Table of squares of natural numbers

Table of squares of natural numbers from 1 to 100. Exponent chart 2. Mathematics For Blondes
In the table of squares of natural numbers the squares of numbers are presented from 1 to 100. Index of exponent of number as a number two writtenin on the right in the top of that number that is involuted. If a number is erected in a square, it means that this number needs to be increased on a such is exact number.

For example, ten in a square equals one hundred, because if ten to multiplication on ten, for us will turn out one hundred. Five in the second degree equals twenty five, because all know that five at five "again twenty five". And two in a square it not that another, "as twice two four".

Why was such thought of is a exponent of number? For short of record of mathematical expressions, whatever to repeat several times multiplication of identical numbers. If at erection of number in the second exponent we save a quite small place, then at erection of numbers in great exponents the economy of place turns out rather significant.

By the feature of squares of numbers, as well as all other even degrees of number, there is that after involution negative number a positive number ensues. In fact minus on minus gives a plus at an multiplication, how many times we this focus did not repeat. A negative number is in the second, fourth, sixth, eighth et cetera degrees gives a positive number as a result.


Multiplication and division by zero

As a zero is not a number, all mathematical operations on multiplying and dividing by a zero take place in area of units of measurements. In relation to the operation of divizion by zero of unit of measuring can be real and virtual. Units of measurements of length behave to the real units of measurements. All other units of measurements, probably, are virtual. Dividing by the zero of virtual units of measuring is impossible, as a result of division by zero of such units of measuring does not make sense.

In the special group it is necessary to distinguish natural unit of measurement of speeds (speed of light) and mathematical unit of corners (corner in 45 degrees). These units of measurements hatch through mathematical methods and their mathematical properties require a further study. More detailed study is required similarly by units of measurements of time.

Virtual units of measurements appear as a result of process that mathematically can be written down as dividing of zero by zero.


where а – is virtual unit of measurement.

The described mathematical properties of virtual units of measurements allow to enter any units of measurements us and use them without influence on the surrounding world. These units of measurements are used both for description of surrounding reality and for everyday needs. The examples of virtual units of measurements can be units of measurements of money, temperature, many physical sizes or applied in a technique and commerce. The process of exit from everyday life of virtual units of measurements of measuring can be mathematically represented as multiplying by a zero. Mathematical properties of similar units of measurements are tested by practice of their use during many millenniums.

In multidimensional space, division by zero increases the amount of the spatial dimensions, multiplcanion by zero diminishes this amount.

In rectangular cartesian coordinates it will look so:

x/0 = xy
xy/0 = (x/0)y = x(y/0) = xyz

At multiplication by zero it is necessary to take into account project properties of space, as a result of such increase depends on that, which one component is multiplied by zero.

xyz*0 = 0 and xy or xz or yz
xy*0 = 0 and x or y

In physical equalizations, division by zero requires introduction of new unit of measurement to examined by equalization the physical co-operation expressed by a mathematical action by an increase (probably, another unit of measurement of length). For example, if to divide unit of measurement of length into a zero, unit of measurement of area will ensue. If to divide unit of measurement of area into a zero, unit of measurement of volume will ensue et cetera.

m/0 = m²
m²/0 = m³

Algebraically it can be presented in a next kind:

a/0 = ab
ab/0 = abc

where а, b, c - mutually perpendicular units of measurements of length.

At multiplication by a zero one of the components of co-operation, described by physical equalization, from co-operation is eliminated. The primary result of co-operation grows into a zero. Remaining components continue to co-operate.

m³*0 = 0 and
m²*0 = 0 and m

Algebraically it can be presented in a next kind:

abc*0 = 0 and ab or ac or bc
ab*0 = 0 and a or b

where а, b, c - mutually perpendicular units of measurements of length.

Expl for blondes: It only began division and multiplication by zero. More in detail we will consider it other time.


Space evolution

Probably, in the process of the evolution, space generates different multidimensional universes with the even amount of dimension. Development originates from spaces with less of dimension to spaces with plenty of dimension. For basis of existence of space it is possible to accept principle of existence of speeds. Our 6-dimension Universe in the chain of space evolution will look like the following.

Space evolution. Mathematics for blondes. Nikolay Khyzhnjak.
Quite possible, that to Big Bang putting beginning of our Universe, there was a 4-dimension universe in that energy had one dimension of length, and a matter two dimension of length in that part of universe that corresponds our slower-than-light. A black hole, the consequence of that in 6-dimension space was Big Bang, giving beginning of our Universe, appeared in the process of evolution of this 4-dimension universe. A matter with two dimension of length in our Universe grew into 2-dimensional energy. Question about transformation of unidimensional on length energy in the process of transition through a black hole, remains open.

In the process of evolution of our Universe energy with two dimension of length partly passes to the matter with three dimension of length. A matter generates in our Universe black holes that give beginning to the new universes in the eightmeasured space. After Big Bang in the eightmeasured space our matter with three measuring of length grows into energy of the eightmeasured universe. Et cetera. The process of space evolution can develop for ever and ever.

During an evolution one 4-dimension universe generates the great number of 6-dimension universes. In turn every 6-dimension universe generates the great number of the 8-dimension universes. It look like the process of spawning. All 6-dimension universe, generated by one 4-dimension universe, can be in the mutually perpendicular dimension, that eliminates their cross-coupling on each other. Each of universes is mapped to all other universes as a point. The 8-dimension universes can be formed just.

In regard to faster-than-light part of universes of any type the theory of symmetry deserves attention in relation to velocity of light. Energy and matter of slower-than-light part freely move in three dimension of length and hardly fixed in the continuous stream of three dimension of time. Dark energy and dark matter freely move in three dimension of time and hardly fixed in the continuous stream of three dimension of length. Even if it not so, possibility of existence of universes of similar type does not need to be thrown down from accounts.

A transition process from a black hole to Big Bang requires an additional study. Mathematically he can be described by the operations of multiplication and division by zero. There are grounds to suppose that the trigger mechanism of gravitational collapse, resulting in appearance of black hole, is space geometry. This question will be considered additionally.

Trigonometric dependences of corner of scale

We will consider trigonometric dependences of corner of scale in two rectangular triangles - at the increase of scale and at diminishing of scale.

