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Showing posts with label division. Show all posts
Showing posts with label division. Show all posts

12.20.2020

Plus on minus what gives?

Positive and negative numbers were invented by mathematicians. The rules for multiplying and dividing positive and negative numbers were also invented by mathematicians. We need to learn these rules in order to tell mathematicians what they want to hear from us.

It's easy to remember the rules for multiplying or dividing positive and negative numbers. If two numbers have different signs, the result will always be a minus sign. If two numbers have the same sign, the result will always be a plus. Let's consider all possible options. 

What gives plus for minus? When multiplying and dividing, plus by minus gives a minus.

What gives plus for minus? Mathematics For Blondes.
What gives plus for minus?

What gives a minus on plus? When multiplying and dividing, we also get a minus sign as a result.

What gives a minus on plus? Mathematics For Blondes.
What gives a minus on plus?

As you can see, all the options for multiplying or dividing positive and negative numbers have been exhausted, but the plus sign has not appeared.

What gives a minus on a minus? There will always be a plus if we do multiplication or division.

What gives a minus on a minus? Mathematics For Blondes.
What gives a minus on a minus?

 

What gives plus for plus? It's quite simple here. Multiplying or dividing plus by plus always gives plus.

What gives plus for plus? Mathematics For Blondes.
What gives plus for plus?

 

If everything is clear with the multiplication and division of two pluses (the result is the same plus), then with two minuses, nothing is clear.

Why does a minus and a minus make a plus?

I can assure you that, intuitively, mathematicians have correctly solved the problem of multiplying and dividing the pros and cons. They wrote down the rules in textbooks without giving us any reasons. For the correct answer to the question, we need to figure out what the plus and minus signs mean in mathematics.

One mathematics teacher told us in the classroom: "Mathematics is an exact science, if you lie twice, you get the truth". This statement was very useful to me. Once I was solving a difficult problem with a long solution. I knew exactly what the result should be. But the result was different. I have been looking for an error in the calculations for a long time, but I could not find it. Then, a few steps before the final result, I changed one number so that the result was correct. I lied twice in the calculations and got the correct result. This is very similar to the minus on minus equals plus rule, isn't it?

2.10.2018

Trapezoid

In Wikipedia there is a page of a trapezoid. In the drawing it is shown that any trapezoid can be turned into a rectangle.

Trapezoid. Mathematics For Blondes.
Trapezoid
Let's look at algebra. These are formulas for calculation of lengths of diagonals of a trapezoid.

Diagonals of trapezoid. Mathematics For Blondes.
Diagonals of trapezoid
We substitute in these formulas data for a rectangle.

Diagonals of rectangle. Mathematics For Blondes.
Diagonals of rectangle
If to trust these formulas, the rectangle has no diagonals. Even schoolboys or schoolgirls can make what mathematicians couldn't make – to execute verification of the solution of a task. It is the actual level of modern mathematics – any statement of mathematicians can be false.

Height of a trapezoid

Height of a trapezoid is determined by the area of a triangle. The area of a triangle is calculated on Heron's formula. The sizes of the parties of a trapezoid allow to receive a triangle which has the same height as a trapezjid. The cunning trick of mathematicians allows to calculate length of diagonals of a trapezoid.

Height of a trapezoid. Mathematics For Blondes.
Height of a trapezoid
 When the legs of a trapezoid are parallel, the triangle disappears and the cunning trick ceases to work. If to determine height of a trapezoid by the area of a trapezoid, then no problems arise upon transition to a rectangle or a parallelogram.

Height of a rectangle. Mathematics For Blondes.
Height of a rectangle
 Conclusion: cunning tricks of mathematicians can result in false results.

2.06.2017

How to divide fraction into fraction?

When performing calculations often there is a question: "How to divide fraction into fraction?". Mathematicians rules tell so:

To divide a number by a fraction, multiply that number by the reciprocal of that fraction.

How to divide fraction into fraction. Mathematics For Blondes.
How to divide fraction into fraction?
Transform fraction to division if in numerator and a denominator there are fractions. Be attentive! The numerator and denominator of fraction need to be known. The result depends on it.

The numerator and denominator. Mathematics For Blondes.
The numerator and denominator

As fraction to divide into fraction. Formula of division of fractions.

Formula of division of fractions. Mathematics For Blondes.
Formula of division of fractions
You believe that in mathematician there is a division? The fraction cannot be divided into fraction. They can only be multipled.

