Time complexity is one of the most important factors in the evaluation of an algorithm’s efficiency. Classical algorithms for multiplication and division of numbers use quadratic time for computation. Therefore, they are good enough for execution of arithmetic operations on relatively small numbers. However, asymptotically fast algorithms are needed to multiply or divide relatively big ones. The Karatsuba algorithm, described in this paper, is one of those algorithms. It has better, or smaller time complexity than the classical algorithm for multiplication of natural numbers. We can notice a significant improvement for relatively big numbers with numerous digits. That is why we say that the algorithm is asymptotically fast. The algorithm for calculating the reciprocal value of a number is linearly dependent on the multiplication algorithm it uses. That means if we use an asymptotically fast multiplication algorithm when dividing two numbers, we have an asymptotically fast division algorithm for natural numbers. Given algorithms use the positional notation for representing natural numbers in an arbitrary base \( b \geq 2 \). An algorithm’s complexity does not depend on our choice of the base. An alternative way of representing numbers is the modular notation. When moduli are chosen correctly, the modular arithmetic is executed in linear time. However, when using the modular arithmetic, we must take into account the conversion from modular to positional notation and vice versa. Schoenhage has shown that there is a multiplication algorithm for natural numbers, that uses modular arithmetic. The algorithm includes necessary conversion from one notation to another and has a better complexity than the classical algorithm, but slightly worse one than the Karatsuba algorithm. We can conclude that the right choice of algorithm depends on our requirements and conditions in which it will be used, so the optimal algorithm is often the one that is just good enough.