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Showing posts with label addition. Show all posts
Showing posts with label addition. Show all posts

## 11.02.2019

I will not tell you recipes for cooking borsch, I will talk about mathematics. What is borsch? These are vegetables cooked in water according to a specific recipe. I will consider two source components (vegetable salad and water) and the final result is borsch. Geometrically, this can be thought of as a rectangle in which one side is a salad, the other side is water. The sum of these two sides is borsch. The diagonal and area of such a rectangle are mathematical concepts and are never used in recipes.

 Borsch trigonometry

How do salad and water turn into borsch? How does the sum turn into trigonometry? To understand this, linear angular functions will help us.

The linear angular functions

There are no linear angular functions in mathematics textbooks. But without them there can be no mathematics. Mathematical laws work whether we know them or not.

The linear angular functions are the laws of addition. This is how algebra turns into geometry, and geometry turns into trigonometry.

The main law of addition, which mathematicians do not like to talk about, requires that the summands have the same units of measurements. For salad, water and borsch it can be units of weight, volume, value or others.

The figure shows two levels of difference. The first level is differences in the field of numbers. This is what mathematicians do. The second level is the difference in the area of units of measurement (shown in square brackets). Physicists are doing this. We can understand the third level - the differences in the field of described objects. Different objects can describe the same number of identical units of measurements. How important this is, the trigonometry of borscht will show us. Here is how linear angular functions for borsch will look.

 The law of addition for borsch

Part of the water and part of the salad give one serving of borsch. Now we can see what will happen when the angle of linear angular functions changes.

 The angle is zero

The angle is zero. We have a salad but no water. You can’t cook borsch. The amount of borsch is also zero. This does not mean that zero borsch is equal to zero water. Zero borsch can be at zero salad (right angle).

 The angle is greater than zero, but less than the right angle

The angle is greater than zero, but less than forty-five degrees. We have a lot of salad, but not enough water. The result is a thick borsch.

The angle is forty-five degrees. We have an equal amount of water and lettuce. Perfect borsch (forgive me the cook, it's just math).

The angle is more than forty-five degrees, but less than ninety degrees. We have a lot of water and a little salad. Get liquid borsch.

 Right angle

Right angle. We have water. From the salad there were only memories. We cannot cook borscht. The amount of borscht is zero. In this case, drink water.

Something like this. If you're tired of borscht, check out a few other stories.

Percentage.

 Percentage

Cell division.

 Cell division

Two friends had their shares in the general business. After killing one of them, everything went to another.

The emergence of mathematics on our planet.

 The emergence of mathematics

All these stories in the language of mathematics are presented using linear angular functions. I will tell you separately about the real place of these functions in the structure of mathematics. In the meantime, we return to the trigonometry of borscht and consider the projections.

## 1.30.2017

### Decomposition on items

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

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

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

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

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

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
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.

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

Lesson 15

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.

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

## 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

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:

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

 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:

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

## 3.01.2016

### How to simplify?

Help simplify the mathematical expressions:

1) 5.25x+7.56+4.98x-1.25x+0.02x
2) 12.78x+3.84x+2.22x+1.16x-0.3x

It is not hard. We believe that X's - it's rabbits. Why rabbits are not the whole? Their evil Putin nibbled. The numbers - it's just a numbers. Now we add up the rabbits with rabbits. The numbers add up separately. Perform a permutation terms, make it easier to summarize.

1) 5.25x+7.56+4.98x-1.25x+0.02x=
=(5.25x-1.25x)+(4.98x+0.02x)+7.56=
=4x+5x+7.56=9x+7.56

2) 12.78x+3.84x+2.22x+1.16x-0.3x=
=(12.78x+2.22x)+(3.84x+1.16x)-0.3x=
=15x+5x-0.3x=19.7x

Here's how it's simple.

## 4.27.2015

### Increased - that's a plus or multiply?

If you have a question: "Increased - is that a plus or multiply?", then definitely no answer to it. You need to read or listen to what's next. If "increased by as much as something something", then this addition and the plus sign. For example, the number 7 is increased by 2 units. What are we doing? To previous value we added increase and get new value:

7 + 2 = 9

 Increased

If "increased in so many times," it is multiplication. For example, the number 7 is increased in 2 times. It is necessary to multiply the previous value of the specified number of times, and we get a new meaning:

7 * 2 = 14

As you can see, the numbers in the above examples are the same, and the result is different. It all depends on how to formulate a thought.