9.03.2016

Division

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

Lesson 13

Division


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

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

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

At the following lesson we will consider
The linear angular functions

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

4.11.2013

Division by zero in physics

Switch. Dividing by a zero in physics. Example of application of mathematics. Mathematics for blondes. Mathforblondes.
All laws can be divided into two groups - invented by us and the laws of mathematics. Laws invented by us may not work in spite of the fact that we invented them. The laws of mathematics, which are displaying the laws of nature, always work, regardless of whether we know them or do not know. That is the case with the law of multiplication and division by zero.

There is an old student's joke that the device that performs mathematical operations of multiplication and division by zero is an ordinary switch. Personally, I more trust the not fresh views of students, than the "scientific" opus of different "scientists". Usually, the first impression is correct.

All mathematical letups, that were written here, I deleted at first, because I believed that the harm from them will be more than good. Comments I cleaned similarly. But then changed my mind. If I will not tell about mathematical principles of work of electric switch, then others will not soon do it. And so, to begin a comment (to the page in Russian language):

"Idiot, Ohm's law correctly write down. Why is current is not equal to zero at zero voltage? And why does a resistance of burned out bulb is zero? You're either blind or can not to read. All is excellent divided by zero when you know the function of dependence. In most cases, the result is infinity. And about the real uncertainties you, probably, did not hear in general. I'm not going to talk about all the options, where it will not work. Even if not to pay attention to the serious errors in the examples. So I conclude you don't have a brain."

This is a typical reaction of a person to coach at zero, as a trained dog on a command "Attack!". It is thus needed to remember that the math is considering abstract concepts that aren't to need understood. In the result, we all turn into trained animals who think exactly as they were taught. I know from own experience how hard it is to get rid from the generally accepted stereotypes. So I explain that throughout the following discussion I will talk not about numerical values of physical quantities, and about the units, which are usually not considered in general mathematics.

Each of us in our daily lives every day many times uses multiplication and division by zero. Engineers, as well as we, suspecting nothing, created a special device that allows to multiply and divide by zero. And all of this is so firmly established in our lives, that without these devices is impossible to imagine our life around. But let's begin one after another, from the math.

All of you know the mathematical law of multiplication:

a*b = c

On the pages of this web-site I told about the mathematical rules of multiplication and division by zero. Take from this page these formulas, which we will use:

ab*0 = 0 and a or b

a/0 = ab

We will rewrite these formulas in a that kind, in what we will use them in our concrete example:

a*b*0 = (a*0)*b = a*(b*0) = c*0

c*0 = {c=0; a=0; b≠0} = {c=0; a≠0; b=0}

a/0 = b/0 = a*b = c


Now we'll check how these algebraic letups correspond to reality. For this, we represent the most ordinary domestic situation: you're sitting in the evening in the livingroom and suddenly light shuts off. What is your first thought? Correctly, either electricity disappeared or a bulb burned out. Or if trying is possible to think of two others variants: either you suddenly became blind or you suddenly died. Since the latter two options are more to biology, we do not consider they. But as far as exactly the first two variants can be described by our algebraic formulas, let's look.

Luminescence of bulb in physics is described by the Ohm's law that looks so:

I*R = U

In this kind the Ohm's law fully coincides the law of increase presented by us in an algebraic kind:

a*b = c

According to this, in the further reasoning, we can replace the algebraic elements of formulas by physical quantities:

a = I - it is a current that flows on wires, it is measured in Amperes;

b = R - it is a resistance to the electric current is in the spiral of bulb, it is measured in Ohms;

c = U - it is a voltage in an electrical circuit, that compels a bulb to shine, it is measured in Volts.

The first variant of the apocalyptic gloom involves to shutdown of knife-switch by some evil man (for to save energy but without our consent), as a result an electric current stops to enter on wires. Or broken wire in an accident on electrical networks.

U*0 = {U=0; I=0; R≠0}

As we see, this mathematical result reports us, that a bulb really left off to burn, as an electric current disappeared in wires, but with our bulb all is normal and she is ready again to begin to shine, as soon as a current will appear.

Now we will look at the second variant of total eclipse, when a bulb simply burned out for us, and with a current in electric networks all is normal:

U*0 = {U=0; I≠0; R=0}

As see, unlike traditional "any number increased on a zero equals a zero", we got establishment of fact of extinct bulb not only U=0, but also two possible reasons of this annoying incident : {I=0; R≠0} and {I≠0; R=0}.