Triangles of corner of scale. Mathematics for blondes. Nikolay Khyzhnjak.
Trigonometric correlations of parties of the got rectangular triangles for diminishing and increase of scale we will take in a table.

Trigonometric dependences of corner of scale. Nikolay Khyzhnjak. Mathematics for blondes.
The got results can be compared to the relativistic radical from the theory of relativity of Einstein. Radicals in these equalizations very look like a sine and secant of diminishing of corner of scale. If to execute not difficult transformations, it is possible to get unit of measurement.

Mathematical transformations. Mathematics for blondes. Nikolay Khyzhnjak.
The conducted transformations specify on that natural unit of measurement of speeds is speed of light. There are no grounds to suppose that there is a change of trigonometric dependences at passing through the point of symmetry of coefficient of scale. On it, it is possible to suppose that for faster-than-light speeds a relativistic radical takes on next values.

Theory of relativity of Einstein for faster-than-light speeds. Nikolay Khyzhnjak. Mathematics for blondes.

On the basis of foregoing it is possible to suggest a next hypothesis about the structure of our Universe. Speed of set is a natural barrier separating slower-than-light part of Universe from faster-than-light part. Slower-than-light part of Universe we have possibility to look after. By virtue of specific properties of velocity of light, faster-than-light part of Universe can not be observed directly. It is possible to suppose that velocity of light is the axis of symmetry of distribution of substance in Universe. Dark matter and dark energy, that render affecting our part of Universe, can be in faster-than-light part of Universe.

If in Universe there are highly developed reasonable civilizations that captured faster-than-light technologies, then for an information transfer they will use not hertzian waves possessing velocity of light, and hard carriers of information on the basis of dark matter, transmissible with speeds, considerably excelling velocity of light.

If to equate trigonometric dependences of corner of scale with the values of trigonometric functions at 90° (it 1; 0 and 1/0), then for diminishing of scale they are taken to equality 0=1, for the increase of scale - to equality of k=0.

Our Universe has three limitations. In space by the border of universe явля-ется area, where velocity of light equals a zero. Outwardly our Universe is a point in space. Mathematical equalization of universe is equality 0=1 - any physical quantity with the unit of measurement in the scales of Universe equals a zero. This law of maintenance is confirmed by some researches of physicists, in particular, about it talked in the lecture of Andrei Linde.

Except spatial limitation, I exist limitations of speed. In slower-than-light part of Universe it is expressed in limitation long - distance between two positions of any point of space in time can not equal a zero. In physics this limitation it is accepted to name the absolute pitch of temperature. Faster-than-light part of Universe has limitation at times - time between two positions of any point in space can not equal a zero. The instantaneous moving in space without moving in time is impossible. Geometrically it can be expressed so: projection of speed on length and for a time can not equal a zero. Implementation of these terms is provided by the presence of rotation on the most different levels: atomic, planetary, galactic. It is possible to suppose that our Universe is similarly revolved in space.

More exact idea about principles of existence of Universe it is possible to get after the detailed study velocities of light as a physical process. For basis it is possible to accept position that velocity of light in our Universe is the result of co-operation of three dimension length with three dimension of time. Mathematically this co-operation is described by an multiplication. Physically our Universe has six dimension - three dimension of length and three dimension of time.

Expl for blondes: Now time to draw our Universe to look, what place she occupies in the space evolution.


Change of corner of scale

The increase of corner of scale can be presented as a change of quantity at unchanging unit of measurement. Diminishing of corner of scale can be presented as a change of unit of measurement at an unchanging quantity. Geometrically in the system of rectangular triangle it will look like the following.

Change of corner of scale. New Math. Mathematics for blondes. Nikolay Khyzhnjak.
An increase and diminishing of scale in the identical amount of one times correspond to one value of corner of scale. On this property of corners of scale trigonometric dependences are based in a rectangular triangle.

From the point of view of mathematical result does not matter, as a change of quantity is described in relation to unit of measurement. A variable quantity at permanent unit of measurement and variable unit of measurement at a permanent quantity will give the identical value of corner of scale.

In a general view the change of scale of quantity can be presented as a turn of unit of measurement on the size of corner of scale. An increase or diminishing of scale depends on that, what position of unit of measurement is taken for basis at comparison and from the type of projection. The increase of scale can be presented as a radial projection on a number ray, perpendicular to the unit of measurement taken for basis at comparison. Diminishing of scale can be presented as a perpendicular projection of the compared unit of measurement on the unit of measurement taken for basis.

Turn of unit of measurement on the corner of scale. Mathematics for blondes. New Math. Nikolay Khyzhnjak.
Expl for blondes: A boring entry is farther closed and interesting begins are trigonometric dependences of corner of scale.


Quantity as basis of mathematics

Co-operation of numbers and units of measurements takes place in a point "unit" and expressed by a mathematical action by an multiplication. Geometrically unit of measurement is perpendicular to the numerical ray. The result of multiplication of numbers on unit of measurement in future will be named "quantity". All quantities are identical mathematical characteristics initially.

Geometrical image of any quantity. Co-operation of numbers and units of measurements. Geometrically unit of measurement is perpendicular to the numerical ray. Mathematics for blondes.
All units of measurements in the surrounding us world it is possible to depict two methods: with a general point "unit" and with a general point "zero". The method of image does not influence on properties of making elements.

If to take unit for a general point, then this will be a circumference with a numerical ray going out the center of circumference. The radiuses of circumference will be units of measurements.

Geometrical image of quantities with a general unit. Represent any universe with all present in her units of measurements foto. Portrait of mathematics. Mathematics for blondes.
Approximately it is so possible to represent any universe with all present in her units of measurements. The image of all units of measurements as radiuses of circumference underlines a that circumstance, that all units of measurements are identical mathematical characteristics initially. (Expl for blondes: And you does not it seem to that this portrait of mathematics very reminds the ancient invention of man - wheel? Are you exactly sure that a wheel was invented exactly by a man? Maybe, did someone try to explain to the man, what mathematics, but he so nothing and did not understand? In memory of meeting with the unknown teachers of mathematics history left us only a wheel... Why did unknown teachers begin the story exactly with it? Because not knowing and not understanding such elementary things, understanding to the mathematician is practically impossible.That the previous generations of mathematicians were brilliantly demonstrated us.)

