Have you ever wondered how the world would be if there was no way to represent the values of birthdays, scores, money, time or weight? This would make life a lot harder as there would be no means of representation of magnitude or quantity to calculate these values.

**Numbers** are an important part of our daily lives. We use numbers during the day for counting items, money, time, etc. Different types of number systems have been used over time for different purposes, but the decimal number system is the most common.

In this article we will discuss numbers and types of numbers in the decimal number system, to make you understand various types of numbers in mathematics and what they are used to express.

**Number Systems**

Before we go in-depth, let’s define a number.

A **number **in mathematics can be defined as an arithmetic value that is assigned to an object to represent quantity. A number can be represented either with special figures called digits such as “11” or written in words such as “eleven”.

As mentioned earlier, there are various types of **number systems** with different ways of representing them. A number system represents a given set of numbers by using digits or other symbols in a specific order. The various types of number systems are explained below.

**Decimal or Base 10 Number System**

The decimal number system is the most common type of number system. It is what we generally use all over the world for everyday operations. It is called base 10 because it consists of 10 single digits from 0 to 9 i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each digit represents a power of 10 and the position of the digits in a given number determines the digit’s value.

The decimal number system is also known as the base 10 number system. The position of the digit could determine if it is of units, tens, hundreds or thousands value counted from right to left. For example, the number 1996 can be broken down as:

1 0 0 0 – One Thousand (Thousand Value)

9 0 0 – Nine Hundred (Hundred Value)

9 0 – Ninety (Tens Value)

6 – Six (Unit value)

**Unary or Base 1 Number System**

The unary or base 1 number system is the simplest type of number system. It uses a tally symbol (a vertical stroke) to represent natural numbers. In this system, numbers are represented by a symbol repeated a couple of times to specify a value.

The numbers 1, 2 and 3 can be represented as:

1 = /

2 = / /

3 = / / /

**Binary or Base 2 Number System**

The binary number system is also known as the base 2 number system. It has a base of 2 and consists of only two digits: 0 and 1 to represent numbers. This is the base which most computing systems use to perform operations.

Here are a few decimal numbers and their binary equivalents:

Decimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Binary | 0 | 1 | 10 | 11 | 100 | 101 | 110 | 111 | 1000 | 1010 |

**Octal or Base 8 Number System **

The octal or base 8 number system has a base of 8. It uses the digits 0 – 7 to represent numbers consisting of 0, 1, 2, 3, 4, 5, 6 and 7. Octal numbers are often used in the development of computer applications.

**Hexadecimal or Base 16 Number System**

The hexadecimal or base 16 number system has a base of 16. It uses the first numbers from the decimal number system i.e. 0 – 9, with the following numbers represented by the first six letters of the Alphabet from A – F.

The following table is a representation of hexadecimal numbers with their equivalent in the decimal system:

Hexadecimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |

Decimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

**Types of Numbers in Mathematics**

The different types of numbers in mathematics depend on the properties they have and the method by which they are represented on the number line. **Numbers** can be grouped into 7 major types which are **integers, natural numbers, rational **and** irrational numbers, whole numbers, real numbers **and** complex numbers**.

Here are the types of numbers with examples:

**Natural Numbers**

**Natural numbers** are the most basic type of numbers we learnt as kids and did math operations on. Natural or counting numbers are made up of a set of positive integers from 1 to infinity. A set of natural numbers is represented with the letter “**N**” and can be denoted as:

N = {1, 2, 3, 4, …}

Some examples of natural numbers are 13, 78, 98 and 46.

**Integers**

**Integers** are a combined set of all whole numbers and the negative set of natural numbers. Integers consist of all numbers lying between negative infinity and positive infinity. The integer set is represented by the letter “**Z**” and can be denoted as:

Z = {-4, -3, -2, -1, 0, 1, 2, 3, 4}

Some examples of integers are 0, -87, 9, 20 and 28.

**Whole Numbers**

**Whole numbers** consist **of all natural numbers including zero.** Whole numbers are also regarded as a set of non-negative integers i.e. all numbers from 0 to infinity. Whole number sets are represented by the letter “**W**” and can be denoted as:

**W = {0, 1, 2, 3, 4, 5, …}**

Some examples of whole numbers are 0, 87, 90 and 72.

