Number System

A number system, or the numeral system, is a mathematical way of representing a set of values using digits or symbols. It uniquely represents a number and helps perform mathematical operations: addition, subtraction, multiplication, and division.

It also helps to determine the value of any digit based on its position in the number and the base of the number system. 

Types

Based on the base of the number system, different types of number systems are used in calculations. However, 4 of them are most commonly used.

Here is a chart of the 4 main types of number systems with their base values and digits.

Decimal Number System (Base 10)

The decimal number system consists of digits 0 to 9 with a base of 10. Here, the first few successive places left of the decimal point represent units, tens, hundreds, and thousands. It is the number system we commonly use in real life.

45₁₀, 289₁₀, and 5781₁₀ are examples of decimal numbers. Here, the number 5781₁₀ can be represented based on its place value:

(5 × 10³) + (7 × 10²) + (8 × 10¹) + (1 × 10⁰)

= (5 × 1000) + (7 × 100) + (8 × 10) + (1 × 1)

= 5000 + 700 + 80 + 1

= 5781

Binary Number System (Base 2)

The binary number system uses digits 0 and 1. The numbers have base 2. 0 and 1 are called bits, and 8 such bits make a byte. Computers and other digital devices store data in bits and bytes. It is thus the number system used in computers.

10101₂, 111001₂, and 101010₂ are some examples of binary numbers.

Octal Number System (Base 8)

The octal number system uses numbers 0 to 7 with a base of 8. Thus, it has fewer digits than the decimal number system. Along with the binary numbers system, octal number systems are used in computers.

7₈, 47₈, and 1681₈ are examples of octal numbers.

Hexadecimal Number System (Base 16)

The hexadecimal number system uses sixteen digits and alphabets. It includes numbers 0 to 9, as in the decimal number system, and alphabets A to E with a base of 16. Here, the letters A to E represent numbers from 10 to 15. This number system is also used in computers to reduce the large string size of binary numbers.

9A5₁₆, 4E₁₆, and 6C2B₁₆ are some examples of hexadecimal numbers.  

Number System Conversion

A number can be converted from one number system to another according to our needs.

To convert numbers in binary, octal, or hexadecimal systems to other systems, we first convert them to decimal systems and then to the system we want to convert.

Check our articles on binary, decimal, octal, and hexadecimal number systems to learn how to convert numbers from one number system to another.

The table shows numbers 1 to 20 in binary, decimal, and hexadecimal form:

Binary	Decimal	Hexadecimal
0	0	0
1	1	1
10	2	2
11	3	3
100	4	4
101	5	5
110	6	6
111	7	7
1000	8	8
1001	9	9
1010	10	10
1011	11	A
1100	12	B
1101	13	C
1110	14	D
1111	15	E
10000	16	F
10001	17	11
10010	18	12
10011	19	13
10100	20	14

Solved Examples