Efficient realization of basic arithmetic operations on big natural numbers is usually performed by representing big numbers as arrays of digits in a chosen base, so that arithmetic operations on digits are performed exactly and quickly in computer arithmetics. Addition and subtraction algorithms, which simulate ”manual” addition and subtraction, have linear complexity, depending on number length. Also, these algorithms are optimal. On the other hand, ”manual” multiplication and division algorithms have quadratic complexity, and they are not optimal. At the beginning of this thesis, we described the classic multiplication algorithm and proved that the complexity of this algorithm is quadratic. We got the first improvement in complexity by designing an algorithm with the ”divide and conquer” strategy. By splitting numbers in 2 or more pieces, and with usage of polynomial interpolation and evaluation, we got an algorithm whose complexity is less than quadratic. This algorithm is called Karatsuba’s algorithm, in honor of A. Karatsuba, who invented this algorithm. We have also implemented the classic and Karatsuba’s algorithm in Java programming language, and showed an example of improvement in speed by using Karatsuba’s algorithm, instead of the classic algorithm. An even better multiplication algorithm is the Schönhage–Strassen algorithm, which uses the Fast Fourier Transform for polynomial multiplication. We have one chapter dedicated to describing this algorithm which has the complexity \(O(n \log n \log \log n)\). This is the fastest multiplication algorithm, so far. At the end, we have described a general division algorithm, based on fast multiplication, which uses Newton’s method with progressive accuracy improvement. We have proved that the complexity of this algorithm depends linearly on the complexity of the chosen multiplication algorithm