Showing posts with label multiplicatoin. Show all posts
Showing posts with label multiplicatoin. Show all posts


Multiplication by zero and zero element

Mathematicians claim that when multiplied by zero, multiplication occurs and "multiplying by zero (zero element) gives a number equal to zero." Let's look at a couple of examples.

Zero on the football field

We go to the stadium to watch European football. This is exactly what blondes need. Firstly, beautiful gerls are shown on TV. Secondly, in European football, twenty-two millionaires kick the same ball with varying degrees of success. Where else will blondes find so many rich suitors in one place? And so, the game is in full swing. One player has broken the rules and is removed from the field. What is left on the field instead of this player? An empty space that cannot take part in the game. If the players are numbers, then the empty space is zero. Empty space is not a player, zero is not a number.

Now consider the "null element", which is no different from the "numeric elements". The same football match, the same situation - the player was sent off. And here is the main trick - instead of a remote player, a "zero player" with the number "zero" on a T-shirt enters the field. He joins the game and soon scores a goal. This is where "higher mathematics" begins. One team proves that the zero player is exactly the same as the rest of the players, therefore, has the right to score goals. The other team proves that this is a removed player and he does not have the right to score goals. Here's a great excuse for you to start a "special military operation" like a "football war" between El Salvador and Honduras in 1969. The idiocy of such a situation needs no comment.

Multiplication by zero in the store

Zero dollars. Mathematics For Blondes.
Zero dollars

Another example from our life. We all go to the store and buy something. What is the buying process? This is the exchange of the money we have for the goods available in the store. The buying process itself can be compared to multiplication. If the buyer has money, and the store has goods, there are no problems. If the customer doesn't have the money or the store doesn't have the product, then you can't make the purchase. You wouldn’t go to the store with an empty wallet to hear from the seller that you can’t buy anything without money? This situation can be seen as an example of multiplication by zero.

Now consider the process of buying with "zero element". Imagine that you have a bill in your wallet that says "zero dollars". You go to the store with this bill and exchange it for a piece of paper with the inscription "zero goods". Technically, you made a purchase without having a cent in your pocket and without buying anything. Mathematicians tell us about a similar “multiplication by zero (zero element)”.

Multiplication by zero is not possible

The substitution of concepts can change our logic beyond recognition. This modified logic forces us to look for multiplication by zero where it cannot be - in the results of multiplication. Since multiplication by zero does not occur, then you need to look not into the void (after all, there is no result of multiplication by zero), but into the initial conditions of multiplication. Two apples both lay and will remain lying, even after casting the spell "Apples, I multiply you by zero." Mathematically, this is written to the point of banality simply:


All this happens because our mathematicians have not learned to adequately describe reality with the help of mathematics. If you want to look at multiplication examples taken from real life, then you can do it here.

We will consider multiplication by zero in geometry and physics in more detail, to be continued.

P.S. What should you do? Remember that zero is not a number. And when it comes to zero in mathematics, discard your logic and common sense and open the Holy Mathematical Scripture. What is written about the case you are interested in, then tell the mathematicians. You will not begin to assert in a theological seminary that there is no God. So I do not recommend arguing with mathematicians - this is fraught with serious consequences for you. When you become adults and mathematicians disappear from your life, then you can say what you think is right.


Multiplication by zero

In the comments to the article "Multiplication by zero" of the Russian version of this site, I was asked an interesting question:

Nikolai, I read the article halfway, but still ... There are two apples in front of me (fact). Further, I, like a "sorcerer", multiply them by zero and still see two apples in front of me! Although, according to the laws of arithmetic, they should have disappeared from me! What does mathematics say about this? Thanks for the answer.

Two apples. Mathematics For Blondes.
Two apples

Here they are, beauties. They lie down and smile. Like, well, what do you say to that? So what is multiplication by zero? Let's try to figure this out.

Pay attention, the question is formulated very cleverly: not "what do mathematicians say?", but "what does mathematics say?". The first question is the easiest to answer. Preachers say, "Read the Bible," mathematicians say, "Read the Definition." They answer stupidly. Nobody needs to explain anything. And the meticulous can always, with a smart look, hang noodles on their ears.

Next, we will consider the situation from the position of the "sorcerer". The sorcerer declares that he will multiply the apples by zero. Then the sorcerer says: "Close your eyes and do not open." While your eyes are closed, the sorcerer hides the apples. "Open your eyes. See the apples? They are not there. The great miracle of multiplication by zero has happened - the apples have disappeared!" The sorcerer-mathematician will surely add: "What was required to be proved."

Now a few words about mathematicians. They, like proud eagles, soar high in the clouds of their abstract ideas. Mathematicians descend to our sinful earth only when they see food - a problem that they can solve. Mathematicians have learned very well to tear numbers from reality and perform various manipulations with them. When it becomes necessary to bring numbers back to reality, sometimes very big problems arise. Multiplying by zero is one such problem.

Let's start from the very beginning. The Russian-language Wikipedia page in one of the versions wrote: "Multiplication is one of the main binary mathematical operations (arithmetic operations) of two arguments (multiplicand and multiplier), the result of which is a new number (product). ... Multiplication by zero (zero element) gives a number equal to zero: x ⋅ 0 = 0".

