1.30.2017

Decomposition on items

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Addition

Lesson 16

Decomposition on items


If only the result of addition is known and items are unknown, then the sum can be spread out to items by means of the linear angular functions.

Decomposition of the sum on items. Mathematics For Blondes.
Decomposition of the sum on items

Transformation of a square of quantity to the work of two sums (see an example above), it is possible to execute with application of decomposition of result of multiplication to multipliers and items.

Transformation of a square to the work of the sums. Mathematics For Blondes.
Transformation of a square to the work of the sums

Similitudes can be useful when studying various natural phenomena to their best comprehension. Let's review an example of reproduction of living beings.

Asexual reproduction of live organisms can be described by means of decomposition of the sum on items. As a result of division of an organism A two self-contained organisms B and C.

Asexual reproduction. Mathematics For Blondes.
Asexual reproduction

Decomposition on items with corners about 45 degrees, is characteristic of unicells. For metaphytes the range of an angle of decomposition varies in wider limits (a vegetative reproduction, budding, fragmentation). The unit of measure at decomposition can be considered a physical body of an organism.

The beginning of life (zero) of similar organisms can be considered the moment of division of a parental organism. The termination of life (unit) can be considered characteristic division or death.

Sexual reproduction is described by means of multiplication. The moment of emergence of sexual reproduction can be described by means of the linear angular functions. At simultaneous reproduction of organisms A and B there was the common stream C which had signs of two parents.

Emergence of sexual reproduction. Mathematics For Blondes.
Emergence of sexual reproduction

What have to be the angles of decomposition for emergence of the common offspring? The most probable candidates in "invented" sexual reproduction it is a larger cage and a virus. The virus breeds in a cage. Along with cell division there was a division of a virus. The result was a new body. Or two body - male M and female W.

Male and female. Mathematics For Blondes.
Male and female
Molecule DNA which is available both for a cage, and for a virus could act as a basis for addition (unit of measure).

It is only one of a set of options of possible succession of events. From the moment of emergence of life on Earth until emergence of sexual reproduction in the Nature was enough time for the most different experiments.

Closing part

Further studying of properties of units of measure will help to understand better and more precisely to describe by mathematical methods various phenomena in the world around.

The separate ideas published in this work will be considered in more detail in the subsequent publications.

Gratitude

I express sincere gratitude to the parents and the daughter Inna for financial support of my working on with mathematics.

Addition

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
The linear angular functions

Lesson 15

Addition


As a result of addition of two different quantities the third quantity turns out. At addition of change occurs in a number domain, the area of units of measure does not change. Addition is possible only for parallel quantities with identical units of measure. Addition reflects the quantitative changes of quantities.

5а+3а=(5+3)а=8а

For realization of addition of two different quantities with units of measure in different scales (the corner of scale of units of measure is not equal to zero), it is necessary to change the scale of units of measure so that the scale corner between them equaled to zero. At the same time does not matter, the first item, the second or both changes at once.

It is impossible to put two identical numbers with different units of measure as the result does not make sense.

5а+5b=5(a+b)

Transformation of result of addition of pieces to the parties of a rectangle looks so.

Addition and rectangle. Mathematics For Blondes.
Addition and rectangle


Items can be presented as the party of a rectangle, then a half of perimeter of a rectangle is result of addition. For any sum it is possible to define the linear angular functions if items are known.

At the following lesson we will consider
Decomposition on items

9.03.2016

The linear angular functions

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Division

Lesson 14

The linear angular functions


If to consider terminating trigonometric functions as coordinates of points of a unit circle in a Cartesian coordinate system, then the linear angular functions are coordinates of points of the chord connecting circle cross points to a coordinate. The sum of coordinates of any point of this chord is always equal to unit.

The linear angular functions. Mathematics For Blondes.
The linear angular functions


In mathematician of a concept, similar to the linear angular functions, are used since ancient times - it is division whole on a part. In the modern world an analog are percent.

The explanation for readers of this website. For myself I called the linear angular functions "linos" and "loses". How I thought up these names? Took designation of a sine and cosine. Visually they quite well differ. In each designation I replaced the first letter with the Latin letter "l" from the word "line". It turned out quite nicely. But to solve to you. Whether these functions in mathematician will get accustomed and as they will be called - time will show. I just offer one more mathematical tool for the description of reality.

At the following lesson we will consider
Addition

Division

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Decomposition on factors

Lesson 13

Division


Contrary to the standard opinion, division is not mathematical operation. This solution of a standard task of finding of one of factors if other factor and result of multiplication is known. In ancient Babylon the fraction was considered as result of multiplication of number to inverse other number. Even in the modern mathematics there is no division of one fraction into other fraction, this operation is replaced with multiplication of a dividend to fraction, inverse to a divider.

