Textbook

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!**

A real number is any number from $−∞$ (negative infinity) to $∞$ (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.

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., $−16 $.

A rational number is one that can be considered “neat.” Rational numbers include decimals that end (like $2.5$), decimals that repeat (like $3.555555…$ or $3.252525…$), 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 $12 $. These types of numbers are considered *irrational* numbers (like pi, it goes on forever without repeating or making a pattern $3.14159265…$). 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, $21 =0.5$, therefore $21 $ is a rational number. However, the number $40 /3=2.10818510678…$, since it is without a pattern, does not end, and is not an integer divided by an integer, can be considered irrational.

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 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 $∞$ (positive infinity). You can remember that you must include $0$ because it is a whole number.

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 $∞$ (positive infinity).

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 $24 $ are true?

- It is a real number.
- It is a rational number.
- It is an integer.
- It is a whole number.
- 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 $2$.

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

Sign up for free to take 3 quiz questions on this topic

All rights reserved ©2016 - 2024 Achievable, Inc.