Types of numbers

Complex numbers

A complex number is any real or nonreal number, so it’s the most general category of numbers. A complex number has two parts:

• a real part
• an imaginary part

You can write a complex number in the form $a+bi$, where:

• $a$ is the real part
• $b$ is the real number coefficient of the imaginary part
• $i$ is the imaginary unit

A number like $4$ might not look like it fits this form, but it does. Its imaginary part is just $0$:

• $4=4+0i$

So every number you can write has a real part and an imaginary part (even if the imaginary part is $0$). Every number is a complex number.

Real numbers

A real number is any number from $-\infty$ (negative infinity) to $\infty$ (positive infinity). In other words, a real number is any number that is not imaginary, meaning it does not include the variable $i$ (for imaginary).

Non-real, imaginary numbers

A non-real (or imaginary) number is any number that includes the variable $i$.

Although the rules for imaginary numbers say that $i^2=-1$, the expression $i^2$ is still written using $i$, so it is still an imaginary number.

Another example is the square root of a negative number, such as \sqrt-16$.

Rational numbers

A rational number is any number that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b\ne 0$.

Rational numbers include:

• decimals that end (like $2.5$)
• decimals that repeat (like $\mathrm{3.555555...}$ or $\mathrm{3.252525...}$)

Rational numbers do not include decimals that go on forever without repeating. Those “messy” decimals often come from square roots of non-perfect squares, like $\sqrt{12}$.

Numbers with decimals that go on forever without repeating are called irrational numbers (like pi, which continues without repeating or forming a pattern: $\mathrm{3.14159265...}$).

It’s also important to think of rational numbers in terms of fractions. You may need to decide whether a number written in fraction form is rational or irrational.

• Any number written as a ratio of integers is always rational.

For example, $\frac{1}{2}=0.5$, so $\frac{1}{2}$ is rational.

However, the number $\sqrt{40}/3=\mathrm{2.10818510678...}$ does not end and does not repeat. Since it cannot be written as a ratio of integers, it is irrational.

Irrational numbers

An irrational number is a number that cannot be written as a fraction of integers. Its decimal form goes on forever without repeating.

Examples:

• $\sqrt{2}$
• $\sqrt{12}$
• π
• 0.101001000100001… (no pattern)

Irrational numbers combine with rational numbers to form the set of real numbers.

Integers

Integers are any negative or positive whole number, including zero. To be an integer, a number must not contain a decimal.

Whole numbers

Whole numbers are any non-negative number without a decimal. For example, $20$ is a whole number, but $20.5$ is not.

$0$ is included in the list of whole numbers, but negative numbers are not. So, the range of whole numbers is any non-decimal number from $0$ to $\infty$ (positive infinity).

Natural numbers

Natural numbers are the most specific category of numbers. A natural number is any positive whole number not including $0$. So, the range of natural numbers is any non-decimal number from $1$ to $\infty$ (positive infinity).

Diagram

This diagram represents the categories of numbers that have been mentioned. The first split is between real and non-real numbers. Then the categories become more specific until they reach natural numbers.

Let’s try an example:

Which statements listed below about the number $\frac{4}{2}$ are true?

1. It is a real number.
2. It is a rational number.
3. It is an integer.
4. It is a whole number.
5. It is a natural number.

Let’s check which categories the number fits. First, simplify the fraction:

• $\frac{4}{2}=2$

Now classify $2$:

1. $2$ is a real number because it has no $i$.
2. $2$ is a rational number because it can be written as a fraction of integers (for example, $2=\frac{2}{1}$).
3. $2$ is an integer because it is a positive whole number (and integers include negative numbers, positive numbers, and $0$).
4. $2$ is a whole number because it has no decimal and is non-negative.
5. $2$ is a natural number because it is positive and is not $0$.