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 one that 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 answer a question that asks you to find the irrational number presented as fractions. 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.

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 $0$.