Geometrical image of quantities with a general zero. The moment of appearance of universe, that it is accepted to name Big Bang. Mathematics for blondes.
Approximately it is so possible to represent the moment of appearance of universe, that it is accepted to name "Big Bang". In this case a numerical ray graphicly can be presented as a numerical cone.

In an algebraic kind any quantity can be presented by multiplying of coefficient of scale by unit of measurement. Numbers come forward as a coefficient of scale.

Equalization of quantity. Any quantity can be presented by multiplying of coefficient of scale by unit of measurement. Mathematics for blondes.
Geometrically any quantity that is the result of multiplication of coefficient of scale on unit of measurement, it is possible to present as a hypotenuse of rectangular triangle the cathetuses of that are unit of measurement and part of numerical ray.

Geometrical image of any quantity of kind ka. Mathematics for blondes.
If the coefficient of scale is equal to unit, then a size is equal to unit of measurement.

Expl for blondes: Here now we got to one of types of mathematical corners we will consider and farther, what corner of scale and as there is a change of corner of scale.


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.


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. Mathematics for blondes.
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.


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. Number ray. Mathematics For Blondes.
Expl for blondes: Farther we will consider relativity of concept "Any number".


Multiplication chart

Multiplication chart. Printable multiplication table. Multiplication tables chart 1 to 10. Mathematics for blondes.

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.

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 in mathematics.


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.


Basic axioms of mathematics

Mathematics is laws there is the surrounding world on that. The laws of mathematics are identical for any universes with any amount of measuring.

Mathematics - it governed without exceptions. If an exception appears in a mathematical rule - this rule must be changed. This statement is the universal formula of the scientific discovery in mathematics.

Mathematics is abstraction. The abstract of mathematics consists in that the laws of mathematics operate always and everywhere identically.

Mathematics is the closed system. If a correct mathematical result is got, then there is an infinite amount of ways resulting in an exactly such result.

Mathematics is symmetry. Absolute symmetry in mathematics is a limit of development of mathematics as sciences.

Mathematics is relativity. Positive and negative numbers do not exist in the wild. Positive and negative numbers are this reflection of our personal opinion in mathematics. A negative number is a sign of the relative system of coordinates, position of that depends exceptionally on our choice of her center. A the same point can have different signs and different numerical values in the different relative systems of coordinates.

Mathematics is basis of commonunication and mutual understanding of reasonable creatures from different civilizations. Geometry translators does not need. Mathematics is closed wherein human logic begins.

Expl for blondes: in more detail we will consider each of these axioms a bit later, and while we will continue an acquaintance with mathematics and will look at some concepts that will be used in future.


Bases of mathematics

Bases of mathematics are a cycle of my reasons. Main task of "Bases of mathematics" - to complement mathematics the absent fragments of mathematical knowledge and set intercommunications between some copy-book maxims already known to us.

Most useful innovations in mathematics will be units of measurement and divizion by zero. Clear that to explain it it will be not simply. For understanding will be thoroughly to understand some generally accepted mathematical concepts to set that there is a true in them, and that is lie.

Why are units of measurement needed in mathematics? Here imagine such situation. Does a child go near you, hands to you an object asks: "That will happen, if to drop this object?" Using principles of modern mathematics, you need to take the list of all great numbers of objects to find, to what great number from existing this object belongs. If this great number of the broken up objects, then this concrete object will be broken up. If this great number of jumpings up objects, then this object will jump up. In the existent lists of great numbers of objects you will have to be long and boring dug, before you will be able to find an answer for a question.

Is it possible to decide a problem simpler? It is possible. If by sight to determine material out of that an object is made and to know properties of this material - then no problems. A glass object will be broken up, a rubber ball will jump up, a ferrous ring will do "drin" and jelly will do "tuff".

Just in mathematics there is business with units of measurement. If you know mathematical properties of unit of measurement, you will say without effort, that can be expected and what it is impossible from a that physical parameter that this unit of measurement belongs to.

Introduction to mathematics of units of measurement as a mathematical element equivalent to the numbers, allows to determine mathematical methods many fundamental properties of the surrounding world.

If you think that as a result of such innovations of mathematician will become yet tangled, you wrong. Mathematics will be simpler, more slender, clearer. Look at the basic axioms of mathematics.


How to memorize trigonometric functions?

How to memorize trigonometric functions? This question is set to itself very many at the study of trigonometric functions. On own experience I know that this business is not simple. Through many years I succeeded to carry two moments only, that behave to the trigonometric functions. First, that well burned into my memory, this determination of trigonometric functions generally: a trigonometric function is this relation something to something. Thus in memory at once there is a shot, at that I do not remember that stands in a numerator forgot that was in a denominator. From all trigonometric relations I remember only a tangent: a tangent is a sine on a cosine. A sine costs on the first place, he means in the numerator of shot. A cosine stands on the second place, he means in the denominator of shot. Sines with cosines I generally on life always mixed up. To the last time...

I succeeded to find a that failing element that so was not enough in the days of my educating. Today even I can faultlessly dab with a finger in a cosine to tell that on what in him divided.

The portrait of tangent appeared this magic stick. Yes, yes, not surprised - exactly portrait, at a look to that faultlessly know a tangent. This portrait of tangent is only work by oil of genius artist of beginning twenty the first century, that on auction of Sothebys in 2137 was sold for one hundred million dollars. Oddly enough, but a picture is written was on reverse side of canvas. Where do I it know from? I am a mathematician and I am able to see the future. By the way, yesterday I spoke with this artist, he asserts that such picture did not yet write, but an idea very pleased him. Specially for you I publish reproduction of this masterpiece here.

Masterpiece. Tangent. Mathematics For Blondes

Now we have all for creation of our own trigonometric Bible that will need to be learned by heart. Do not be frightened, in our Bible there will be only one line:

tangent - is sine by cosine

One-only picture from this Bible I already showed you. Now, as the most real preacher, I will teach you all of it it is correct to read. I think, the best method of educating will be a gape-seed of trigonometric comics. Let in the near time by it and we will engage.


Why are sines and cosines needed?

Why are sines and cosines needed? Really, interesting question. In comments to the trigonometric circle of sines and cosines such question appeared:

where will sin and cos be useful in life?
p.s why are they needed sines cosines?

