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

Examples of multiplication

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

Lesson 9

Examples of multiplication

Multiplication in space. Mathematics For Blondes.
Multiplication in space

At interaction of two one-dimensional spaces (in figure D is quantity of spacelike dimensions) the two dimensional space turns out. At addition to interaction of perpendicular measurement, the quantity of spacelike dimensions of result increases. It is necessary to understand any quantities participating in multiplication as spacelike dimensions. Multiplication of two one-dimensional lengths gives the two dimensional area, multiplication of the two dimensional area at one-dimensional length gives three-dimensional volume and so on. The possibility of receiving four-dimensional space as a result of multiplication of two two dimensional spaces is theoretically possible, but demands specification from the point of view of physical reality.

Examples of multiplication. Mathematics For Blondes.
Examples of multiplication

Voltage is result of interaction of current intensity and resistance. If to consider current intensity not as one-dimensional quantity and as two dimensional (the area of current), then as a result of interaction with a resistance power will turn out. Units of measure of the specified quantities (Volts, Ampere, Ohms, Watts) are household (convenient in daily use), but demand mathematical representation for comprehension of an essence of the physical processes expressed by these units of measure. The system of physical quantities offered R.O. di Bartini can be considered as attempt of transition from household units of measure to mathematical. Interaction between quantities is carried out due to the processes studied in physics.

The arrangement of threads in the perpendicular direction allows to receive cloth. Interaction between threads is carried out at the expense of frictional force. If in the perpendicular directions to arrange reinforcing rod stock, then it is possible to receive a two dimensional reinforcing grid or a three-dimensional reinforcing framework. Interaction of separate rod stock in clusters is provided with electric welding or a knitting wire. Plywood is the fibers of a wood neveer located perpendicularly. Interaction is provided by pasting of separate layers of wood neveer.

At multiplication the quantity of goods on unit price, as a result turns out the cost of a consignment of goods (sum). It is an example of application of mathematical model of multiplication to the virtual units of measure. In this case physical interaction between goods and the price is absent.

Sexual reproduction it is possible to present in multiplication form. As a result of interaction of a male and a female there is a posterity which has genetic signs of two parents. Partial transfer of genetic signs from each of parents can be described by means of the linear angular functions.

For more precise reflection of reality in mathematician it is necessary to consider three stages of multiplication: beginning of multiplication, interaction process, end of multiplication. In electric circuits the switch is the device which operates interaction. Creation of cloth, plywood, a reinforcing grid is the beginning of interaction. Physical destruction of these objects is the end of interaction.

At the following lesson we will consider
Distinctions between multiplication and addition

8.11.2016

Multiplication

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

Lesson 8

Multiplication


As a result of multiplication of two different quantities the third quantity turns out. At multiplication of change occurs both in a number domain, and in the field of units of measure. Multiplication is possible only for perpendicular quantities. Multiplication reflects high-quality changes of quantities.

Multiplication of the parties of a rectangle allows to receive the area.

Multiplication of the parties of a rectangle. Mathematics For Blondes.
Multiplication of the parties of a rectangle

For transition from the area of a rectangle to the simple area it is necessary to divide the square into itself. In this case two options of algebraic transformations are possible. It is necessary to highlight that the recommendation of mathematicians about reduction of identical quantities in numerator and a denominator of fraction is consciously controlled for identification of quantity of the received result. The first option of transformations looks as follows.

First option of the unit area. Mathematics For Blondes.
First option of the unit area

The tangent and cotangent are connected by the inverse proportion. The parties of a rectangle can change somehow, from infinitely big quantity to infinitesimal. If change of the parties is carried out with keeping of the inverse proportion, then the area of a rectangle will remain invariable.

The second option of transformations is presented in an algebraic form and in physical units of measure.

Second option of the unit area. Mathematics For Blondes.
Second option of the unit area

This transformation shows that as a result of multiplication of two different units the third unit turns out. Multiplication is an interaction of two perpendicular units of measure as a result of which the new unit of measure turns out. As any quantity is result of multiplication of number and a unit of measure, under figure "one" can the number, a unit of measure or result of their interaction is meant.

Perpendicular units of measure are not connected among themselves in any way and can have the arbitrariest scale. The scale of units of measure of factors scales work units of measure, but does not influence a process essence – multiplication leads to high-quality change of initial units of measure.

The area is a result of interaction of two perpendicular measurements of length. If to multiplication inches by meters or millimeters on meters, then the area will be expressed in inches on meter or millimeters on meter. Traditionally, the area can be expressed in identical units of measure of length and width. In practice frames with different scales on a vertical and a horizontal are often used.

At determination of the area of a square the size of the party can be built in the second degree. But it does not mean at all that the party of a square is multiplied by itself. The area can still be determined only by multiplication of length to width, just at a square they have identical numerical values. We never multiply rectangle length by length or width on width because the result of such actions does not make sense. The involution is a multiplication of the different perpendicular quantities having identical numerical values and units of measure.

