A Number System is a method of expressing numbers using a consistent set of symbols.
- The classification of numbers based on how they are used and represented is called the number system.
Each number system has a base or radix, which determines:
- The type of number system,
- The total number of unique digits it can contain,
- The range of individual digits.
Radix/Base:
The base of a number system tells us how many unique digits it uses.
It also helps identify:
- The type of number system,
- The range of possible digits.
Formula to find the range of any number system:- For the Decimal number system (base 10):
- Range= 0 – (base – 1)
- eg. 0 – (10-1)
- =0 – 9
Types of number system:-
1.) Binary Number System (Base 2):
The binary number system is a number system that uses only two digits, 0 and 1, to represent all numbers.
- It is the fundamental language of computers and digital electronics, where each digit represents a power of 2.
- Example:
- The binary number 1011₂ equals 11 in decimal, calculated as:
- (1×2³ + 0×2² + 1×2¹ + 1×2⁰) = 8 + 0 + 2 + 1 = 11
2.) Octal Number System (Base 8):
The octal number system is a number system that uses eight digits, from 0 to 7, and is based on powers of 8.
- It was commonly used in older computing systems, especially for memory addressing.
- Example:
- The octal number 725₈ equals 469 in decimal, calculated as:
- (7×8² + 2×8¹ + 5×8⁰) = 448 + 16 + 5 = 469
3.) Decimal Number System (Base 10):
The decimal number system is the standard number system used by humans, which consists of ten digits, 0 to 9.
- It is based on powers of 10 and is used in everyday counting and calculations.
- Example:
- The decimal number 259₁₀ is already in base 10, and it represents:
- (2×10² + 5×10¹ + 9×10⁰) = 200 + 50 + 9 = 259
4.) Hexadecimal Number System (Base 16):
The hexadecimal number system is a base-16 number system that uses sixteen symbols, including digits 0 to 9 and letters A to F, where A represents 10 and F represents 15.
- It is widely used in computer programming and digital systems, especially for representing memory addresses and color codes.
- Example:
- The hexadecimal number 3F₁₆ equals 63 in decimal, calculated as:
- (3×16¹ + 15×16⁰) = 48 + 15 = 63
Number Base Conversion:
