Decimal
42
Binary
101010
Hex
2A
Octal
52
42
101010
2A
52
ASCII Character
*
8-bit — 3 bits set
Repeatedly divide by 2, record remainders (read bottom to top)Divide the decimal number by 2 repeatedly. The remainders (0 or 1) read from bottom to top form the binary representation.
Where:
n= The decimal number to convertremainder= n % 2 at each step (0 or 1), forms binary digitsdecimal = Σ(bit × 2^position)Multiply each binary digit by 2 raised to its position (right to left, starting at 0), then sum all products.
Where:
bit= Each binary digit (0 or 1)position= Position index from right, starting at 0Inputs
Result
42 ÷ 2 = 21 R0, 21 ÷ 2 = 10 R1, 10 ÷ 2 = 5 R0, 5 ÷ 2 = 2 R1, 2 ÷ 2 = 1 R0, 1 ÷ 2 = 0 R1. Read remainders bottom to top: 101010.
Inputs
Result
1×128 + 1×64 + 1×32 + 1×16 + 1×8 + 1×4 + 1×2 + 1×1 = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255. This is the maximum value for an 8-bit byte.
Inputs
Result
1024 = 2¹⁰. In binary, this is a 1 followed by 10 zeros: 10000000000. This is exactly 1 KB (kilobyte) in computing. Hex: 400, Octal: 2000.
Inputs
Result
42 in binary = 00101010, 15 in binary = 00001111. AND compares each bit: both must be 1 to produce 1. Result: 00001010 = 10 decimal. AND is commonly used for bit masking.
Binary is base-2 using only 0 and 1. Computers use it because transistors have two states: on (1) or off (0). Everything in a computer (text, images, programs) is ultimately stored as binary. Each digit is called a "bit."
| Number System | Base | Digits Used | Example (decimal 10) |
|---|---|---|---|
| Binary | 2 | 0, 1 | 1010 |
| Octal | 8 | 0-7 | 12 |
| Decimal | 10 | 0-9 | 10 |
| Hexadecimal | 16 | 0-9, A-F | A |
Repeatedly divide by 2, record remainders. Read remainders from bottom to top. Example: 13 → 13÷2=6 R1, 6÷2=3 R0, 3÷2=1 R1, 1÷2=0 R1. Read up: 1101. So 13 in binary is 1101.
| Decimal | Conversion Steps | Binary |
|---|---|---|
| 5 | 5→2R1, 2→1R0, 1→0R1 | 101 |
| 10 | 10→5R0, 5→2R1, 2→1R0, 1→0R1 | 1010 |
| 25 | 25→12R1, 12→6R0, 6→3R0, 3→1R1, 1→0R1 | 11001 |
| 100 | 100→50→25→... | 1100100 |
Multiply each digit by its power of 2 (right to left: 1, 2, 4, 8, 16...), then add. Example: 1101 = 1×8 + 1×4 + 0×2 + 1×1 = 8+4+0+1 = 13.
| Binary | Calculation | Decimal |
|---|---|---|
| 101 | 1×4 + 0×2 + 1×1 | 5 |
| 1010 | 8 + 0 + 2 + 0 | 10 |
| 1111 | 8 + 4 + 2 + 1 | 15 |
| 10000 | 16 + 0 + 0 + 0 + 0 | 16 |
| 11111111 | 128+64+32+16+8+4+2+1 | 255 |
Key powers of 2: 2=10, 4=100, 8=1000, 16=10000, 32=100000, 64=1000000, 128=10000000, 256=100000000. Also: 255=11111111 (max 8-bit), 1023=10 ones (max 10-bit).
| Decimal | Binary | Significance |
|---|---|---|
| 1 | 1 | 2⁰ |
| 2 | 10 | 2¹ |
| 4 | 100 | 2² |
| 8 | 1000 | 2³ |
| 16 | 10000 | 2⁴ |
| 255 | 11111111 | Max 8-bit (1 byte) |
| 256 | 100000000 | 2⁸ |
Same as decimal but only 0 and 1. Addition: 0+0=0, 0+1=1, 1+1=10 (carry 1). Subtraction: Similar, with borrowing. Example: 101 + 11 = 1000 (5+3=8).
| Operation | Binary | Decimal Check |
|---|---|---|
| 101 + 10 | 111 | 5 + 2 = 7 |
| 101 + 11 | 1000 | 5 + 3 = 8 |
| 1111 + 1 | 10000 | 15 + 1 = 16 |
| 1010 - 11 | 111 | 10 - 3 = 7 |
Hexadecimal (hex) is base-16 using 0-9 and A-F. Each hex digit = 4 binary bits. F = 1111 = 15. Hex is easier to read than long binary. Color #FF00FF = 11111111 00000000 11111111.
| Hex | Binary (4 bits) | Decimal |
|---|---|---|
| 0 | 0000 | 0 |
| 5 | 0101 | 5 |
| 9 | 1001 | 9 |
| A | 1010 | 10 |
| F | 1111 | 15 |
| FF | 11111111 | 255 |
Binary is the foundation of all digital computing systems.
Every piece of data in a computer is ultimately stored as binary digits (bits).
Understanding binary helps with programming, computer science, and digital electronics.
Last Updated: Feb 13, 2026
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