If we translate the above quote into ordinary human language, then two elements (multiplicand and multiplier) are needed for multiplication. After multiplying them, a new element will be obtained, which is the result of multiplication. It is customary to write it like this:


On the left side of the equal sign is written what precedes the multiplication. The result of the multiplication is written on the right side. One element is multiplied by another element, resulting in a third element.

If we consider the logic of mathematicians, then calling zero the "zero element", all the "laws" of multiplication are observed - when multiplied by the zero element, all other elements turn into the zero element. There is only one question left: "Where do the apples go?".

Now I will present my own view on the problem of multiplication by zero. First read my reasoning, and at the end I will give practical recommendations on how to use my new knowledge. So what does the math say about multiplying by zero?

From the point of view of mathematics, multiplication by zero is impossible, since the multiplication itself does not occur. If in my earlier works I stated something else, then I was mistaken. The process of cognition is continuous and what seemed right to me yesterday may look completely different today.

The positional notation of numbers looks like this: units, tens, hundreds... If there is a number in the positional notation, then we write it down. For example, 324 is three hundred, two tens, four ones. And if there is no number in a separate position? What then? We write zero instead of the number that is missing. For example, 304 is three hundred, no tens, four units. I affirm that the absence of a number cannot be a number. In other words, zero is not a number and the rules of numbers do not apply to it.

In the multiplication example, zero represents an empty space in the place of one of the factors and an empty space in the result of the multiplication. Multiplication, as a mathematical operation, does not occur. It's like trying to clap with one hand. To get sound, there must be two palms. You see how smart we have become: we have determined that clapping is a binary operation that can be described by the mathematical operation of multiplication:

[one palm]*[other palm]=[applause]

Do you want to add numbers here? Please:


Now let's remove one palm. In our mathematical expression, we will replace one of the palms with zero and look at the result.



In order for the multiplication to occur, we need two completely different palms, and not the same one. Mathematicians tell us that when a number is raised to the second power, it multiplies itself. It is just as impossible to multiply a number by itself as it is impossible to create an applause with one palm.

You can say that in audio equipment, one speaker can produce sound, the second speaker is not needed for this. In the case of a speaker, there is another multiplication formula: the speaker is multiplied by the electric current and the result of the multiplication is sound.

1[speaker]*1[electric current]=1[sound]

If there is no speaker (for example, it is broken), the electric current cannot produce sound.

0[speaker]*1[electric current]=0[sound]

If there is no electric current (Putin cut off the wires), the speaker cannot produce sound.

1[speaker]*0[electric current]=0[sound]

In ordinary human language, the operation of multiplication by zero can be translated as follows:


Multiplication does not occur because there is nothing to multiply, the result of multiplication is absent.


Multiplication does not occur because there is nothing to multiply by, the result of multiplication is absent.

Something like this. Next time I will tell you about the zero element in multiplication.


Cube of 101

Today we will look at an example of how you can find one hundred and one cubed using the abbreviated multiplication formulas. In other words, how to raise cube of 101 using the abbreviated multiplication formula.

The following cry for help was heard in the comments:

Help please, problem (example) from the physics and mathematics school (Russia), grade 8:


I don't understand how to solve this with abbreviated multiplication formulas.

It is very easy to cube the number 101 - you need to type the number 101 on a calculator and multiply twice by the same number 101.


If you don't have a calculator at hand (you never know, the phone has just been stolen), then you can calculate on a piece of paper in a column (the picture will be at the end, like checking both the calculator and the formula).

Cube of 101
We apply the cube of sum formula.

The problem statement says that you need to find the cube of 101 using the abbreviated multiplication formula. According to math teachers, everyone should know these formulas. Naive. Where to find this formula?

You can search the Internet for a cube of sum. Google to help you. You can find these formulas in a reference book on mathematics, in a textbook on mathematics, you can ask a classmate who knows the formulas for abbreviated multiplication by heart. The formula for the abbreviated multiplication we need is called cube of sum. You will see it below.

And now the answer to the most tricky question: how to get the sum of numbers from one number? It is necessary to expand this number into terms. From the point of view of mathematics, the number of terms can be any, but ... I found the abbreviated multiplication formulas for raising a sum to a cube only for two and three terms. The formula for the cube of the sum of three terms is very complicated, I wish you never to face such a thing. But the cube of the sum of the two terms looks nice. The basic principle of expansion into terms for applying abbreviated multiplication formulas is that numbers can be easily multiplied in your head without using a calculator. For the number 101, the best option would be 101 = 100 + 1. 100 and 1 numbers is easy to multiply without a calculator. Let's see what we get.

Cube of 101. 101 cubed. Cube of sum. Mathematics For Blondes.
Cube of 101

I don't know about you, but I couldn't do without a piece of paper. Yes, I wrote everything down in a line, not a column, but nevertheless. And in conclusion, let's check our solution by multiplying in a column on a piece of paper.


