Number Systems
What is a Numbering System?
A numbering system is a way to represent numbers using specific symbols and rules. We use numbering systems in everyday life for counting, measuring, and performing calculations. Different numbering systems exist, each with a unique base that determines the number of symbols used. Understanding numbering systems is essential, especially in computing, where different systems are used for various applications.
Types of Numbering Systems
There are several numbering systems, but the most commonly used ones are:
2.1 Decimal Numbers (Base-10)
The decimal system is the numbering system we use daily. It consists of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The value of a digit depends on its position and the base (10) of the system.
Example: The decimal number 247 can be expanded as:
(2 × 10²) + (4 × 10¹) + (7 × 10⁰) = 200 + 40 + 7 = 247
2.2 Binary Numbers (Base-2)
- The binary system is used in computers and digital electronics. It consists of only two digits: 0 and 1. Every binary digit (bit) represents a power of 2.
- The binary number system is also known as the base 2 number system.
Example: The binary number 1011 can be converted to decimal as:
(1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11
2.3 Octal Numbers (Base-8)
- The octal system has eight digits: 0 to 7. It is used in computing as a shorthand for binary numbers since three binary digits correspond to one octal digit.
- The octal number system is also known as the base 8 number system.
Example: The octal number 345 can be converted to decimal as: (3 × 8²) + (4 × 8¹) + (5 × 8⁰) = 192 + 32 + 5 = 229
2.4 Hexadecimal Numbers (Base-16)
- The hexadecimal system has sixteen symbols: 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15). It is widely used in programming and digital systems for representing large binary numbers compactly.
- The hexadecimal number system is also known as the base 16 number system.
Example: The hexadecimal number 2F can be converted to decimal as:
(2 × 16¹) + (15 × 16⁰) = 32 + 15 = 47
3. Number Conversion
Decimal to Binary: Divide the decimal number by 2 repeatedly and note the remainder.
Example: Convert (52)10 into binary.
The Result is (110100)2
Binary to Decimal Conversion: Multiply each binary digit by powers of 2: Example
Binary to Octal conversion:-
- For conversion from binary to octal, the binary numbers are divided into groups of 3 bits each, starting at the binary point and proceeding towards left and right.
Binary to Hexadecimal conversion:-
- For converting a binary to a hexadecimal number, the binary number starting from the binary point, groups of 4 bits each, on either side of the binary point.
Dicimal to octal Conversion:
- To convert a decimal whole number to octal, divide the number by 8 again and again until the quotient becomes 0.
- To convert a decimal fraction to octal, multiply the fraction by 8 again and again until the result becomes 0 or until you get the required number of decimal places.
Decimal to Hexadecimal Conversion:
- The decimal to hexadecimal conversion is same as octal.
octal to binary Conversion
- To convert a given a octal number to binary, replace each octal digit by its 3- bit binary equivalent.
octal to decimal conversion:
- For converting an octal to a decimal number, multiply each digit in the octal number by the weight of its position
and add all the product terms.
Octal to hexadecimal conversion
- For conversion of octal to Hexadecimal, first convert the given octal number to binary and then to hexadecimal.
Hexadecimal to binary conversion:
- For conversion of hexadecimal to binary, replace hexadecimal digit by its 4 bit binary group.
Hexadecimal to decimal conversion:-
- For conversion of hexadecimal to decimal, multiply each digit in the hexadecimal number by its position
weight and add all those product terms.
Hexadecimal to Octal conversion:-
- For conversion of hexadecimal to octal, first convert the given hexadecimal number to binary and then binary number to octal.
1’s Complement
1’s complement is a method of representing negative numbers in binary. It is found by flipping all the bits of a binary number (changing 0s to 1s and 1s to 0s).
Example: Binary: 1010 → 1’s complement = 0101
2’s Complement
The 2’s complement is found by adding 1 to the 1’s complement.
Example: Binary: 1010 → 1’s complement = 0101 Add 1 → 0110 (2’s complement)
Binary Addition
Binary addition follows these rules:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (carry 1 to the next column)
Binary Subtraction
Binary subtraction is done using:
Direct subtraction method.
2’s complement method.
Example: 1101 – 101 = ? Convert 101 to 2’s complement: 101 → 010 → add 1 → 011 Now add: 1101
011 +1101 = 1000
Binary Multiplication
Binary multiplication follows these rules:
0 × 0 = 0
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1
Compiled By Er. Basant Kumar Yadav