Division can be considered as a projection of result of multiplication along one of factors. For example, length is a projection of the area along width, width is a projection of the area along length.

The most interesting in this plan is the speed which is measured by the lengthiest, divided into time. If to assume that length is result of multiplication of two perpendicular directions of time, then speed is a projection of length (the area of time) along one of the directions of time. For a comprehension of the nature and an essence of light velocity, this approach can be the very useful.

At the following lesson we will consider
The linear angular functions

Decomposition on factors

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Zero and infinity

Lesson 12

Decomposition on factors


Mathematical operation, opposite on sense to multiplication, decomposition on factors is. It is carried out with application of the infinite trigonometric functions.

Decomposition on factors. Mathematics For Blondes.
Decomposition on factors
 The most prime example of decomposition on factors at an angle in 45 degrees is root squaring. As both factors in this case are identical, as result of decomposition it is accepted to write down only one of factors.

Decomposition on factors can be applied when only the result of multiplication is known and any of factors is not known. Units of measure as a result decomposition on factors should be selected intuitively so that as a result of their multiplication the tentative unit of measure turned out. The quantity of spacelike dimensions in units of measure of factors at decomposition can be a miscellaneous. For example, three-dimensional volume can be spread out to one-dimensional factors by means of two operations of decomposition, one of options looks so:

Decomposition of volume. Mathematics For Blondes.
Decomposition of volume

In this example corners α and β are not bound among themselves. If to display volume in a cube (a=b=c), then α≈35° is a angle between the diagonal of a cube and diagonal of the basis, β=45 ° is a angle between the diagonal of the basis and its party.

At the following lesson we will consider
Division

8.19.2016

Zero and infinity

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Distinctions between multiplication and addition

Lesson 11

Zero and infinity


If the angle is equal to zero or 90°, then the two dimensional rectangle disappears and there is a one-dimensional piece. From here the sense of infinity follows: as if we did not change the party of a rectangle, it will never turn into a piece. Unit divided into zero is not equal to infinity. Infinitesimal size is not equal to unit divided into infinity.

Zero and infinity. Mathematics For Blondes.
Zero and infinity

Difference between elements in these inequalities same as difference between the point lying on a straight line and the point which is not lying on a straight line.

Multiplication to zero and division by zero do not fall into to mathematical operations with numbers, they are carried out in the field of units of measure. It is possible to call these values of trigonometric functions non-numerical.

In addition to the materials about multiplication and division by zero explained earlier it is necessary to add the following. In positional notation zero designates lack of number of the particular category. Lack of number number cannot be. Here zero is similar to punctuation marks in writing which have the graphic form, but are not said when reading.

Generally zero should be understood as lack of the considered unit of measure. For example, zero value of a angle means that the angle is absent. Division by zero should be considered as need of introduction of a unit of measure, perpendicular to already existing, for the further solution of a task. Division by zero does not mean automatic transition to multiplication. For example, it is impossible to describe turn of a piece in one-dimensional space, for this purpose it is necessary to enter padding measurement and to consider a task in two dimensional space.

At the following lesson we will consider
Decomposition on factors

8.14.2016

Distinctions between multiplication and addition

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
Examples of multiplication

Lesson 10

Distinctions between multiplication and addition


Multiplication is a change of quality, that is change of units of measure. Contrary to the standard opinion, multiplication cannot be presented in the form of addition. When multiplication is substituted for addition, mathematical properties of multiplication are used. With units of measure substitution looks so:

Multiplication substitution by addition. Mathematics For Blondes.
Multiplication substitution by addition

Algebraic expressions with use of letters show distinction between addition and multiplication:

Multiplication in algebra. Mathematics For Blondes.
Multiplication in algebra

If at addition items have identical numerical values, then it is possible to apply mathematical model of multiplication for addition. At the same time we assume that units of measure at multiplication do not change:

Addition replacement with multiplication. Mathematics For Blondes.
Addition replacement with multiplication

In mathematician possess separate properties of units of measure:

- number systems of numbers – it is impossible to addition the numbers presented in different number systems, a question of a possibility of multiplication of numbers in different number systems the author it was not studied;

- denominators of common fractions – it is impossible to addition fractions with different denominators, at multiplication of fractions denominators are multiplied;

- alphabetic references in algebraic expressions – it is impossible to addition numbers with different designations, at multiplication new designation of result turns out;

- legends of functions (for example, trigonometrical).

At the following lesson we will consider
Zero and infinity