How to learn multiplication table in 20 minutes. Description of the online simulator

It's not a secret for anyone how important it is to know the multiplication and division tables, in particular when performing arithmetic calculations and solving examples in mathematics.

However, what if the child is frightened by this huge set of numbers, called the "Multiplication and Division Table", and even knowing it by heart seems to be a completely overwhelming task?

Then we hasten to calm down - Learning the entire multiplication table is easy! To do this, you need to remember only 36 combinations of numbers (a bunch of three numbers). Here we do not take into account multiplication by 1 and 10, since this is an elementary action that does not require much effort in memorization.

Description of the online simulator

This simulator works on the basis of a specially developed algorithm for increasing the complexity of examples: starting with the simplest numbers "2 x 2", gradually increasing the difficulty to "9 x 9". Thus, smoothly luring into the learning process.

Thus, you will have to memorize the multiplication table in small portions, which will significantly reduce the load, since children will direct their attention to just a few examples, forgetting about the entire "large" volume.

The simulator has a settings menu for choosing the mode of studying the table. There is a choice of action - "Multiplication" or "Division", the range of examples "Entire table" or "By some number". All this is the extended functionality of the site and is available after payment.

Each new example is accompanied by help tip, so it will be easier for the child to start his study and memorize new combinations unknown to him.

If, in the course of training, any example causes difficulty, you can quickly remind yourself of its result by using additional hint, this will help you more effectively cope with memorizing difficult examples.

Percentage scale will quickly make you understand what level of knowledge of the multiplication table you have.

An example is considered fully learned if the correct answer was given. 4 times in a row... However, upon reaching 100% , we urge you not to quit studying, but to return the next day and refresh your knowledge by re-going through all the examples. After all, it is regular classes that develop memory and consolidate skills!

Description of the online simulator interface

Firstly, the simulator has a “quick access panel” that includes 4 buttons. They allow you to: go to the home page of the site, enable or disable sound signals, reset learning outcomes (start over), and also get to the reviews and comments page.

Secondly, this is the basic structure of the program.

Above all is percentage scale, showing the approximate level of knowledge of the multiplication table.

Below is example field, to which you need to answer. During the answer, it will change its color: it will turn red - if an incorrect answer was given, green - if the answer is correct, blue - after using the hint, and yellowish - when showing a new example.

Next is message line... It displays text information about errors, correct answers, as well as help and additional tips.

At the end is screen keyboard, containing only the buttons necessary for operation: all numbers, "backspace" - if you need to correct the answer, the "Check" and "Additional hint" buttons.

We are confident that this Multiplication Table in 20 Minutes simulator will help.

Learning fast with the best free game. Check it out for yourself!

Learn multiplication table - game

Try our educational e-game. Using it, you will be able to solve math problems in the classroom at the blackboard without answers tomorrow, without resorting to a sign to multiply the numbers. One has only to start playing, and in 40 minutes you will have an excellent result. And to consolidate the result, train several times, not forgetting to take breaks. Ideally, every day (save the page so you don't lose it). The play shape of the simulator is suitable for both boys and girls.

See the full cheat sheet below.

Multiplication directly on the site (online)

How to multiply numbers with a column (video on math)

To practice and learn quickly, you can also try column-multiplying numbers.

Multiplication table or the Pythagorean table is a well-known mathematical structure that helps students learn multiplication, and also just solve specific examples.

Below you can see it in its classic form. Pay attention to the numbers from 1 to 20, which are the headings of the lines on the left and the columns above. These are multipliers.

How to use the Pythagorean table?

1. So, in the first column we find the number that needs to be multiplied. Then in the top line we are looking for the number by which we will multiply the first. Now we look at where the line and column we need intersect. The number at this intersection is the product of these factors. In other words, this is the result of their multiplication.

As you can see, everything is pretty simple. You can see this table on our website at any time, and if necessary, you can save it to your computer as a picture in order to have access to it without an Internet connection.

2. And again, note that below there is the same table, but in a more familiar form - in the form mathematical examples... To many, this form will seem easier and more comfortable to use. It is also available for download to any medium in the form of a convenient picture.

Finally, you can use our calculator, which is present on this page, at the very bottom. Just enter the numbers you need for multiplication in the empty cells, click on the Calculate button, and a new number will appear in the Result window, which will be their product.

We hope this section will be useful to you, and our Pythagoras table in one form or another it will help you more than once in solving examples with multiplication and just for memorizing this topic.

Pythagoras table from 1 to 20

Multiplication table in standard form from 1 to 10

Multiplication table in standard form from 10 to 20

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The multiplication table is a basic concept in mathematics, which we get to know in elementary school and which we then use all our lives, regardless of profession. But the children are in no hurry to memorize the endless columns by heart, especially if the task was during the holidays.

site will give advice on how to easily learn the table with the children and make this process fun.

