Why is 2s complement of 000 equal to 111, but 9s complement of 000 is not 888?

I'm pretty confused so I hope I don't mix up the different terms here.

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The two's complement representation of decimal 0 is simply 000

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The two's complement of 000 is 111

• I imagine that complementing a number is equivalent to flipping bits in binary

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The nine's complement of 000 is 999

• This is what confuses me. Are two's complement and nine's complement similar (except for the base change obviously)?

• If they are, then I'd expect the nine's complement of 000 to be 888 because 8 is the biggest digit in radix 9 and therefore the complement operation would assign the highest digit (8) to the lowest value input (0) [I imagine a folding from the center]

Obviously this is totally wrong but I'm not sure which part I've misunderstood.

You are very confused due what is simply poor terminology, to be honest. Both your statements 2 and 3 are false due to the same misunderstanding.

For each base $b$ there are two mainstream variants of the 'complement', the radix complement and the diminished radix complement.

The two most common bases in computer science are base $2$ and base $10$. Confusingly, the definitions usually used are:

• one's complement: the diminished radix complement of base $2$

• two's complement: the radix complement of base $2$

• nine's complement: the diminished radix complement of base $10$ (not $9$!)

• ten's complement: the radix complement of base $10$.

StackExchange Computer Science Q#135048, answer score: 32