Types of Numbers

Numbers are symbols we use to count things. Numbers have been in use since the antiquity. We can’t live a normal life without numbers. We can’t even order a pizza, say our age or count the money without knowing the numbers. In absence of numbers, we’d be in the mental state of a 2-3 year toddler who has not yet learnt to count. In short, our live would be much difficult (to not say impossible) without numbers.

Natural Numbers
Natural Numbers Diagram

Natural Numbers (N)

The oldest category (set) of numbers is the set of Natural Numbers. The symbol of natural numbers is N. They are also known as Counting Numbers. Natural numbers therefore, are used to count things we use in daily life. They start from 1, then continue with 2, 3, 4 and so on. Natural numbers are always positive. We can list the natural numbers as below: 

N = {1, 2, 3, 4, ...}

The three dots on the right mean the set of Natural Numbers has a beginning (1) but no end.

Examples of the use of natural numbers in daily life: 

There are 16 girls and 12 boys in a class. 

Teddy has earned ₤250 during the last week.  

Integers (Z)

Integers include positive and negative whole numbers including 0. They can be listed as below: 

{..., -3, -2, -1, 0, 1, 2, 3, ...}

The three dots in the left and in the right mean this set of number has neither a beginning, nor an end. They extend from negative infinity to positive infinity. 

Examples of the use of integers in daily life:

Today is very cold. The temperature outside is -50C. 

Some places in Netherland are below the sea level. For example, Zuidplaspolder is 7m below the sea level. Thus, its altitude is -7m. 


Rational Numbers Diagram

Rational Numbers (Q)

Rational Numbers are also known as fractions. The symbol of rational numbers is denoted by Q. All rational numbers include 3 parts: 

The upper part which is known as numerator. It can be an integer, i.e. numerator can take positive, negative or zero value.

The lower part which is known as denominator. It can only be a natural number, i.e. can take only a positive value. 

The fraction bar is a symbol that is used to separate the two other parts of the fractions. 

When rational numbers have to be written as decimals, the decimal part is either a finite number or an infinite periodical one.

For example, 3.2759 represents a rational number as the decimal part is finite. If we write it as a fraction, we have:

Rational Numbers Example

7.8181818181... is also a rational number as the decimal part is a periodical number, although it is infinite. This is known as a recurring decimal. If written as fraction, it takes the form:

Rational Numbers Example

The method used to convert a recurring decimal into fraction is:

First, we write the original number by x. Thus, 7.81818181... = x

Then, we multiply it by a number such that we obtainagain the same numbers in the decimal part. Here, we multiply it by 100 because 100x = 781.81818181...

Then we subtract the first number from the second one to remove the recurring part:

Rational Numbers Examples

Thus, x = 774/99. 

Integers are also kind of rational numbers. They can be written as fractions with denominator 1. For example, 

Rational Numbers More Examples


Irrational Numbers (I) 

They are numbers that can neither be written as finite decimals nor as infinite periodical ones. For example, √5 = 2.2360679... is an irrational number. 

Real Numbers (R) 

Both rational and irrational numbers form a general set of numbers known as Real Numbers (R). 

The diagram of the sumber sets is shown below: 

The diagram of the sumber sets


Surds are numbers left in the form $\sqrt { n }$ where n is a positive integer that is not a square number. Surds are irrational numbers left in their root form. Most of the time this is in the form $\sqrt { x }$, although occasionally it can be $\sqrt [ 3 ] { x }$.