Hex to binary
| Decimal | Binary | Octal | Hex |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 8 | 1000 | 10 | 8 |
| 10 | 1010 | 12 | A |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
| 64 | 1000000 | 100 | 40 |
| 100 | 1100100 | 144 | 64 |
| 255 | 11111111 | 377 | FF |
| 256 | 100000000 | 400 | 100 |
| 1024 | 10000000000 | 2000 | 400 |
| 65535 | 1111111111111111 | 177777 | FFFF |
Hex to binary needs no arithmetic at all: just substitution. Each digit becomes its own four bits: 2 is 0010, A is 1010, F is 1111, so 2AF is 0010 1010 1111. Going back is the same in reverse: group the bits into fours from the right and read each group as one hex digit.
Each hexadecimal digit becomes exactly four binary digits, so no arithmetic is needed. Just substitution. 2AF becomes 0010 1010 1111. The mapping is exact because 16 is 2 to the fourth power.
How to convert hex to binary
This exact four-to-one mapping is the whole reason hexadecimal exists in computing. It gives a compact notation that a person can translate to and from binary without calculation, which matters when reading a register value or a memory dump. Octal has the same property with three bits, which is why it was common on machines with 12, 24 or 36-bit words, and why it faded once eight-bit bytes won.
Questions
1010. Each hex digit is exactly four bits.
Because 16 equals 2 to the fourth power, so one hex digit holds precisely four bits with no remainder.
Group the bits into fours from the right, then read each group as one hex digit.
11111111: eight bits, one byte, all set.
Yes, with three bits per digit. It suits word sizes divisible by three rather than by four.