Number Systems in Computers

Computers use different number systems to represent and process data. The most common are:

1. Decimal Number System (Base 10)

Digits used: 0 to 9 (10 symbols).
It is the number system we use in daily life.
Example: 325 (3×10² + 2×10¹ + 5×10⁰).

2. Binary Number System (Base 2)

Digits used: 0 and 1 only.
Each digit is called a bit.
It is the language of computers (machine language).
Example: (1011)₂ = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 11 (Decimal).

3. Octal Number System (Base 8)

Digits used: 0 to 7.
Example: (537)₈ = 5×8² + 3×8¹ + 7×8⁰ = 351 (Decimal).

4. Hexadecimal Number System (Base 16)

Digits used: 0–9 and A–F (where A=10, B=11, …, F=15).
Used in computer memory addresses, color codes (like HTML).
Example: (2F)₁₆ = 2×16¹ + 15×16⁰ = 47 (Decimal).

Inter-Conversions of Number Systems

A. Decimal → Other Systems

Decimal → Binary
- Divide the decimal number by 2 repeatedly, write remainders in reverse.
- Example: (13)₁₀ → divide by 2 → 1101₂.
Decimal → Octal
- Divide by 8 repeatedly.
- Example: (125)₁₀ → 175₈.
Decimal → Hexadecimal
- Divide by 16 repeatedly.
- Example: (254)₁₀ → FE₁₆.

B. Binary → Other Systems

Binary → Decimal
- Multiply each digit by 2ⁿ (from right to left).
- Example: (10101)₂ = 21₁₀.
Binary → Octal
- Group binary digits in 3 bits (right to left).
- Example: (110101)₂ → (110)(101) → 65₈.
Binary → Hexadecimal
- Group binary digits in 4 bits.
- Example: (10111100)₂ → (1011)(1100) → BC₁₆.

C. Octal ↔ Hexadecimal (via Binary)

Convert Octal → Binary → Hexadecimal.
Example: (745)₈ → Binary: 111100101 → Group in 4 bits → (1 1110 0101) → 1E5₁₆.

Quick Reference Table

Decimal	Binary	Octal	Hexadecimal
0	0000	0	0
1	0001	1	1
2	0010	2	2
3	0011	3	3
4	0100	4	4
5	0101	5	5
6	0110	6	6
7	0111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F

✅ Summary for Exams

Binary = Base 2 (0,1)
Octal = Base 8 (0–7)
Decimal = Base 10 (0–9)
Hexadecimal = Base 16 (0–9, A–F)
Conversion → Repeated division or grouping method.