Pythagoras table

Despite the fact that the task is to learn, that is, memorize, the table by heart, first of all it is important to understand the essence of the action itself. To do this, you can replace multiplication by addition: the same numbers are added as many times as we multiply. For example, 6 × 8 would be to fold 8 times 6.

Highlight the same values

An excellent assistant for learning multiplication will be the Pythagorean table, which also demonstrates some patterns. For example what about When the multipliers are changed, the product does not change: 4 × 6 = 6 × 4. Mark such "mirror" answers with a certain color - this will help to remember and not get confused when repeating.

It is better to start studying the Pythagorean table with the simplest and most understandable parts: multiplication by 1, 2, 5 and 10. When multiplied by one, the number remains unchanged, while multiplying by 2 gives us twice the value. All answers to multiplication by 5 end in either 0 or 5. But multiplying by 10, in the answer we get a two-digit number from the digit that was multiplied and zero.

Table to consolidate the result

To consolidate the results, draw an empty Pythagoras table with the child and invite him to fill in the cells with the correct answers. To do this, you just need a piece of paper, a pencil and a ruler. You need to draw a square and divide it into 10 parts vertically and horizontally. And then fill in the top line and the leftmost column with numbers from 1 to 9, skipping the first cell.

Of course, all children are individual and there is no universal recipe. The main task of a parent is to find an approach and support his child, because we all once started with such at the same time simple and complex steps.

Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take as an example an infinite set of natural numbers, then the considered examples can be presented in the following form:

For a visual proof of their correctness, mathematicians have come up with many different methods. Personally, I look at all these methods as dancing shamans with tambourines. Essentially, they all boil down to the fact that either some of the rooms are not occupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of the century. Of course, the time factor can be stupidly ignored, but it will already be from the category "the law is not written for fools." It all depends on what we are doing: adjusting reality to match mathematical theories or vice versa.

What is an "endless hotel"? An endless hotel is a hotel that always has any number of vacant places, no matter how many rooms are occupied. If all the rooms in the endless visitor corridor are occupied, there is another endless corridor with the guest rooms. There will be an endless number of such corridors. Moreover, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, however, are not able to distance themselves from commonplace everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. Here are mathematicians and are trying to manipulate the serial numbers of hotel rooms, convincing us that it is possible to "shove the stuff in."

I will demonstrate the logic of my reasoning to you on the example of an infinite set of natural numbers. First, you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves, in Nature there are no numbers. Yes, Nature is excellent at counting, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers there are. Consider both options, as befits a real scientist.

Option one. "Let us be given" a single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have it. And if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:

I wrote down the actions in the algebraic notation system and in the notation system adopted in set theory, with a detailed enumeration of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one subtracts from it and adds the same unit.

Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:

Subscripts "one" and "two" indicate that these items belonged to different sets. Yes, if you add one to the infinite set, the result will also be an infinite set, but it will not be the same as the original set. If we add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

Lots of natural numbers are used for counting in the same way as a ruler for measurements. Now imagine adding one centimeter to the ruler. This will already be a different line, not equal to the original.

You can accept or not accept my reasoning - it's your own business. But if you ever run into mathematical problems, think about whether you are not following the path of false reasoning trodden by generations of mathematicians. After all, doing mathematics, first of all, form a stable stereotype of thinking in us, and only then add mental abilities to us (or, on the contrary, deprive us of free thought).

Sunday, 4 August 2019

I was writing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... the rich theoretical foundation of Babylonian mathematics did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."

Wow! How smart we are and how well we can see the shortcomings of others. Is it hard for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:

The rich theoretical basis of modern mathematics is not holistic and is reduced to a set of disparate sections devoid of a common system and evidence base.

I will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious blunders of modern mathematics. See you soon.

Saturday, 3 August 2019

How do you divide a set into subsets? To do this, it is necessary to enter a new unit of measurement that is present for some of the elements of the selected set. Let's look at an example.

Let us have many A consisting of four people. This set was formed on the basis of "people" Let us denote the elements of this set by the letter a, a subscript with a digit will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sex" and denote it by the letter b... Since sexual characteristics are inherent in all people, we multiply each element of the set A by gender b... Note that now our multitude of "people" has become a multitude of "people with sex characteristics." After that, we can divide the sex characteristics into masculine bm and women bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sex characteristics, it does not matter which one is male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.

After multiplication, reduction and rearrangement, we got two subsets: the subset of men Bm and a subset of women Bw... Mathematicians think about the same when they apply set theory in practice. But they do not devote us to the details, but give a finished result - "a lot of people consist of a subset of men and a subset of women." Naturally, you may wonder how correctly the mathematics is applied in the above transformations? I dare to assure you, in fact, the transformations were done correctly, it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I'll tell you about it.

As for supersets, you can combine two sets into one superset by choosing the unit of measurement that is present for the elements of these two sets.

As you can see, units of measurement and common mathematics make set theory a thing of the past. An indication that set theory is not all right is that mathematicians have come up with their own language and notation for set theory. Mathematicians did what shamans once did. Only shamans know how to "correctly" apply their "knowledge". They teach us this "knowledge".

