Types of numbers

Complex numbers

A complex number is any real or nonreal number and is thus the most general category of numbers. It is made up of both a real part and an imaginary part, like this: $a+bi$ (or in normal words, real number $+$ imaginary number!).

You may think that the number $4$ doesn’t look similar to this definition, but in this case, the imaginary number is just equal to $0$! It looks like this: $4+0i$. So, every number that could be written actually includes both parts, although oftentimes the imaginary number ($bi$ in the formula above) is equal to $0$. Just remember that 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 considered imaginary, meaning that it does not have the variable $i$ (meaning imaginary) in it.

Non-real, imaginary numbers

A non-real or imaginary number is any number including the variable $i$ in it. Although the rules for imaginary numbers may say that $i^2=-1$, the expression $i^2$ is still an imaginary number. Another example is a square root of a negative number, e.g., $\sqrt{-16}$.

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 $b\ne 0$. A rational number can be considered “neat.” Rational numbers include decimals that end (like $2.5$), decimals that repeat (like $3.555555\dots$ or $3.252525\dots$), but do NOT include decimals that infinitely continue without a pattern. This kind of messy decimal always comes from a square root of a non-perfect square like $\sqrt{12}$. These types of numbers are considered irrational numbers (like pi, it goes on forever without repeating or making a pattern $3.14159265\dots$). It is also important that you think of rational numbers in terms of fractions. You may have to determine whether a number written in fraction form is rational or irrational. Any number written as a ratio of integers is always rational. For instance, $\frac{1}{2}=0.5$, therefore $\frac{1}{2}$ is a rational number. However, the number $\sqrt{40}/3=2.10818510678\dots$, since it is without a pattern, does not end, and is not an integer divided by an integer, can be considered 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. In order to be an integer, a number must not contain a decimal .

Whole numbers

Whole numbers are just as they sound, any non-negative number without a decimal. $20$ is a whole number, but $20.5$ is not a whole number. $0$ 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). You can remember that you must include $0$ because it is a whole number.

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 is a diagram that represents the categories of numbers that have been mentioned. You see that the first difference is between real and non-real numbers, then the categories get more and 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 see if the number fits the categories given. First, we simplify the fraction in order to see it easier, and it comes out to be $2$.

1. $2$ is a real number because it has no $i$ variable.
2. $2$ is a rational number because it is not an infinite decimal without pattern.
3. $2$ is an integer because it is a negative or positive whole number.
4. $2$ is a whole number because it has no decimal and is positive.
5. $2$ is a natural number because it is positive and is not $0$.