Introduction to Number System
A number system is a group of digits or symbols used to express quantities for performing calculations. Long ago, people used fingers, sticks, pebbles, or knots in ropes to count. Later, calculators and computers were used for faster and more accurate calculations.
The base or radix of a number system defines the total number of digits used in that system. Different types of number systems are used depending on the application.
Types of Number System
Number systems are mainly classified into four types based on their base:
| Number System | Base (Radix) | Digits Used | Example |
|---|---|---|---|
| Decimal | 10 | 0,1,2,3,4,5,6,7,8,9 | 539 |
| Binary | 2 | 0,1 | 1011 |
| Octal | 8 | 0,1,2,3,4,5,6,7 | 345 |
| Hexadecimal | 16 | 0–9, A–F (A=10, B=11, … F=15) | 1AC |
1. Decimal Number System (Base 10)
Used in daily life.
Each digit has a place value: units, tens, hundreds, thousands, etc.
2. Binary Number System (Base 2)
Used by computers to store and process data.
Only two digits: 0 and 1.
3. Octal Number System (Base 8)
Used in computing for byte representation.
Digits: 0–7.
4. Hexadecimal Number System (Base 16)
Used in computers for memory addresses and data representation.
Digits: 0–9, A–F (A=10, B=11, … F=15).
Number System Conversions
Since humans use decimal but computers use binary, octal, or hexadecimal, conversions are important.
Decimal → Binary: Divide by 2, write remainder, repeat until quotient = 0, then read remainders bottom to top.
Binary → Decimal: Multiply each digit by its positional power of 2 and add.
Decimal → Octal: Divide by 8, write remainders, read bottom to top.
Octal → Decimal: Multiply each digit by its positional power of 8 and add.
Decimal → Hexadecimal: Divide by 16, write remainders, read bottom to top.
Hexadecimal → Decimal: Multiply each digit by its positional power of 16 and add.
Binary ↔ Octal: Group binary digits in 3s (right to left) → convert to octal.
Binary ↔ Hexadecimal: Group binary digits in 4s → convert to hex.6
Octal ↔ Hexadecimal: Convert via binary as an intermediate step.
Binary Calculations
Addition: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (carry 1)
Subtraction: 0-0=0, 1-0=1, 1-1=0, 0-1=1 (borrow 1)
Multiplication: 0×0=0, 1×1=1, 1×0=0
Division: 0÷1=0, 1÷1=1, 1÷0=undefined
Bits, Nibbles, Bytes, and Words
Bit: Smallest unit of data, 0 or 1.
Nibble: 4 bits.
Byte: 8 bits (1 character).
Word: Number of bits a processor can handle at once (8, 16, 32, 64 bits).
Data Measurement Units:
| Unit | Size |
|---|---|
| 1 Bit | 0 or 1 |
| 1 Nibble | 4 bits |
| 1 Byte | 8 bits |
| 1 KB (Kilobyte) | 1024 bytes |
| 1 MB (Megabyte) | 1024 KB |
| 1 GB (Gigabyte) | 1024 MB |
| 1 TB (Terabyte) | 1024 GB |
| 1 PB (Petabyte) | 1024 TB |
| 1 EB (Exabyte) | 1024 PB |
| 1 ZB (Zettabyte) | 1024 EB |
| 1 YB (Yottabyte) | 1024 ZB |
Bit < Byte < KB < MB < GB < TB < PB < EB < ZB < YB
Summary
Number system: A set of symbols used to express quantities.
Decimal (0–9), Binary (0–1), Octal (0–7), Hexadecimal (0–9, A–F).
Computers mainly use binary, octal, and hexadecimal systems.
Bits, nibbles, bytes, and words are basic units of data in computers.
Conversion between number systems is essential for human-computer interaction.
Technical Terms
Number system: Set of symbols for calculations.
Binary number: Digits 0 and 1.
Octal number: Digits 0–7.
Decimal number: Digits 0–9.
Hexadecimal number: Digits 0–9 and letters A–F.
Base/Radix: Total digits in a number system.
Byte: 8 bits.