Binary to decimal
| 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 |
From the right, the places are 1, 2, 4, 8, 16, 32, 64, 128. Add the ones where a bit is set: 10101010 has bits at 128, 32, 8 and 2, which totals 170. With practice a byte is readable directly, which is why the eight powers of two up to 128 are worth memorising.
Each binary place is a power of two, doubling from the right: 1, 2, 4, 8, 16 and so on. Add the values where a bit is set. 10101010 is 128 + 32 + 8 + 2 = 170.
How to convert binary to decimal
Grouping matters more than it looks. A long binary string is nearly unreadable, which is why it is conventionally written in groups of four or eight. 11001010 rather than the same eight digits run into their neighbours. Four bits is a nibble and maps to one hex digit; eight is a byte and maps to two. Read that example by its places: 128 + 64 + 8 + 2 = 202.
The conversion here reads the bits as an unsigned magnitude, and that assumption is the thing to check. 11111111 comes out as 255. If those eight bits came out of a signed byte, the top bit is a sign bit and the identical pattern means −1. Nothing in the digits says which reading is right; you have to know what produced them, and if the answer is a signed field, the two’s complement calculator is the tool that reads it properly.
The useful sanity check is the width: n bits hold values up to 2ⁿ − 1, so eight bits stop at 255 and sixteen at 65535, and a run of n ones is always exactly that maximum. If a decimal result is larger than the digit count allows, you have miscounted the bits somewhere.
Questions
Ten. The places are 8, 0, 2, 0.
255 — the largest eight-bit value.
As an unsigned value, yes. Read as a signed byte the same pattern is −1, because the top bit then carries the sign.
2ⁿ − 1. Eight bits reach 255, sixteen reach 65535, thirty-two reach 4294967295.
The bits row shows it. Each extra bit doubles the range.
Yes. Separators are ignored and the whole run is read as a single number. If the bits are one character per group, the binary to ASCII tool is the one you want.
Because a circuit reliably distinguishes two states but not ten. Everything above binary is a convenience for humans reading it.