Let us will call a spade a spade. To swingeing majority from you they will be never useful. Unless, when will your children go out into school and will begin to study trigonometric functions, they too will put question you "Why are sines and cosines needed?" and, in addition, will ask to explain, what is it.

Money we use every day already not alone thousand years and perfectly we do without every sines, cosines and other elegant mathematical pieces. I assure you, and through millions of years in the count of money nothing will change. Not because we are such dull, and because such are mathematical properties of money : it is impossible to increase roubles on roubles and with money in the second degree to hurry in a motor show to buy "Lamborghini".

On a kitchen, in culinary recipes, you will meet neither sines nor cosines too. If to give a glance soberly on our everyday life, then all our everyday mathematics remains somewhere at the level of knowledge of Ancient Greece. We are enough with a head.

So why are sines and cosines needed? As compared to Ancient Greece, we have very much different pieces about that ancient Greeks could not dream even today. Even their Gods did not ride on machines, did not use mobile communication, did not communicate on the Internet. But we have all of it and we use this constantly. Did all this extraordinary riches undertake from where? He was created by us. At first scientists did the scientific opening. Then engineers, on the basis of done by the scientists of opening, created every useful things. We use these things today, not having not the least concept about that is into these things and what scientific laws are fixed in basis of their work. So, if there were not sines and cosines, there would not be all these useful things.

Sines and cosines are used most effectively scientists and engineers. I will not say that they continuously trigonometric functions are used only. No, they use them rarely, but well-aimed. Sines and cosines often are in the formulas of different calculations an engineer or scientific.

Often with sines and cosines it is necessary to clash to the geodesists. They have the special instruments for an exact goniometry. Through sines and cosines corners can be converted into lengths or coordinates of points on an earth surface.

The teachers of mathematics on the sort of the duties constantly deal with trigonometric functions. This year they told about sines and cosines to you, the next year the teachers of mathematics will tell the same to other students. Such for them work - to teach.

Schoolchildren and students study trigonometric functions on the lessons of mathematics. Personally I got through tortures sines and cosines at school, техникуме, institute.

Adults sometimes engage in sines and cosines then, when their toschoolchildren need a help at preparation of homeworks.

All! Sines and cosines do not need other generally! In everyday life most people they are not used hardly ever. If I wrong, remedy me.

So why then generally to teach these sines and cosines? Well, firstly, such is the school program. Secondly, if you in life may need apply a sine or cosine, you know already, what is it and where it is needed to search information about them. The knowledge gained at school will fully have you, what is independent in everything to understand.

So what such the sines, cosines and other trigonometric functions? It is a mathematical instrument it is needed that to be able to use. That we this instrument we do not use hardly ever, talks not that studying them is not necessary, and that efficiency of application of the knowledge gained by us is practically equal to the zero. But it is quite another theme already.


How to calculate the area of surface of direct three-cornered prism

How to calculate the area of surface of direct three-cornered prism? For this purpose the best of all to imagine a three-cornered prism in all her beauty. How does a three-cornered prism look? Look at a picture

Three-cornered prism. Area of surface of three-cornered prism. Mathematics for blondes.
Now next question: do you have a tube of lipstick in form three-cornered prism? If you answered this question (and or no) though as, means now you already know about what three-cornered prism and as she looks. Here those two triangles, from above and from below, are named the grounds of prism. Three rectangles on each side are named the verges of prism. As a triangle lies in founding of prism, verges for us exactly three. If there will be a pentagon in founding, verges it will be five. And if is there 1234-cornered in founding of prism? Correctly, verges there will be 1234 things. With the construction of direct prism we understood, after it it is possible to take up the mathematical calculations of area of surface of three-cornered prism.

As a prism is a geometrical body, her structure can be investigated on the example of body of blonde. That is included in the area of surface of prism? If you will come a heel on something sharp or knocked by the обо-что top of the blond head, it will be badly you. That is why both heels and top of head enter in the complement of surface of your body. Just overhead and lower triangles are included in the area of surface of three-cornered prism. If to dab with you a finger in a side, you will say "Ouch"! because a side belongs to your body. Here those three rectangles, that are from the sides of three-cornered prism, make the area of her side.

By the result of our scientific researches of bodies of blonde and three-cornered prism, we came to the conclusion, that the area of surface of three-cornered prism consists of areas overhead and lower grounds (they are equal) and area of side. The area of triangle needs to be found on one of formulas. I remind that the area of triangle found on a formula at the calculation of area of surface of prism needs to be taken two times, id est to increase her on two.

The area of side of prism is determined as a sum of areas of rectangular verges. It is needed to increase length of every side of triangle (that triangle that is in founding) on the height of prism and lay down three got areas together. It is possible to act simpler: to increase the perimeter of triangle (a perimeter of triangle is a sum of lengths of all his parties) on the height of prism. But adding up not to avoid you in any case: either to the increase (we fold long parties) or after an increase (we fold the areas of rectangular verges). Another trouble that you can lie in wait on a way to the area of surface of three-cornered prism is this absence of values of all lengths of parties. It is not mortal for the calculation of area of triangle. And for the area of side of prism you will have preliminary to find lengths of all three parties of triangle, applying the mathematical knowledge.


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.


Mathematics forever!

Mathematics forever!

You can congratulate me, my first scientific publication went out in light. There are "The papers of independent authors" in a magazine, producing № 18, to the page 110 the little article is printed under the modest name "About symmetry of mathematical actions".

At the last time (in a blog in Russian language) I promised you to show that needs to be done in order that impossible in mathematics became possible. In this little article, only on two pages of text, it is shown, as a commutative law works at deduction and division. If someone wants to look, can pass on this page (in Russian language), there is reference for a flush-off. Nothing difficult in this article is present - the half of text is occupied by examples on addition, deduction, increase and division. All at the level of middle classes of school. Such mathematics any blonde will understand.

