One byte, one digit

Which bit is which bar

The eight bits of the byte map to the eight LEDs in order, starting at the bottom:

A is the top bar. B, C, D, E, F go round the outside clockwise from the top right. G is the bar across the middle. H is the dot.

So a 0 needs everything except the middle bar and the dot — A, B, C, D, E, F — which is 0b00111111. A 1 needs the two bars on the right, B and C, which is 0b00000110.

Slide through the characters and read the boxes against the bars. The drawing is made from the byte, not from a list of names, so what you see is what that number does.

A to F, and what seven bars cannot do

The sixteen characters in the figure are the hexadecimal digits, which is as far as a single seven-segment digit goes. Six of them are letters, and two are lower case — b and d — because an upper-case B on seven bars is an 8 and an upper-case D is a 0.

Everything else is a compromise or an impossibility. There is no K, no M, no V, no W and no X. Words on a seven-segment display are a party trick, and HELLO works only because those five letters happen to be among the ones that do.

The common-anode byte

Switch the figure to the 5161BS and the top row does not change. The bottom one does.

A segment table is written once, for a common-cathode digit, where a 1 bit means a lit bar. A common-anode digit lights on LOW, so the byte that goes down the wire is the same table with every bit flipped. You do not keep two tables; you keep one and invert it in the one place the byte leaves:

const uint8_t kDigits[10] = {
  0b00111111, // 0
  0b00000110, // 1
  0b01011011, // 2
  0b01001111, // 3
  0b01100110, // 4
  0b01101101, // 5
  0b01111101, // 6
  0b00000111, // 7
  0b01111111, // 8
  0b01101111, // 9
};

Adding the decimal point is a bit-or, not a second table: kDigits[3] | 0x80 shows 3., and it works before the inversion, because the inversion is the last thing that happens.

What the glass is telling you covers the shape you get when the inversion is set the wrong way, which is the easiest fault here to recognise and the easiest to fix.