big o - Why is multiplication n^2 time? -

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i've read operations such addition/subtraction linear time, , multiplication n^2 time. why true?

isn't addition floor(log n) times, when n smaller operand? same argument goes subtraction, , multiplication, if make program long multiplication instead of adding integers together, shouldn't complexity floor(log a) * floor(log b) , b operands?

the answer depends on "n." when addition o(n) , multiplication (with naïve algorithm) o(n^2), n length of number, either in bits or other unit. definition used because arbitrary precision arithmetic implemented operations on lists of "digits" (not base 10).

if n number being added or multiplied, complexities log n , (log n)^2 positive n, long numbers stored in log n space.


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