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.

Expl for blondes: Now a turn came to look, as numbers and units co-operate in mathematics. This piece I named quantity. More interesting things on the page "New Math".

Relativity of concept is "Any Number"

For the receipt of numerical axis does not matter, what from numbers are taken for any number: positive anymore units, positive less than units negative anymore minus units or negative less than minus units. Imposition reverse and mirror symmetries on any of these groups of numbers results in the receipt of all row of the real numbers.

From the choice of group of numbers as any number the results of mathematical actions will depend are different combinations of increase or diminishing of any number as a result of concrete mathematical action. In a table below the possible variants of concept "Any number" are marked just as there are corresponding to them fragments of numerical axis in traditionally assumed an air. For evidentness the increase of any number is doubled by a sign "+", diminishing - by the sign of "-", corresponding cells are distinguished by a different color.

Relativity of concept is Any Number

As be obvious from a table, addition and deduction are mirror symmetric in relation to a point "zero". An increase and division are mirror symmetric relatively two points are points "unit" and points "zero", here reverse symmetry is mirror symmetric in relation to a point "zero". All reasoning about priority and secondaryness of mathematical actions are an error. Symmetry of mathematical actions is considered in the separate article.

Expl for blondes: Farther we will consider units of measurements and mathematical actions. More interesting things on the page "New Math".

Number line

In mathematics it is accepted to represent numbers as a number line. We will consider transformation of number ray to the number line.

Reverse symmetry allows to get numbers less unit. As a point of reverse symmetry is unit, this symmetry does not depend on units of measurement. Reverse symmetry reflects relativity of concepts "greater than unit" and "less than unit" at comparison of two numbers. In case of comparison of two numbers without fail it is needed to accept one of these numbers as unit of measurement.

After introduction of unit of measurement we get the absolute system of coordinates for any unit of measurement. Unit of measurement on a picture is represented in the traditionally accepted variant - with imposition on the area of reverse numbers. At imposition of mirror symmetry the point of that is a zero, we enter negative numbers and get the relative system of coordinates. All enumerated transformations are represented on a picture below, where the sign of endlessness is mark any number.

Number line

Expl for blondes: Farther we will consider relativity of concept "Any number". More interesting things on the page "New Math".

Multiplication chart

Multiplication chart

Multiplication chart, printable multiplication table, multiplication tables chart 1 to 10 for you from mathematics for blondes.

Look similarly The multiplication table 1 to 20.

Relativity is in mathematics

All distinctions between two numbers or two units of measurements come to light only at comparison of two numbers or two units of measurements. All results of comparison are relative, as depend on what from two elements takes up basis at comparison. Relativity of results of comparison is represented different kind by symmetries. If up basis of symmetry a point takes "zero", then mirror symmetry ensues. If up basis of symmetry a point takes "unit" - reverse symmetry ensues. For units of measurements of corners reverse symmetry is transformed in perpendicular symmetry that is possessed by the values of trigonometric functions.

All distinctions between two numbers or two units come to light only at comparison of two numbers or two units. All results Comparison of two any numbers is not possible without the presence of the general founding unit comes forward as that. For the location of any numbers in order of growth in modern mathematics as unit of numbers the number systems are used: binary, ternary, octal, decimal, sexadecimal et al. Comparing of two numbers to the different grounds is not possible without bringing them over to the general founding.

Comparison of two numbers at different units of measurements becomes possible at the use of the third unit of measurement - one of the number systems, for example, decimal.

Result of comparison of two numbers is described by concepts "greater than" and "less than". Relativity of comparison of numbers is expressed in that the result of comparison depends on that, what number takes up basis at comparison. For example, if to compare numbers 2 and 3, we will get two results:

2 less than 3
3 greater than 2

On the first place it is accepted to write down a number that takes up basis at comparison, on the second is a that number it is compared to that. The results of comparison possess property of mirror symmetry - at the change of founding a result changes on opposite. The point of mirror symmetry is equality of two compared numbers. The results of comparison of two numbers are analogical to the relative system of coordinates:

less than - equal - greater than
minus - zero - plus

Comparison of two units of measurements is possible at presence of general point "zero". The result of comparison of two different units of measurements can be a conclusion about perpendicularity or parallelism of these units of measurements. Parallelism or perpendicularity of one unit of measurement in relation to other is concepts relative.

Expl for blondes: we will examine farther, as well as where a Number line appeared from. More interesting things on the page "New Math".

Some concepts of mathematics are continuation

A point is this reserved space with the radius of curvature equal to the zero (Expl for blondes: I do not understand clearly, that means this phrase. But I know exactly, that she is correct and very useful for us, when we will begin to examine mathematical principles of teleportation. To ride on an own car even prestige - it not so already prestige. In fact you however will escape farther than this planet. And here with you, even with blondes, the same, that happened to the dinosaurs will happen sooner or later - nature you will kill. Where will you escape from a submarine boat, even if this boat measuring with a planet?). Any space consists of endless amount of points. Through any point of space it is possible to conduct an endless amount mutually perpendicular lines. All points of space possess properties of both zero and unit, that allows without difficulty and arbitrarily to impose any relative system of coordinates and apply any type of symmetry in any point of space. In any point of space equality is executed: a zero is equal to unit. Equalization of point 0 = 1. Properties of zero and unit for one point of space can not show up simultaneously in one system of coordinates.

A line is this open-space with the radius of curvature equal to unit divided by a zero, consisting of separate points.

In mathematics it is necessary to distinguish the next types of corners : corner of scale, trigonometric corner, corner of turn.