Algebraic transformations of multiplication of two to a square can be written down so:

Algebraic transformations. Mathematics For Blondes.
Algebraic transformations

If to consider that quantity a at a quadrating is multiplied by itself, then the inverse transformation of a square in work of two sums will be impossible. As it is possible to execute a similitude, is shown in the partition Decomposition of sume below.

Diagonal of a rectangle and angle between the diagonal and the party are padding characteristics of the interacting quantities.

At the following lesson we will consider
Examples of multiplication

8.08.2016

Variable units of measure

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

Lesson 7

Variable units of measure


Example of variable units of measure are exchange rates of the different countries. The main world currencies are in continuous fluctuation. They are connected by laws of the inverse symmetry – rise in price of one currency means reduction in cost of other currency.

One more example. Why people never celebrate such significant day how "The Middle of Life"? Because nobody knows the precise duration of characteristic life. After death such important parameter as "life expectancy", ceases to change. Similar indeterminacy is characteristic of all living beings.

If to take duration of existence of an alive organism for unit, then the description in variable units of measure will be the most precise mathematical description of any life. As the beginning of this unit of measure it is necessary to consider the conception moment (at sexual manifolding), the end is the death. It is necessary to notice that increase in life expectancy of concrete organisms is not a priority in natural selection.

Quantity, the back symmetric life expectancy, the age is. The age allows to express the variable characterizing life in constant units of measure of time.

From the point of view of mathematics, emergence of life can be considered as transition from constant units of measure (inanimate nature) to variables (wildlife).

At the following lesson we will consider
Multiplication

8.05.2016

Quantity

Subject of occupations:
TRIGONOMETRIC FUNCTIONS IN A RECTANGLE
Subject of the previous lesson
The infinite trigonometric functions

Lesson 6

Quantity

Tangent and cotangent. Mathematics For Blondes.
Tangent and cotangent

Mathematical interpretation of the drawing given above will look as follows.

Quantity. Number and unit of measure. Mathematics For Blondes.
Quantity

Both options of representation of quantity are absolutely equal and yield identical result. On a numerical axis the parity of numbers and units of measure looks so.

Numbers on a straight line. Mathematics For Blondes.
Numbers on a straight line

In this case zero and unit divided into zero act as the horizons which cannot be reached by means of numbers. For further transformation of quantity to a look habitual to us, to the abstract mathematical concepts it is necessary to enter homocentrism elements.

If we consider that across units of measure settle down, and numbers are perpendicular them, then quantity will have two options of representation.

Two options of quantity. Constant unit of measure and variable number. Variable unit of measure and constant namber. Mathematics For Blondes.
Two options of quantity

The numerical axis takes the following form.

Numerical axis with units of measure. Mathematics For Blondes.
Numerical axis with units of measure

Introduction of the following element of a homocentrism allows to pass to habitual representation of quantity. If we consider that the unit of measure always remains to a constant, then for the adequate description of quantity it is necessary to enter inverse numbers. The inverse symmetry of numbers is result of transition from variable units of measure to constants.

Quantity in constant units of measure. Inverse number
Quantity in constant units of measure

The numerical axis will be transformed as follows.

Numbers and inverse numbers. Mathematics For Blondes.
Numbers and inverse numbers

In this case the numerical axis is represented without imposing of a reflecting symmetry. Unit acts as a point of the inverse symmetry. By the inverse symmetry numbers and units of measure in any quantity are connected. At invariable quantity, decrease of number leads to increase in a unit of measure, increase in number – to decrease of a unit of measure. The algebra of similitudes looks so.

Decrease of number Increase in number. Mathematics For Blondes.
Decrease of number Increase in number

Here elements, the falling into number domains, are represented in round brackets. Elements, the falling into areas of units of measure, are represented in square brackets.

At the following lesson we will consider
Variable units of measure

8.03.2016

The infinite trigonometric functions

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

Lesson 5

THE INFINITE TRIGONOMETRIC FUNCTIONS


If to divide rectangle elements with simple diagonal into terminating trigonometric functions, the infinite trigonometric functions will turn out.

Transition from terminating trigonometric functions to the infinite. The sine and cosine, a tangent and a cotangent. Mathematics For Blondes.
Transition from terminating trigonometric functions to the infinite

The tangent and cotangent can be presented as the party of a rectangle when as a unit of measure of length length of the perpendicular party is accepted. This geometrical representation of a tangent and cotangent can be considered as visual display of interaction between numbers (the infinite trigonometric functions) and units of measure (unit) as a result of which size appears. Other option of geometrical representation of size is shown in earlier published work.

The tangent and cotangent. Mathematics For Blondes.
The tangent and cotangent

Similar representation of numbers and units of measure possesses property of a rectangular symmetry. Value of a angle the alpha in this case characterizes a dependence angle between number and a unit of measure.

At the following lesson we will consider
Quantity