Laws of arithmetics

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Laws of arithmetics are the specific set of axioms about operations with numbers.

Usually, the Laws of arithmetics are formulated for the natural numbers. However, many more complicated objects (integer numbers, rational numbers, real numbers, etc) are constructed in such a way, that for them, the same laws of arithmetics are valid; this required careful definition of operations with newly constructed numbers.

In general, the objects that obey the laws of arithmetics are called numbers. HOwever, the term number can be used for an object from a set, where not all element satisfy, for example, the commutativity; in Quantum Mechanics, such objects are called q-numbers, to distinguish them from the c-numbers, that are supposed to commute and can be interpreted as just complex numbers.