Here so, unnoticed, we with you began to live in a completely another epoch - Epoch of Great Mathematical Opening. Doing these mathematical opening coming to you, I can only broadly speaking explain to you, what mathematics, where in mathematics, opening is hidden and as they need to be searched. By the way, my article is the first step on a way to dividing by a zero. Mathematical actions are symmetric: if in mathematics there is multiplying by a zero, means there is under an obligation to be dividing by a zero. If dividing by a zero is impossible, means multiplying is impossible by a zero. The third variant (that is known by us all) can not be. When I will show you, where and as there is dividing by a zero, you will understand that mathematics is this not паханное field on that we engage in the most primitive collector. To throw open and take the crop on the mathematical field, at a desire, any of you can.


Four-valued mathematical tables for blondes

Four-valued mathematical tables of Bradis - this was the basic mathematical reference book of soviet schoolchildren, students, engineers to appearance of calculators.

A trigonometric table for blondes is done by me to the navigation more informatively saturated in a plan. It what you did not lose way and did not entangle trigonometric functions.

Trigonometric table of sines and cosines
- from 0 to 90 degrees punctually to the minute corner.

Trigonometric table tangent cotangent in degrees - from 0 to 90 degrees punctually to the minute corner.

Trigonometric table in radians - sin, cos, tan.

We will do justice to work of Bradis and we will remember a bit history. His book "Tables of the four-valued logarithms and natural trigonometric sizes" went out in 1921. This book was repeatedly reprinted, but already under more simple name "The Four-valued mathematical tables". This bestseller looked approximately so.

Four-valued mathematical tables of Bradis for blondes. The basic mathematical reference book of soviet schoolchildren, students, engineers
It is possible to say without a false modesty, that on these tables all Soviet Union was built, a man started to fly in space, a soviet nuclear club was created et cetera. Schoolchildren, engineers, scientists - all used the tables of Bradis. We will remember those distant times - the Internet is not present, mobile telephones are not present, computers and calculators are not present. Even televisions were not then! There were only books. In many books formulas were written for a calculation, and necessary for calculations numerical values were taken from the tables of Bradis. And what did numbers multiply by then? Not on accounts... Well and time was! As it was then possible normally to live??? But lived somehow.

An interesting question arises up. What mathematical tables did Americans build famous sky-scrapers on and created the nuclear club? In fact did not they steal for us tables of Bradis? Omniscient Wikipedia is quiet on this occasion, and information about the table of Bradis in Wikipedia in English language I did not find. There is there a mathematical reference book of Abramowitz, Milton and Irene A. Stegun with tables, but he is published was only in 1964. And that did Americans have to these tables?

I used the table of Bradis once. This whole art, to search the value of trigonometric functions punctually to the minute corner. Fortunately, today we have calculators.

Unfortunately, work of great soviet writer of mathematical tables Bradis me does not interest. Therefore the four-valued mathematical table into language of blondes transferred will not be. In the simplified kind the table of Bradis will be presented in the tables of sine, cosine, tangent and cotangent.


Trigonometric circle - comments 2

Trigonometric circle - comments 2. This continuation of publication of comments to the page the "Trigonometric circle", left by visitors in the Russian version of blog "Mathematics For Blondes".

Comment - I am very thankful you, I skipped these themes and there is nothing could understand, and now though that I understand)))

Comment - I in general understood nothing)

Nikolay Khyzhnjak - Nothing, it is not needed to be disordered. All do not can everything it is identically good to understand. A trigonometric circle someone understands well, someone can not understand in general.

Comment - Class site))))))

Nikolay Khyzhnjak - Concerning a site not quite true: here almost is not that it must be are mathematicians. Simple, clear, evident. I will try to reform)))

Comment - Your table very pleased me all is simple, evidently and clear. Thank you!!!

Nikolay Khyzhnjak - It is my little strokes to stranger work. The trigonometric circle I will draw together with you straight in this blog. I hope, soon this we will take up)))

Comment - I thank you. I do not present although, how all of it I will teach... But for a crib it very well))

Nikolay Khyzhnjak - And really do the salespeople of estimations require cramming yet?))) Personally I prefer to understand mathematics, but not to learn by rote ridiculous rules)))

Comment - cos pi/12 as possible to present?

Nikolay Khyzhnjak - cos pi/12 = cos (180/12) degrees = cos 15 degrees = 0,9659 Finding this value is possible on the table of cosines.

Comment - cos corner in intervals from pi to 3pi/2 does take on negative values??

Nikolay Khyzhnjak - Yes, in the third fourth at a cosine negative values, as well as in fourth, from 3pi/2 to 2pi.

Comment - how to find the cosine of pi on 12

Nikolay Khyzhnjak - It is hardly higher written in comments, that on the table of cosines it is needed to find a cosine 15 degrees.

Comment - you are dimwits

Comment - Thank you for the name "Mathematics for blondes". I 30 years engage in that I translate school mathematics into the language of "mathematics for blondes". Actuality grew with introduction of single state examination. I add to the line of tangents and cotangents (students understand it).

Nikolay Khyzhnjak - Accept my sincere condolences))) Unlike you, I have a right to be quiet. As a teacher of mathematics, you such right do not have. you are under an obligation to explain ALL themes, I can simply not talk about separate moments.
Yes, lines of tangents and cotangents simply are under an obligation to be on a trigonometric circle. Although I and lazy, but I will try in the near time to draw such picture.
very interesting me, how do you explain negative and complex numbers in language of blondes?

Comment - creator, for you alike at with a head badly, if you consider blondes fools.

Nikolay Khyzhnjak - I think that my articles "Mathematics - the blonde searches x", "http://www.mathforblondes.com/2010/11/wunderkindes-and-cretin-with-blondes.html" testify to reverse.

Comment - thank you that who did everything I am a not blonde and not girl but all is even

Nikolay Khyzhnjak - I write for all, who wants to read it)))

Comment - good site, to the author respect and respect, though I and not blonde, I badly understand in mathematics, but all is normally written with a clear language, thank you

Nikolay Khyzhnjak - Thank you for a comment. Not everybody can understand mathematics - it firstly. Secondly, someone must and mathematicians to feed, dress, write their portraits...

Comment - seems, you similarly are blond

Nikolay Khyzhnjak - When I became a mathematician, I, seems, began to understand logic of thinking of blondes. Paradox, but for blondes, alike, mathematical character of thinking.

Trigonometric circle - comments

Math blonde. A Paris Hilton on a picture reads a book Discrete Mathematics. Mathematics For Blondes.
The Russian specialists of Photoshop compelled Paris Hilton to read a book "Discrete mathematics".

Comment - and to me reached!)

Nikolay Khyzhnjak - it is stranger work, this picture of trigonometric circle I only adorned))) it will be Needed to draw the trigonometric circle, by sight clear at first blush)))

Comment - you are a fool bald, explain normally! I am a hard blonde, I understand nothing!

Nikolay Khyzhnjak - If you are a hard fool, then medicine is here powerless. Mathematics - moreover. Now try once again, only already politely, to answer a question: that you are not clear exactly?

Comment - what if at being of value alpha equal 4pi, for me 720 degrees, how farther to find a sine and cosine alpha??

Comment - however I hate geometry, though I am a not blonde=)

Nikolay Khyzhnjak - concerning a hatred. It is your problems((( to Mathematics on a drum, blonde you or no, you love geometry or hate, in mathematics twice two however will equal four, and the laws of geometry will continue to work, as well as before. It is needed simply to reconcile oneself to with this fact. All people can not identically masterly own geometry. But it is desirable to know about basic concepts. In fact you are a reasonable creature, but not animal. It not needed the animal of knowledge, for life by him there are fully enough instincts.
Concerning 720 degrees now I will write on a separate page.

Comment - on my you engage in unavailing enough business. To teach blondes, it too same, that to try to drive away a tram... Although, good luck)

Comment - better will explain on an example - how to decide cos(x - pi/2)

Nikolay Khyzhnjak - Concerning educating of blondes. Blondes are a simply advertising trick))) Although in some places I understand their logic perfectly)))
In business-literature a phrase pleased me: "If not able to do business, teach other people, as needed to do business. If not able to teach people to business, teach teachers, as needed to teach people to business"))) As I in trigonometry small that I understand, I teach other and study. Blondes - it my level just)))
About the decision of examples. Examples without решебника I I am not able to decide. I am simpler to set about in a fuzz and ashes all trigonometric circle, what to decide a предложнный example in due form educational-bureaucratic art))) I will Find a decision - I will publish)))

Comment - is pi divided by 12 exists? If yes, then what is it equal to?

Nikolay Khyzhnjak - Yes, pi divided on 12 is a not mirage, it exists. For that, what finding, what it is equal to, is needed to divide 180 degrees on 12. 15 degrees ensue.
pi/12 = 180/12 = 15 degrees
next equality is Here used for translation of radian measure of corners in a degree:
pi radian = 180 degrees

Comment - Thank you, then yet: (Pi/6+2Pi)/2=13Pi/6. If is it calculated right, then how to bring this expression over? and whether it is possible to simplify him not taking away Pi.

Nikolay Khyzhnjak - It is calculated not correctly. After equal sign the result of calculation of expression is writtenin in brackets, and he needs to be yet divided in half. In the total it must turn out 13pi/12. With the value of corner already nothing will do, unless to write down as 1 and 1/12 pi. And trigonometric functions with the corner of such size it is possible to lead on the special formulas. Every trigonometric function, has the formulas of coersion. All trigonometric functions of any corners can be driven away in a stall measuring from 0 to 90 degrees. The formulas of bringing stray sheep over we will consider in the nearest time)))


Mathematics of blondes and not only...

Today for blondes we will conduct a lesson of biological mathematics. Surprisingly, but the fact:

If to count up quantity of hair on a head of the person (and it is an abstract mathematics without any prejudices) results of calculation will appear such: at their blondes nearby 150,000 (hundred fifty thousand), chestnut hair on a head we will count about 110,000 (hundred ten thousand), black hair it will appear about 100,000 (hundred thousand), red - 90,000 (ninety thousand). Blondes! You possess the thickest hair-do among all people! And, it not a leisure invention of the bald author of a blog "Mathematics for blondes", and the fact confirmed with mathematics.

Mathematics for blondes. Curious  facts. Humour. Curious facts  about numbers. A few  curious numbers are from  life of pilose people.
The number of the top eyelashes makes from 150 to 200, bottom - from 50 to 100 (horror, but even at blondes number of the bottom eyelashes in 2 - 3 times are less, than number of the top eyelashes - here even MaхFactor is powerless);

It is long the eyelashes, measured by the most ordinary ruler for measurement are long eyelashes, for the top eyelashes yields result 8 - 12 millimetres, for bottom equals 6 - 8 millimetres (here already is where to clear up not only Maksfaktoru, but also to false eyelashes);

The size of day growth of hair on a head makes 0.5 - 0.7 millimetres (specially for blondes I will write with letters these terrible numbers - the five tenth and seven tenth millimetre) is less than the tiniest division into your school ruler almost twice, and here day growth of hair of a beard already on the one tenth millimetre is more - from 0.6 to 0.8 for a day (it does not threaten blondes the same as to me - growth of hair on a head).

And some more the sad facts from life of the hairy. A day hair fall on a head - from 50 to 120 pieces (it is interesting, why I bald???!!! Where the mathematics looks!). Life expectancy of hair makes from 2 till 4 years. Life expectancy of eyelashes makes only 150 (hundred fifty!) days. You represent?! Already in half a year on your eyes there will be no familiar eyelash - all new! Horror!!!

Photo from Ledi Truth.

Sine and cosine 11 degrees and 32.7 minutes how to calculate?

When I went to school, I had to use the table of Bradisa for a finding of sine, cosine, tangents and cotangents. Has already forgotten, as this table to use. But today we live in a computer century, and what such the computer? Correctly, it is such big calculator. And in each big calculator there should be a calculator small. Here this calculator also needs to use. At me operating system Windows XP, on the screen the glory, below, is a button "Start-up". Press this button, then in the menu choose "All programs", from all programs choose "Standard". In standard programs the calculator of sine necessary to us and cosine hides.

The calculator usually has no sine and cosine. It is necessary to press a button "Kind" in top panel the calculator and to choose "Engineering". In the engineering calculator there are buttons necessary to us a sine "sin", cosine "cos" and a tangent "tg".

After that it is necessary to track that in the calculator decimal notation and degrees for corners would be included. For this purpose it is necessary to press "Dec" and "Degrees" as on a picture it is shown. Our tool for a finding of sine and cosine is ready. Now we will start directly process of extraction of useful trigonometrical minerals.

