a real number.

What's NOT a real number?

Imaginary numbers! When you try to take the square root of a negative number like $\sqrt{-1}$, you get an imaginary number (represented by $i$). But we'll save that topic for later.

Classifying Numbers

Let's practice identifying which categories a number belongs to.

Example 1: Classify the number 5.

Natural? Yes (counting number)
Whole? Yes (natural numbers are whole)
Integer? Yes (no fraction part)
Rational? Yes (can be written as $\frac{5}{1}$)
Real? Yes (all of the above are real)

Example 2: Classify the number -3.

Natural? No (negative)
Whole? No (negative)
Integer? Yes (whole number, including negatives)
Rational? Yes (can be written as $\frac{-3}{1}$)
Real? Yes

Example 3: Classify the number $\frac{2}{3}$.

Natural? No (not a counting number)
Whole? No (not a whole number)
Integer? No (has a fractional part)
Rational? Yes (already in fraction form)
Real? Yes

Example 4: Classify $\sqrt{7}$.

Natural? No
Whole? No
Integer? No
Rational? No (7 is not a perfect square)
Irrational? Yes
Real? Yes

Example 5: Classify 0.

Natural? No (natural numbers start at 1)
Whole? Yes (whole numbers include 0)
Integer? Yes
Rational? Yes (can be written as $\frac{0}{1}$)
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 also behave differently on number lines, in equations, and in real-world applications. Knowing your number types is like knowing the difference between tools in a toolbox—you need the right type for the job.

Practice Problems

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

8
-12
$\frac{5}{6}$
$\sqrt{25}$
$\sqrt{10}$
0
$\pi$
-0.75

Solutions

8: Natural, whole, integer, rational, real
-12: Integer, rational, real
$\frac{5}{6}$: Rational, real
$\sqrt{25} = 5$: Natural, whole, integer, rational, real
$\sqrt{10}$: Irrational, real
0: Whole, integer, rational, real
$\pi$: Irrational, real
-0.75 = $\frac{-3}{4}$: Rational, real

What's Next?

Now that you understand the different types of numbers, you're ready to dive deeper into specific concepts. In the next lesson, we'll explore absolute value—a way of measuring the distance of any number from zero on the number line.