Converting a decimal number to binary is a basic skill in computing and digital electronics. It is necessary to understand Unix permissions (chmod 755 = 111 101 101), network addresses, bitwise operations and the internal representation of data.
The successive division by 2 method is the simplest for converting decimal to binary. Divide the number by 2 repeatedly, note the remainders (0 or 1) at each step, and read the remainders from bottom to top to get the binary representation. Alternatively, you can subtract successive powers of 2.
📐 Formula
Method: divide by 2 repeatedly → read remainders from bottom to top
📊 Reference table
| Decimal | Binary | Verification |
|---|---|---|
| 0 | 0 | — |
| 1 | 1 | 1 |
| 5 | 101 | 4+1 |
| 10 | 1010 | 8+2 |
| 15 | 1111 | 8+4+2+1 |
| 16 | 10000 | 16 |
| 42 | 101010 | 32+8+2 |
| 100 | 1100100 | 64+32+4 |
| 255 | 11111111 | 128+64+32+16+8+4+2+1 |
💡 Practical examples
Example 1: convert 13 to binary
13 ÷ 2 = 6 remainder 1 → 6 ÷ 2 = 3 remainder 0 → 3 ÷ 2 = 1 remainder 1 → 1 ÷ 2 = 0 remainder 1. Read remainders bottom to top: 1101. Check: 8+4+1 = 13 ✓
Example 2: Unix chmod 755 permissions
7 → 111 (rwx), 5 → 101 (r-x), 5 → 101 (r-x). chmod 755 in binary = 111 101 101: read-write-execute for owner, read-execute for others.
Example 3: represent an IP byte
An IP address byte like 192: 192 = 128+64 = 11000000 in binary. Subnets and CIDR masks rely on this representation.