Corner of scale is a corner scope from 0 to 90 degrees. The corner of scale can equal a zero, but 90 degrees can not equal. This corner reflects quantitative changes within the limits of one unit of measurement. Any changes of corner of scale can not cause the quality change of unit of measurement.

A trigonometric corner is a corner scope from 0 to 90 degrees. A trigonometric corner can equal both a zero and 90 degrees. This corner reflects dependences between units of measurements (project properties of space) and condition of quality changes of units of measurements. Dependence of quantitative changes of units of measurements on a trigonometric corner is expressed by trigonometric functions. The quality changes of units take place at the values of trigonometric functions equal to the zero and unit divided by a zero.

A corner of rotation is a corner that can have any values. In a range from 0 to 90 degrees the corner of rotation numeral can coincide with a trigonometric corner or corner of scale. The corner of rotation reflects the circular moving without the change of quantitative or quality descriptions of unit.

A direct corner differs from all other corners that the mutual projection of two intersecting lines is a point. For all other values of corner the projection of one line on other is a line. At coal equal to the zero, lines coincide. Mathematical unit of corners is a corner equal 45 degrees. This unit of corners submits to the rules of the binary number system.

Expl for blondes: Thereon the set of clever mathematical words is closed and we pass to consideration of mathematical mechanism - that, as and why works in mathematics. We will begin our excursion with relativity is in mathematics. More interesting things on the page "New Math".

Some concepts of mathematics

Expl for blondes: "Some concepts of mathematics" are my crib on your own, darling. Deciding some mathematical question, I often had to remember the own non-standard decisions of other questions. What is long not to dig in the memory, I collected all most important moments at the beginning of cycle of reasons of "Bases of mathematics". Some articles from this cycle will bring us over to the conclusions that is already writtenin here.

Equal sign reflects dependence of causality in the surrounding world. (Expl for blondes: is the example of application of the first basic axiom of mathematics.) If 2 х 2 = 4, it not nearly means that 4 = 2 х 2. There is an endless great number of decisions resulting in an exactly such result - four. 2 х 2 - only one of these decisions.

In mathematics there are three basic equalities:

0 = 0
1 = 1
0 = 1

All physical laws and mathematical equalizations are taken to one of these equalities. (Expl for blondes: do you think why I so easily succeeded to find the decision of the undecided equalizations (in Rassian)? Because I beforehand know an answer - the decision of any mathematical equalization is taken to one of these equalities. If know a problem specification and right answer, decision to find much simpler. By the way, here one of remarks of physicists, about that I wrote in the article "Zero is equal to unit": "Sum of energy of substance and gravitational energy is saved, but this law of maintenance is unusual: this sum is equal to ZERO"!. Most strikes me circumstance that physicists result in mathematics, as old jade! But it must be quite the reverse - it mathematicians must explain to the physicists: that, as and why works in this world. Physics is an experimental base of mathematics. If physicists will find some exceptions from mathematical rules, means to the mathematicians it will be needed to correct mathematics.)

In mathematics it is possible to distinguish such basic elements: zero, unit, any number and unit of measurement.

Numbers reflect quantitative description anything. Any number is equal to any number - this property of numbers allows to distinguish them in the special group that it is accepted to designate a word "number". All separately taken numbers possess identical mathematical properties. (Expl for blondes: not surprised, most bad dream of any mathematician (all numbers are equal) is cruel mathematical reality. Do not be afraid, I do not gather to take from mathematicians their favourite toy that are numbers. I simply want to say an obvious thing: all of you know many most different toys (and child, and adults), but all unites them one property - it is possible to play by them.)

Any number is the positive real number more unit. If to one any number to add other any number the first number will increase. Just any number will increase at multiplying of him by other any number. If from any number to subtract other any number, the first number will diminish. If to divide one any number into other any number, the first number will diminish.

Unit is a number, but is not any number, as at multiplying and dividing by unit any number remains unchanging. Unit is a neutral element at an increase and division.

Geometrically any number is represented by a point. All numbers form a numerical ray with beginning in a point "unit". A numerical ray does not have an end. Any number can be designated by a sign "infinity", as any number can be how pleasingly great.

Units of measurement reflect quality description anything. Any unit of measurement is equal to any unit of measurement. All units of measurement possess identical mathematical properties. (Expl for blondes: In mathematics units of measurement symmetric to the numbers, will remember an axiom about symmetry.) For numbers universal units of measurement are the number systems: binary, decimal, sexadecimal to and other. (Expl for blondes: I think, for mathematicians it will be the real discovery. I in any way can not get used to that any writtenin number has a tail of unit of measurement is "abstract unit".)

Geometrically any unit of measurement is represented by a segment (by two points): point "zero" is this beginning of unit of measurement, point "unit" is an end of unit of measurement.

Zero is not a number, as at addition of zero to any number and deduction of zero from any number this number remains unchanging. (Expl for blondes: It there is that simple and elegant decision of problem with zero about that I talked before. I agree, it is another act of mocking above mathematical sacred objects. But, there be nothing to be done - beauty of mathematics requires victims. you only present, how many energy and paper we will economize, if we will not in every example on a division write a "denominator does not equal a zero". Environmentalists will be happy here!) Zero is a neutral element at addition and deduction. Zero is beginning of the absolute system of coordinates. In the relative system of coordinates zero is the point of mirror symmetry.

In a point "unit" takes place connection of unit of measurement with any numbers. Unit is the point of reverse symmetry in the absolute and relative systems of coordinates.

Expl for blondes: It is a yet not end. Tomorrow we will continue to examine some mathematical concepts. More interesting things on the page "New Math".