If it was not possible to you extract the calculator from the computer, do not despair! Specially for you I have placed in this blog "Mathematics for blondes" the calculator free of charge which you can use directly here and now!

web 2.0 scientific calculator

At first it is necessary to translate minutes in degrees. For this purpose 32.7 we divide on 60. It is As a result received 0.545 degrees. On 60 we divide because in one degree of 60 minutes. To received циферке it is added 11 degrees which at us already are, and it is received 11.545 degrees. Here from such corner on the calculator it is possible to take already sine and косинусы. For this purpose it is necessary to press simply a button "sin" or "cos".

All process press buttons looks so:

32.7 / 60 + 11 = sin

As a result in a calculator window there will be number 0.20013750391127021629780041181162

For math it registers so:

sin (11° 32.7') = sin 11.545° = 0.2001

For a cosine of the angle of 11 degrees of 32.7 minutes value is equal almost to unit and will register so:

cos (11° 32.7') = cos 11.545° = 0.9798

For a tangent all is carried out precisely also, only right at the end instead of a button "sin" the button "tg" is pressed. Here with cotangent, apparently, a problem. There is no such button in the calculator! But we clever, also remember that trigonometrical function cotangent is return trigonometrical function in relation to a tangent (so much clever words for once - already most terribly!). In practice it looks very simply: at first we find a tangent as it is described above. When numbers a tangent have appeared in a calculator window, we press a button "1/х". Numbers a tangent will exchange on numbers cotangent. And this additional magic button is called "number, the return entered". For the sake of a trick, enter number 2, press this magic button and you will have number 0.5 that equally 1/2.

For transfer of seconds in minutes, seconds as need to be divided on 60, as in each minute of 60 seconds. For transfer in the degrees, the received minutes it is necessary to divide once again on 60:

1" = 0.016667' = 0.00027778°

Here, apparently, all how to calculate a sine and косинус 11 degrees and 32,7 minutes. If someone still had questions, write to comments. If to someone laziness most to press buttons in the calculator, it is not necessary to write to comments! I understand, not imperial this business - to be picked the calculator. Then go on a site of the decision of problems, they will execute any your mathematical whim, naturally, for your money.

For the favourite blondes I can give some small helps. The sine of 6 degrees of 30 minutes needs to be typed on the calculator as 6,5 degrees, then to press a sine button.

Now example it is more difficult, with seconds: cosine 6 degrees of 7 minutes of 9 seconds. 9 seconds we divide on 60, we add 7 minutes, again we divide on 60, we add 6 degrees. Number 6.11916666 should turn out... Degrees. Now we press a button cosine "cos". An order press buttons the such:

9 / 60 + 7 / 60 + 6 = cos

For math recalculation of degrees, minutes and seconds in degrees for 6 degrees of 7 minutes of 9 seconds can be written down so:

(9 : 60 + 7) : 60 + 6 = 6.11916666...

In a general view for a corner in x degrees, y minutes, z seconds the transfer formula in degrees will look so:

(z : 60 + y) : 60 + x = degrees

I hope, this formula is useful to you.

How to find a value:

How do you find the sin of degrees and minutes on a calculator - here is a calculator it is written, how on him to calculate a sine

At me of 720 degrees how further to find a sine and cosine of corner?

"And what if at a finding of value of corner equal 4pi, at me of 720 degrees how further to find a sine and cosine of corner??" - such question has been set in comments. Really, how to find trigonometrical function if a corner the alpha is more than 360 degrees?

Present that all corners 360 degrees there are more or 2 pi is a ball of a yarn. To learn values of trigonometrical functions for such corners, this ball of a yarn needs to be unwound at first. One coil of a yarn equals to a corner in 360 degrees or 2 pi. To unwind a ball it is necessary until value of a corner does not become less than 360 degrees or 2 pi. After that under the table of values of trigonometrical functions or on a trigonometrical circle we find value of the necessary trigonometrical function.

720° - 2 x 360° = 0°

The same focus with corners in radians will look so:

4π - 2 x 2π = 0

For a corner of 720 degrees or 4 pi it turns out that a sine and cosine same, as well as for a corner of 0 degrees. The decision can be written down so:

sin 720° = sin 4π = sin 0 = 0

cos 720° = cos 4π = cos 0 = 1

The sine of 720 degrees or 4 pi is equal to a sine of zero of degrees and is equal to zero. Cosine 720 degrees or 4 pi it is equal cosine zero of degrees and it is equal to unit.

What for all these troubles with such big corners are necessary? I have knowingly mentioned a yarn ball. Let's try together it not only to untangle, but also we will look, as it is reeled up and what for in general is necessary.

Mathematics - the blonde searches x

A blonde searches x. Rectangular triangle, theorem of Pythagoras. Mathematics for blondes. Humour about blondes picture. A photo is a blonde
The blonde searches x is a classical picture at which all Internet laughed. We have laughed also. And now we will try to understand, whether so it is ridiculous. What is told in the task? "Find х". What has the blonde made? It has found x and has shown it. Where it is told, it is necessary what to search for VALUE x? Remember the childhood when mum spoke to you "Show on a picture rabbit". You stuck with a finger in rabbit on a picture, and your mum admired the cleverest child.

The task "Find х" it is possible to understand on a miscellaneous. The blonde, from the engineering point of view, has carried out the task excellent - with the least expenses of time and forces. Give for a minute we will glance in any office. The director speaks to the manager on purchases (now fashionably all to call managers): "Find..." - also names the goods. What, the manager removes all money from accounts department and buys the named goods for all sum? No. Though it also is the manager on purchases, it simply collects all information on the named goods. It transfers this information to the director. And already the director solves, where, how many and at what price the goods need to be bought. Or while not to buy at all. So all normal managers arrive.

Really, surrounding conditions impose some stereotype of behaviour. As at the trained animals. If you the manager on purchases, without direct instructions what exactly to buy, buy it is necessary nothing. That to you would not speak. If you pass the decision of the equations in school, means on command "find х" you should search for value "x", instead of a dagger on a picture. So all arrive clever (and it would be desirable to write "little monkeys") pupils.

