Units of measurements and mathematical actions

Symmetry of addition and deduction in relation to a point a zero testifies that these mathematical actions can be executed only with one unit of measurement. Actually, addition and deduction reflect comparison of three numbers - two present and result. For different unitsof measurements, getting the result of these mathematical actions not maybe, as numbers have different warrants, and their comparison is not possible. The geometrical mapping of addition and deduction will be considered additionally.

Symmetry of multiplication and division in relation to a point "unit" allow to present dividing as multiplying by a number reverse to any number:

а : b = a x 1/b

Just, multiplying by a number reverse to any number, it is possible to present as dividing by any number:

а х 1/b = a : b

Traditional determination of prime fractional number as a result of division of two integers of p and q interchangebly to the result of increase of integer of p on a number reverse to the integer of q:

p : q = p x 1/q

In further exposition term a "multiplication" will imply an increase and division in the generally accepted sense because of their complete symmetry and relativity of these concepts.

Multiplication is this co-operation of two different units of measurements at right angles in a point "zero". As a result of co-operation new unit of measurement appears with beginning in a point "zero", that causes the quality change of interactive units of measurements. A mathematical action opposite on sense to the multiplication is decomposition on factors. Decomposition is executed through trigonometric functions that can have numerical and not numerical (0 and 1/0) values. Simplest similarity of decomposition under a corner in 45 degrees - this square root. Decomposition and trigonometric functions are more detailed will be considered additionally.

An area (for example, area of rectangle) is a result of co-operation of two perpendicular units of measurements of length. The multiplication of parallel units of measurements is not possible (at the multiplication of lengths of two parallel parties of rectangle, measured in meters, it is possible to get meters square, but it is impossible to get an area). Mathematical properties of units of measurements will be considered additionally.

As in mathematics it is accepted to distinguish the separate sets of numbers that is partly included in a concept "Any number", it is at a desire possible to set forth mathematically exact determinations for some from them. For example:

unit and all numbers that can be got addition of units are named natural;

all numbers that can be got addition or deduction of units are named integers (at deduction of the same amount of units, that is present, numbers apply in a zero);

numbers being not whole are named a fractional.

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