## 3.18.2011

### About symmetry of mathematical actions

About symmetry of mathematical actions - is my first official publication. To all appearances, my flaming speech under the name "Mathematics forever!" remained unnoticed. It is clear. Reading a like, I would say that a next idiot rushed about on all Internet with the ridiculous idea. But... All, that is here written, I write exceptionally for you and publish here in an only copy, unlike other authors of raving ideas. With my article about symmetry of mathematical actions you are first can become familiar right here and now. I am herein anything interesting or no - decide. In brackets I will give some comments (specially for you) that in the printed variant of the article are absent.

Annotation: Rules of symmetry of mathematical actions allow to apply a commutative law to all mathematical actions: to addition, deduction, multiplication and division. (An annotation is this obligatory condition for the publication of the article. Such are rules of the bureaucratic playing science)

Changes in the surrounding world are expressed by mathematical actions. Quantitative changes are expressed by addition and deduction. Quality changes are expressed by an increase and division. No quantitative changes can cause the change of quality.

Quantitative changes reflect the change of amount of the separately taken unit. Addition and deduction are symmetric mathematical actions reflecting the quantitative changes of any unit. Addition and deduction are mirror symmetric relatively neutral element are points zero.

An increase and division similarly are symmetric mathematical actions reflecting the quality changes of units. An increase and division are back symmetric relatively neutral element are points one.

Rules of symmetry of mathematical actions:

1. Any mathematical action is begun with a neutral element.

2. A sign of mathematical action is the inalienable attribute of number before that he stands.
(This fragment is distinguished by me by fat text specially for you)

Application of these rules allows to apply a commutative law to all mathematical actions reflecting quality or quantitative changes.

0 + 3 + 7 + 4 = 0 + 7 + 3 + 4 = 14

0 – 3 – 7 – 4 = 0 – 7 – 3 – 4 = –14

0 + 3 – 7 – 4 = 0 – 7 + 3 – 4 = –8

1 х 3 х 7 х 4 = 1 х 7 х 3 х 4 = 84

1 : 3 : 7 : 4 = 1 : 7 : 3 : 4 = 1/84

1 х 3 : 7 : 4 = 1 : 7 х 3 : 4 = 3/28

A commutative law can not be used in the cases of the mixed implementation of mathematical actions reflecting quality and quantitative changes in one mathematical expression.

The change of the mathematical operating on symmetric gives a symmetric result, here the point of symmetry is a neutral element. Application of commutative law does not influence on a result.

0 – 3 – 7 – 4 = 0 – 7 – 3 – 4 = –14

0 + 3 + 7 + 4 = 0 + 7 + 3 + 4 = 14

0 – 3 + 7 + 4 = 0 + 7 – 3 + 4 = 8

1 : 3 : 7 : 4 = 1 : 7 : 3 : 4 = 1/84

1 х 3 х 7 х 4 = 1 х 7 х 3 х 4 = 84

1 : 3 х 7 х 4 = 1 х 7 : 3 х 4 = 28/3

Running the numbers in the mathematical operating on symmetric relatively neutral element of number gives a symmetric result.

0 + (–3) + (–7) + (–4) = 0 + (–7) + (–3) + (–4) = –14

0 – (–3) – (–7) – (–4) = 0 – (–7) – (–3) – (–4) = 14

0 + (–3) – (–7) – (–4) = 0 – (–7) + (–3) – (–4) = 8

1 х 1/3 х 1/7 х 1/4 = 1 х 1/7 х 1/3 х 1/4 = 1/84

1 : 1/3 : 1/7 : 1/4 = 1 : 1/7 : 1/3 : 1/4 = 84

1 х 1/3 : 1/7 : 4 = 1 : 1/7 х 1/3 : 1/4 = 28/3

Simultaneous change of the mathematical operating on symmetric and running the numbers on symmetric relatively neutral element of number abandons a result without changes.

0 – (–3) – (–7) – (–4) = 0 – (–7) – (–3) – (–4) = 14

0 + (–3) + (–7) + (–4) = 0 + (–7) + (–3) + (–4) = –14

0 – (–3) + (–7) + (–4) = 0 + (–7) – (–3) + (–4) = –8

1 : 1/3 : 1/7 : 1/4 = 1 : 1/7 : 1/3 : 1/4 = 84

1 х 1/3 х 1/7 х 1/4 = 1 х 1/7 х 1/3 х 1/4 = 1/84

1 : 1/3 х 1/7 х 4 = 1 х 1/7 : 1/3 х 1/4 = 3/28

The neutral elements of mathematical actions it is not accepted to write at the decision of mathematical problems and examples, as they do not influence on a result. Before application of commutative law introduction of neutral elements allows correctly to apply a commutative law.

All of it is written, certainly, not for blondes, and for mathematicians. In the future we yet not once will call to this article. And while... you know any more mathematician about symmetry of mathematical actions.