Cube of 101 in a column on a piece of paper. Mathematics For Blondes.
Cube of 101 in a column on a piece of paper

Well, something like this. Why do you need all this? So that you know that you can multiply not only on a calculator or in a column, but sometimes you can effectively use abbreviated multiplication formulas. If, of course, you know them.


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?


Decomposition on factors

Subject of occupations:
Subject of the previous lesson
Zero and infinity

Lesson 12

Decomposition on factors

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

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

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

Decomposition of volume. Mathematics For Blondes.
Decomposition of volume

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

At the following lesson we will consider


Distinctions between multiplication and addition

Subject of occupations:
Subject of the previous lesson
Examples of multiplication

Lesson 10

Distinctions between multiplication and addition

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

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

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

Multiplication in algebra. Mathematics For Blondes.
Multiplication in algebra

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

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

In mathematician possess separate properties of units of measure:

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

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

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

- legends of functions (for example, trigonometrical).

At the following lesson we will consider
Zero and infinity

Examples of multiplication

Subject of occupations:
Subject of the previous lesson

Lesson 9

Examples of multiplication

Multiplication in space. Mathematics For Blondes.
Multiplication in space

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

Examples of multiplication. Mathematics For Blondes.
Examples of multiplication

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

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

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

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

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

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



Subject of occupations:
Subject of the previous lesson
Variable units of measure

Lesson 8


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

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

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

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

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

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

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

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

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

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

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

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

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

Algebraic transformations. Mathematics For Blondes.
Algebraic transformations

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

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

At the following lesson we will consider
Examples of multiplication


Increased - that's a plus or multiply?

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

7 + 2 = 9

Increased. This is a plus or multiply. Mathematics for blondes.

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

7 * 2 = 14

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


9 multiply 7 equals

Immediately write a ready answer: 9 multiply 7 equals 63. So we have the multiplication table says that we all have been taught, but to the end, many of us have not learned. I'm not an exception. Natural laziness and chronic multiple sclerosis did not allow me to remember this vital piece of mathematics. But it is knowledge of mathematics allows me to easily cope with this column multiplication tables. I just calculate the result I needed.

We all know that when multiplied by a permutation of the factors does not change the result. If somewhere sometime mathematics will tell you otherwise, do not believe them - they are wrong. Simple math multiplication like to call it, that in fact is not a multiplication. For example, if we call a cow "Multiplication", it then becomes to turn milk and poop in the hay. After all, the cow to the mathematical action has nothing to do.

So, I always use the following equation:

9*7 = 7*9

Next is quite simple. That number, which must be multiplied by 9, I multiply by 10 and take away his once:

7*9 = 7*10-7 = 70-7 = 63

Multiplied by 10 is very simple - have added to the number of the toe and the result is ready. Subtract this number from a small number of possible even without a calculator. Now you can easily cope with any stitch multiplication table 9.

For example, if we need to multiply 9 by 9, proceed as follows:

9*9 = 90-9 = 81

If needed multiply 9 by 3:

9*3 = 3*9 = 30-3 = 27

In all these transformations have no miracles, we just use the normal math.

9 multiply. The multiplication table for 9. Mathematics for blondes.
9 multiply

My personal opinion is: better to use the thinking abilities of their brain than clog the brain remembering any nonsense. Even if this nonsense forced to learn mathematics.


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.


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.


The multiplication table to 20

The multiplication table to 20 square still is called the table of Pythagoras. To download the multiplication table it is possible by means of the right button of the mouse, having chosen in the menu "to Keep the image as...". After preservation, this multiplication table can be unpacked. To learn or not to learn this multiplication table - business voluntary. What for to load the memory the multiplication table to twenty when there is a picture and calculators? In your operative memory, that is in a head, there is enough multiplication table to ten. It three times is less numbers.

The multiplication table to 20, multiplication chart. Mathematics for  blondes.
The multiplication table to 20

table multiplication 11 à 20 - a picture can be looked here.


Multiplication by zero

Multiplication by zero is possible, rules of mathematics multiplication by zero is not forbidden. Any number, multiplication by zero, will equal zero. If a whole or fractional number multiplication by zero, zero will ensue.

We will consider the example of multiplication by zero of integer. How many will it be, if 2 (two) to multiplication by 0 (zero)?

2 х 0 = 0

Decision: if 2 (two) to multiplication by 0 (zero), 0 (zero) will turn out.

Example of multiplication by zero of broken number. How many will it be, if 0,25 (zero whole twenty five hundredth) to multiplication by 0 (zero)?

0,25 х 0 = 0

Decision: if 0,25 (zero whole twenty five hundredth) to multiplication by 0 (zero), 0 (zero) will turn out.

If to multiplication a positive or negative number by zero, zero will turn out. A number zero does not have sign, therefore signs a plus or minus before zero is not put. Examples of multiplication of positive whole and fractional numbers are made a higher.

Example of multiplication by zero of negative number. How many will it be, if -2 (minus two) to multiplication by 0 (zero)?

-2 х 0 = 0

Decision: if -2 (minus two) to multiplication by zero, there will be 0 (zero).