What are Number Systems?
A number system is a system representing numbers. It is also called the system of numeration and it defines a set of values to represent a quantity. These numbers are used as digits and the most common ones are 0 and 1, that are used to represent binary numbers. Digits from 0 to 9 are used to represent other types of number systems.

Definition of Number Systems
A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner. The value of any digit in a number can be determined by a digit, its position in the number, and the base of the number system. The numbers are represented in a unique manner and allow us to operate arithmetic operations like addition, subtraction, and division.

Types of Number Systems
There are different types of number systems in which the four main types are:

Binary number system (Base – 2)
Octal number system (Base – 8)
Decimal number system (Base – 10)
Hexadecimal number system (Base – 16)
We will study each of these systems one by one in detail.

Types of Number Systems

Binary Number System
The binary number system uses only two digits: 0 and 1. The numbers in this system have a base of 2. Digits 0 and 1 are called bits and 8 bits together make a byte. The data in computers is stored in terms of bits and bytes. The binary number system does not deal with other numbers such as 2,3,4,5 and so on. For example: 100012, 1111012, 10101012 are some examples of numbers in the binary number system.

Binary Number System

Octal Number System
The octal number system uses eight digits: 0,1,2,3,4,5,6 and 7 with the base of 8. The advantage of this system is that it has lesser digits when compared to several other systems, hence, there would be fewer computational errors. Digits like 8 and 9 are not included in the octal number system. Just as the binary, the octal number system is used in minicomputers but with digits from 0 to 7. For example: 358, 238, 1418 are some examples of numbers in the octal number system.

Octal Number System

Decimal Number System
The decimal number system uses ten digits: 0,1,2,3,4,5,6,7,8 and 9 with the base number as 10. The decimal number system is the system that we generally use to represent numbers in real life. If any number is represented without a base, it means that its base is 10. For example: 72310, 3210, 425710 are some examples of numbers in the decimal number system.

Decimal Number System

Hexadecimal Number System
The hexadecimal number system uses sixteen digits/alphabets: 0,1,2,3,4,5,6,7,8,9 and A,B,C,D,E,F with the base number as 16. Here, A-F of the hexadecimal system means the numbers 10-15 of the decimal number system respectively. This system is used in computers to reduce the large-sized strings of the binary system. For example: 7B316, 6F16, 4B2A16 are some examples of numbers in the hexadecimal number system.

Hexadecimal Number System

Conversion Rules of Number Systems
A number can be converted from one number system to another number system. Like binary numbers can be converted to octal numbers and vice versa, octal numbers can be converted to decimal numbers and vice versa and so on. Let us see the steps required in converting number systems.

Conversion of Binary / Octal / Hexadecimal Number Systems to Decimal Number System
To convert a number from the binary/octal/hexadecimal system to the decimal system, we use the following steps. The steps are shown by an example of a number in the binary system.

Example: Convert 1001112 into the decimal system.

Solution:

Step 1: Identify the base of the given number. Here, the base of 1001112 is 2.

Step 2: Multiply each digit of the given number, starting from the rightmost digit, with the exponents of the base. The exponents should start with 0 and increase by 1 every time as we move from right to left. Since the base is 2 here, we multiply the digits of the given number by 20, 21, 22 , and so on from right to left.

Number System binary

Step 3: We just simplify each of the above products and add them.

Binary to Decimal Number System

Here, the sum is the equivalent number in the decimal number system of the given number. Or, we can use the following steps to make this process simplified.

100111 = (1×25) + (0×24) + (0×23) + (1×22) + (1×21) + (1×20)

= (1×32) + (0×16) + (0×8) + (1×4) + (1×2) + (1×1)

= 32 + 0 + 0 + 4 + 2 + 1

= 39

Thus, 1001112 = 3910.