And blondes - they others. They well remember, to that them learnt in the childhood. They consider the children's decision as more simple. Really, what for to be soared with any decision when it is possible to show simply on a picture this most whether икс, whether a dagger. You consider yourselves already adult, and laugh at the naive children's decision of the blonde? And let's look at you, the adult, values x's finding, at other lesson...

... And so, you at a geography lesson, stand near with a pointer in hands of a card and the teacher speaks to you: "Find Honduras". You feverishly rummage eyes on a card in search of Honduras, but it anywhere is not present! Here to you nothing can already help - neither ability to find value x, nor knowledge of the Pythagorean theorem... Everything that from you is required it it is stupid to stick with a pointer into this ill-starred Honduras. You perplexed look in a class where tens fingers show in the most different corners of globe and you are even more lost. An unseeing sight having stared in a card, you suddenly understand that the teacher specially for you has brought today on this lesson a card WITHOUT Honduras!!! Doomed having lowered a pointer and eyes, at the very bottom of a card, you distinctly see an inscription tiny letters - even in a magnifier it is impossible to read it! But you already precisely know that there is written "printed in printing house "Revenge of the teacher". Circulation 1 copy (without Honduras)"...


How many will be metres and decimeters?

"How many there will be 22 metres of 2 decimeters a minus of 1 metre of 8 decimeters?" Difficulty of this problem consists that for a designation of one number is used at once two units of measurement: metres and decimeters. If to translate told on normal language of blondes it almost the same that "I have bought yesterday a phrase two jackets and one has presented to the girl-friend" to replace with a phrase "I yesterday has bought two jackets and four sleeves, and then one jacket and two sleeves has presented to the girl-friend". It is natural that last phrase will confuse you at once: "And sleeves that, separately from jackets are on sale?". Certainly, no. Sleeves are sewn to jackets, here we only consider them separately. What for? What you to confuse. Problems in the mathematician always get confused, as a ball of threads. You should untangle and receive them the right answer. It develops ingenuity, here. And simple writing off from the answer book develops dullness. Now we will start the decision.

Means, a problem at us such: how many will be, if from 22 metres and 2 decimeters to take away 1 metre and 8 decimeters? The main thing what it is necessary to know at the decision of this problem - how many decimeters in one metre? Or how many metres in one decimeter.

1 metre = 10 decimeters

1 decimeter = 0.1 metres

One metre is equal to ten decimeters. One decimeter is equal the one tenth metre. Here now we have everything what to solve a problem. Is the whole three ways of the decision of this problem - in metres, in decimeters, in two units of measurement at once.

We solve a problem in metres. For this purpose decimeters reduced (this that number from which subtract) and subtracted (this that number which subtract) it is translated in metres.

2 decimeters = 0.2 metres

8 decimeters = 0.8 metres

We add these tails to our numbers in metres:

22 metres + 0.2 metres = 22.2 metres

1 metre + 0.8 metres = 1.8 metres

Now it is possible to execute subtraction and to receive a difference of these metres. It is possible to translate the answer at once in metres and decimeters:

22.2 metres - 1.8 metres = 20.4 metres = 20 metres of 4 decimeters

The second way is to solve the same problem in decimeters. For this purpose we will translate metres in decimeters:

22 metres = 220 decimeters

1 metre = 10 decimeters

To the received decimeters it is added those decimeters which at us are on a statement of the problem:

220 decimeters + 2 decimeters = 222 decimeters

10 decimeters + 8 decimeters = 18 decimeters

We carry out subtraction and we translate decimeters again in metres and decimeters:

222 decimeters - 18 decimeters = 204 decimeters = 20 metres of 4 decimeters

It was the mathematics. Now for blondes. What to understand that we did, let's replace metres with bows, and decimeters on ruches. Thus we will agree that it is possible to make ten ruches of one bow, and it is possible to make one bow of ten ruches – well to sew. We copy a statement of the problem: «you have 22 bows and 2 ruches. And 8 ruches you should give 1 bow to the girl-friend». With bows all is simple – we take one and we give. And how to be with ruches? They obviously do not suffice us. We will look, as we solved this problem math in two ways. From outside it looks so.

The first way. You take all bows and untie them, smooth all ruches. Take a needle with a thread and sew all it in one long tape. After you have stopped to sew, measure the necessary quantity from a tape and cut off scissors. Solemnly hand over to the gone nuts girlfriend a tape piece. Then you with the girlfriend cut the tapes, do of them bows and ruches. As a result you with the girlfriend, all in bows and ruches, stand, as two marriageable silly women.

The second way. You untie all bows and cut them on ruches. Do a heap of ruches and from this heap select 18 pieces for the girl-friend. After that solemnly hand over to the gone nuts girl-friend a small group of ruches. Then together begin to smooth ruches and to sew from them bows. Thus your girl-friend thinks: «In the silly woman! Because of this ugly creature I should sew now a bow». You think: «In the silly woman! Because of this ugly creature I at first cut, and now should sew already twenty bows». To tell the truth, I do not envy you.

As you can see, math correct ways from outside look is not absolutely practical. How any blonde will arrive in this situation? As the two-nuclear processor. Takes one bow from a small group of bows and will make of it ten ruches. Now you have one small group from twenty one bows and the second small group from twelve ruches. From the first small group you take one bow, eight ruches a beret from the second small group. Hand over to the happy girl-friend one bow and eight ruches. Then you admire the treasures and is simple blah-blah-blah instead of puffing with scissors or needles. It also is the third way of the decision of a problem. Coming back to our metres and decimeters, math it can be written down so:

22 metres and 2 decimeters = 21 metre and 12 decimeters

(21 metre and 12 decimeters) – (1 metre and 8 decimeters) = (21 metre – 1 metre) and (12 decimeters – 8 decimeters) = 20 metres and 4 decimeters

We with two different units of measure carry out two mathematical actions of subtraction simultaneously, in the same way, as two-nuclear processors Intel for computers work. I think, after such decision in your writing-book mathematicians will start to go nuts. Though so all calculations practically become. In the mathematician it is accepted to say that it is necessary to execute at first subtraction of decimeters, and already then subtraction of metres. Basically, mathematicians are right. So it is heavier to get confused. At subtraction of decimeters you can always steal one metre from metres if decimeters do not suffice you. Then from the remained metres subtract metres. Only do not forget about one metre if you have already dissipated it in decimeters.