It is here needed to mark that in traditional mathematics multiplying by a zero what or element of equality taken to one of basic mathematical equalities:

0=0

Usually all mathematics closes thereon. In the variant of multiplying offered by me by a zero this situation means disappearance of primary equality and passing to two inequalities - tension is not equal to strength of current and tension is not equal to resistance of electric chain :

U ≠ I

U ≠ R

In a general view for algebraic expression a*b = c it looks so:

c ≠ a

c ≠ b

For renewal of primary equality it is necessary to execute the mathematical operation of dividing by a zero. In our example it is necessary either to recover an electric current in wires or replace a burneout bulb. Thus there is the following:

I/0 = [I*(R*0)]/0 = I*(R*0/0) = I*(R*1) = I*R = U

R/0 = [(I*0)*R]/0 = (I*0/0)*R = (I*1)*R = I*R = U

Equality is used in our case 0/0=1, where as unit units come forward electric that or electric resistance. Introduction to the formula of any other unit of measuring will not result in a primary result, as electric tension ensues exceptionally co-operation of strength of current and resistance. you can go about in circles, winding meters long and почесывая itself in the back of head. You can get a purse and throw about money. Burneout bulb from it will not begin to shine:

I/0 = [I*(L*0)]/0 = I*(L*0/0) = I*(L*1) = I*L ≠ U

I/0 = [I*($*0)]/0 = I*($*0/0) = I*($*1) = I*$ ≠ U

As see, application of dividing by a zero supposes the presence of reason, but not dull implementation of mathematical actions.

In conclusion I want to say that engineers did switches allowing to execute multiplying and dividing by a zero in electric chains already a long ago. This device is basic custom control by electric chains. Switches are equip practically all electric devices: bulbs, engines, televisions, mobile telephones and other.

3.18.2011

About symmetry of mathematical actions

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

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

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

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

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

Rules of symmetry of mathematical actions:

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

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

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

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

0 – 3 – 7 – 4 = 0 – 7 – 3 – 4 = –14

0 + 3 – 7 – 4 = 0 – 7 + 3 – 4 = –8

1 х 3 х 7 х 4 = 1 х 7 х 3 х 4 = 84

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

1 х 3 : 7 : 4 = 1 : 7 х 3 : 4 = 3/28

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

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

0 – 3 – 7 – 4 = 0 – 7 – 3 – 4 = –14

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

0 – 3 + 7 + 4 = 0 + 7 – 3 + 4 = 8

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

1 х 3 х 7 х 4 = 1 х 7 х 3 х 4 = 84

1 : 3 х 7 х 4 = 1 х 7 : 3 х 4 = 28/3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

10.11.2010

Devidend of fraction as named?

The devidend of fraction is named numerator. A numerator is always written above a fractional line. It is needed nowhere to peep in search of numerator of fraction, type under a bed. A numerator of fraction is always on the most prominent position, from above fractional line, as a prince on a pea. As a numerator there can be numbers or mathematical expressions.

8.23.2010

How is a devidend named, that under a line?

In a fraction under a line a divizor is written. He is named denominator. If you must find the denominator of fraction, it is needed to search him under a fractional line. Type, how under a bed to glance. All denominators are always hidden under a fractional line, as under a bed. As a denominator there can be both numbers and whole mathematical expressions, sometimes very large.

Devidend of fraction as named?

The devidend of fraction is named numerator. A numerator is always written above a fractional line. It is needed nowhere to peep in search of numerator of fraction, type under a bed. A numerator of fraction is always on the most prominent position, from above fractional line, as a prince on a pea. As a numerator there can be numbers or mathematical expressions.

8.17.2010

Table division by zero

Division by zero is forbidden. Any number, positive or negative, whole or shot, to divide by zero is forbidden. Therefore a division table by zero will look so:

1 : 0 = division by zero is forbidden
2 : 0 = division by zero is forbidden
3 : 0 = division by zero is forbidden
4 : 0 = division by zero is forbidden
5 : 0 = division by zero is forbidden
6 : 0 = division by zero is forbidden
7 : 0 = division by zero is forbidden
8 : 0 = division by zero is forbidden
9 : 0 = division by zero is forbidden
10 : 0 = division by zero is forbidden

If to designate any number through а, then a division table by zero for any numbers will consist only of one line:

а : 0 = division by zero is forbidden

8.16.2010

Division by zero

It is accepted to consider in mathematics, that division by zero not possibly, as a result of division of number by zero can not be certain. Yet mathematicians it is said that division of number by zero behaves to the mathematical operations, to not making sense. Wikipedia asserts on this occasion, that in arithmetic, division by a zero is forbidden. Therefore, when in examples there is division by zero, it is said that an example does not have a decision, as division by zero is forbidden. This mathematical rule behaves to all, even to the blondes.

It becomes firmly established in very clever mathematical books, that division by zero possibly. More precisely, mathematicians thought of sly tricks, what this division by zero to go round a side. They are sure that it succeeded them. So, if in conversation with a clever mathematician, you will hear a phrase "I am able to divide by zero!", not surprised, your interlocutor believes sincerely, that it is possible.