Binary, Octal, Decimal, Hexadecimal conversions
Digital Fundamentals - How computers represent numbers
| System | Base | Digits |
|---|---|---|
| Binary | 2 | 0, 1 |
| Octal | 8 | 0-7 |
| Decimal | 10 | 0-9 |
| Hexadecimal | 16 | 0-9, A-F |
(1010)₂ = Binary
(12)₈ = Octal
(10)₁₀ = Decimal
(A)₁₆ = Hexadecimal
Also written as: 0b1010, 012, 10, 0xA
Convert 25 to binary:
25 ÷ 2 = 12 remainder 1
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read remainders bottom-up: 11001
(25)₁₀ = (11001)₂
(1101)₂ = ?
Position: 3 2 1 0
Binary: 1 1 0 1
Value: 2³ 2² 2¹ 2⁰
= 8 + 4 + 0 + 1 = 13
(1101)₂ = (13)₁₀
Binary: 110101
Group: 110 | 101
Octal: 6 | 5
Answer: (65)₈
Binary: 11010101
Group: 1101 | 0101
Hex: D | 5
Answer: (D5)₁₆
Hex values: A=10, B=11, C=12, D=13, E=14, F=15
| Decimal | Binary | Octal | Hex |
|---|---|---|---|
| 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 |
| 15 | 1111 | 17 | F |
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (0, carry 1)
Example:
1011 (11)
+ 0110 (6)
------
10001 (17)
Use 2's complement!
A - B = A + (2's complement of B)
Flip all bits
5 = 0101
1's complement = 1010 = -5
1's complement + 1
5 = 0101
2's complement = 1010 + 1 = 1011 = -5
To subtract: Add 2's complement
7 - 5 = 7 + (-5)
= 0111 + 1011 = 10010
Discard overflow: 0010 = 2 ✓
| Conversion | Method |
|---|---|
| Dec → Bin | Divide by 2, collect remainders |
| Bin → Dec | Sum of (bit × 2^position) |
| Bin → Oct | Group by 3 bits |
| Bin → Hex | Group by 4 bits |
| 2's Comp | Flip bits + 1 |
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