**Rational Numbers**

**Rational Numbers **are numbers that can be expressed in the form of a fraction. Rational numbers can also be referred to as a ratio of one number to another.

If we consider the number 0.9, it can be written in the fractional form of 9/10 making it a rational number. Rational numbers can either be positive or negative and are represented by the letter “**Q**”.

Some examples of Rational numbers are 0.9, -2, 0.75 and 5.

Rational numbers are the foundation of fractions as they are used to represent non-whole numbers as a ratio of two integers. This ratio of two numbers is known as a fraction when written as a quotient. For example, the rational number 0.9 can be written as 9/10 in a fraction. Fractions depend on rational numbers for their existence.

**Deep Dive** – Fractions

**A fraction **denotes parts of a whole piece with a value represented in the form **p/q** where **q ≠ 0**. All fractional numbers are rational numbers but not all rational numbers are** **fractions.** **

Fractions are expressed as a quotient like this: 7/10 where 7 is the numerator, 10 is the denominator and the numerator is divided by the denominator giving 0.7. 0.7 is a rational number.

There are two types of fractions which are: proper fractions where the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.

For example, 22/4 and 7/2 are improper fractions while 1/2 and 3/8 are proper fractions.

**Irrational Numbers**

**Irrational numbers** are numbers that cannot be expressed in the form of a fraction. Irrational numbers cannot be written as a ratio of two integers. They are represented by the letter “**P**” and the digits continue to infinity. Some examples of irrational numbers are π, √2 and √5.

**Real Numbers**

**Real numbers** are the combination of the set of all rational and irrational numbers including all the numbers that can be written in decimal form. Real numbers can be **positive integers,** **negative integers**, **fractions or decimal** numbers. A set of real numbers is represented by the letter “**R**”. Some examples of real numbers are π, √2, -2, 0.75 and 5.

**Complex Numbers**

A **complex number** is the combination of a set of imaginary numbers and a real number. Real numbers can be represented as **a+ bi**, where **a **and **b** are **real numbers** and **i** is an **imaginary** **number.**

A set of complex numbers can be represented with the letter “**C**”. Complex numbers are applied to the concept of periodic motion in physics. Some examples of complex numbers are 2 + 3i, 1 + j and √5 + 4i.

**Imaginary Numbers**

**Imaginary Numbers** are a **subset of complex numbers that are not real numbers**. When imaginary numbers are squared, we get a negative result meaning the imaginary number is the square root of a negative number i.e. i = √-1. The imaginary part of complex numbers is denoted by the letter Z.

**Key Points**

- Numbers are essential as they are used to represent the magnitude or quantity of a lot of things in our daily lives.

- The most common number system is the decimal or base 10 numeral system.

- Numbers are grouped into seven major types based on their properties and how they are represented on the number line.

- Natural numbers are positive integers starting from 1 to infinity. Integers are a combination of whole numbers and negative natural numbers. Whole numbers are natural numbers plus zero.

- Rational numbers are numbers that can be represented in the form of a fraction, irrational numbers are numbers that cannot be represented as fractions, real numbers are a combination of rational and irrational numbers and complex numbers are numbers with real and imaginary parts.

**FAQ**

- What are the types of numbers?

The types of numbers are the different classifications of numeric values used to denote values based on their properties and their position on the number line. The different types of numbers are natural numbers, real numbers, rational numbers, irrational numbers, integers, whole numbers and complex numbers.

- What is the smallest number system?

The smallest number system is the unary number system which uses base 1 to express the numeric value of characters. It is represented with a tally symbol ( // ).

**Quiz**

- A number can be represented in _______.

Answer: digits

- What are the different types of number systems?

Answer:

- Unary number system
- Binary number system
- Decimal number system
- Hexadecimal number system
- Octal number system

- Rational numbers are the foundation of fractions

True or False

Answer: True

- Real numbers are the combination of the set of all __________

- Whole numbers and natural numbers
- Rational and irrational numbers
- Natural numbers and integers
- Rational numbers and real numbers

Answer: B

- What is the difference between a natural number and a whole number?

Answer: Natural numbers are positive integers starting from 1 to infinity while Whole numbers are natural numbers plus zero.