Diophantine and Non-Diophantine Arithmetics: Operations with Numbers in Science and Everyday Life

Science and mathematics help people better to understand world, eliminating different fallacies and misconceptions. One of such misconception is related to arithmetic, which is so important both for science and everyday life. People think that their counting is governed by the rules of the conventional arithmetic and that other kinds of arithmetic do not exist and cannot exist. It is demonstrated in this paper that this popular image of the situation with integer numbers is incorrect. In many situations, we have to utilize different rules of counting and operating. This is a consequence of the existing diversity in nature and society and to represent correctly this diversity people have to utilize different arithmetics. To distinct them, we call the conventional arithmetic Diophantine, while other arithmetics are called non-Diophantine. Theory of non-Diophantine arithmetics is developed in the book of the author "Non-Diophantine arithmetics or is it possible that 2 + 2 is not equal to 4." In this work, some properties of non-Diophantine arithmetics are considered, as well as their connections to numerical computations and contemporary physics are explained.