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I'm taking a course in Automata and I have homework (I'll solve it myself, I just need to know the exact meaning of the statement) asking me to answer what the length of $x$ and $y$ is, where both $x$ and $y$ are binary numbers.

I don't quite understand what "length" refers to in this case. Does '000' mean that the length is $3$ ? If that's the case, then what does "length of xy" mean more specifically? Is that like saying "length of $x$ multiplied by the length of $y$" , so if length of $x$ is $2$, and length of $y$ is $3$, then the length of xy is $6$? Or would you have to multiply the individual numbers, like (000)(111) to get the length in that case (if that makes any sense...)?

If the question seems weird, it is because I don't really understand what is being asked. and I haven't found a sufficient answer in the course litterature.

Any help would be greatly appreciated.

Asaf Karagila
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Lobs002
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    I think it refers to number of bits in the binary representation. An x-bit number multiplied by y-bit number can result in (x+y) bit number – Srini Oct 06 '16 at 14:34
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    I assume that the symbol $xy$ denotes the concatenation of the string which $x$ represents with the string that $y$ represents. If $x=10$ and $y=11$, then $xy$ would be $1011$ instead of the product $110$. – Git Gud Oct 06 '16 at 14:35
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    Typically, the binary length of a number is equivalent to the minimum number of bits required in order to represent that number. So obviously the length of $0$ is $1$ and not $3$ as denoted in your example. That said, it is possible that the Automata takes bit strings of constant length, hence whatever input you inject has to be of a fixed length. In any case, the length of $xy$ is most likely the number o bits required in order to represent the product of $x$ and $y$. Observe that always $|x|+|y|-1\leq|xy|\leq|x|+|y|$. – barak manos Oct 06 '16 at 14:37
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    I second the concatenation interpretation. That's what it meant when I took an automata course. Consult your textbook and/or class notes, as that convention should be established somewhere. –  Oct 06 '16 at 14:37
  • Thanks for the answers. I'll continue searching through the book and class notes, but for now, I'll just assume it denotes the concatenation of the strings. It does make sense since we're talking automata (as opposed to the minimum bits required to represent that number). – Lobs002 Oct 06 '16 at 14:41

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