Types of Numbers

There are, as I’m sure you know, a lot of numbers, infinitely many as it happens. Because of this mathematicians need a way to classify numbers, a way to group then and tell them apart from each other. We are going to take look at the different groups of numbers that exist.

Natural Numbers

Natural numbers – or the counting numbers – are probably the first type of numbers you will have come across. The Natural numbers are whole numbers greater than zero. These numbers are 1, 2, 3, 4, …, numbers that you can physically count. The Natural Numbers do not include negative values. For these we need another group.

Note: Some people include zero as a Natural Number, if unsure it is best to check with your teachers whether they do or not.


Integers are whole numbers, both negative and positive, including zero. These numbers are …, -3, -2, -1, 0, 1, 2, 3, …. Positive integers are integers greater than zero, and negative integers, not surprisingly, are integers less than zero.

Rational Numbers

Rational number are numbers that can be written as a fraction where both the numerator and denominator are integers. This means that both the top and bottom of the fraction are whole numbers. Examples of these would be \frac{1}{3} and \frac{23}{7}.

Irrational Numbers

Irrational Numbers are numbers that can’t be written as fractions, examples of these are ones such as \pi and \sqrt{2}. If we try to write these numbers as decimals they go on forever, with no recurring digits.

Square Numbers

Square Numbers are integers that can be written as the square of some other integers, ie. a product of an integer multiplied by itself. Examples of these are 4, (2×2) and 81, (9×9). Square Numbers can also be written in the form 5^2. This notation means 5 squared, which is 25.


Surds are numbers left in the form \sqrt{n}, where n is a positive integer that is not a square number. For more in-depth information about Surds be sure read our article Surds

Prime Numbers

Prime numbers are numbers greater than 1 that can only be divided by themselves and 1 to give an integer answer. Example of these are 2, 3, 5, 7, 11, 13, 17, …. Note that 2 is the first and only even prime number.

Note: Some courses class the number 1 as a prime number, if unsure it is again best to check with your teacher.

Real Numbers

Real numbers are all the numbers that you have ever come across, all of the rational and irrational numbers. All of these real numbers can be written in a finite or infinite decimal form, such as \sqrt{2} = 1.414213562… and \pi = 3.141592654 .