Finally, I want to show you how mathematicians manipulate with.

Monday, January 7, 2019

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". This is how it sounds:

Let's say Achilles runs ten times faster than a turtle and is a thousand steps behind it. During the time it takes Achilles to run this distance, the turtle will crawl a hundred steps in the same direction. When Achilles has run a hundred steps, the turtle will crawl ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the turtle.

This reasoning came as a logical shock to all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them has become a generally accepted solution to the question ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from magnitude to. This transition implies application instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement either has not yet been developed, or it has not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant units of measurement of time to the reciprocal. From a physical point of view, it looks like time dilation until it stops completely at the moment when Achilles is level with the turtle. If time stops, Achilles can no longer overtake the turtle.

If we turn over the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly catch up with the turtle."

How can you avoid this logical trap? Stay in constant time units and do not go backwards. In Zeno's language, it looks like this:

During the time during which Achilles will run a thousand steps, the turtle will crawl a hundred steps in the same direction. Over the next interval of time, equal to the first, Achilles will run another thousand steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the turtle.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insuperability of the speed of light is very similar to the Zeno aporia "Achilles and the Turtle". We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia Zeno tells about a flying arrow:

The flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is motion. Another point should be noted here. From a single photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs are needed, taken from the same point at different points in time, but it is impossible to determine the distance from them. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.

Wednesday, 4 July 2018

I have already told you that, with the help of which shamans try to sort "" reality. How do they do it? How does the formation of a set actually take place?

Let's take a closer look at the definition of a set: "a set of different elements, thought of as a single whole." Now feel the difference between the two phrases: "thinkable as a whole" and "thinkable as a whole." The first phrase is the end result, the set. The second phrase is preliminary preparation for the formation of a set. At this stage, reality is broken down into separate elements ("whole") from which a set will then be formed ("a single whole"). At the same time, the factor that makes it possible to unite the "whole" into a "single whole" is carefully monitored, otherwise the shamans will fail. After all, shamans know in advance what kind of multitude they want to demonstrate to us.

Let me show you the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, but there are no bows. After that we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.

Now let's do a little dirty trick. Take "solid in a pimple with a bow" and combine these "wholes" by color, selecting the red elements. We got a lot of "red". Now a question to fill in: the resulting sets "with a bow" and "red" are the same set or are they two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We have formed a set of "red solid into a bump with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a pimple), ornaments (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics... This is what it looks like.

The letter "a" with different indices denotes different units of measurement. Units of measurement are highlighted in brackets, by which the "whole" is allocated at the preliminary stage. The unit of measurement, by which the set is formed, is taken out of the brackets. The last line shows the final result - the element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, not the dancing of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing it “by evidence,” because units of measurement are not included in their “scientific” arsenal.

It is very easy to use units to split one or combine several sets into one superset. Let's take a closer look at the algebra of this process.

Saturday, 30 June 2018

If mathematicians cannot reduce a concept to other concepts, then they do not understand anything in mathematics. I answer: how do the elements of one set differ from the elements of another set? The answer is very simple: numbers and units.

Today, everything that we do not take belongs to some set (as mathematicians assure us). By the way, have you seen on your forehead in the mirror a list of those sets to which you belong? And I have not seen such a list. I will say more - not a single thing in reality has a tag with a list of the sets to which this thing belongs. The multitudes are all the inventions of shamans. How do they do it? Let's look a little deeper in history and see what the elements of a set looked like before shamanic mathematicians pulled them apart into their sets.

A long time ago, when no one had ever heard of mathematics, and only trees and Saturn had rings, huge herds of wild set elements roamed physical fields (after all, shamans had not yet invented mathematical fields). They looked something like this.

Yes, do not be surprised, from the point of view of mathematics, all the elements of sets are most similar to sea urchins - from one point, like needles, units of measurement stick out in all directions. For those who, I remind you that any unit of measurement can be geometrically represented as a segment of arbitrary length, and a number as a point. Geometrically, any value can be represented as a bunch of segments sticking out in different directions from one point. This point is point zero. I won't draw this piece of geometric art (no inspiration), but you can easily imagine it.

What units of measurement form an element of the set? Anyone describing this element from different points of view. These are the ancient units of measurement that were used by our ancestors and which everyone has long forgotten about. These are the modern units of measurement that we use now. These are also unknown units of measurement that our descendants will invent and which they will use to describe reality.

We figured out the geometry - the proposed model of the elements of the set has a clear geometric representation. What about physics? Units of measurement are the direct connection between mathematics and physics. If shamans do not recognize units of measurement as a full-fledged element of mathematical theories, this is their problem. I personally cannot imagine the real science of mathematics without units of measurement. That is why, at the very beginning of my story about set theory, I spoke of it as the Stone Age.

But let's move on to the most interesting thing - to the algebra of elements of sets. Algebraically, any element of a set is a product (the result of multiplication) of different quantities. It looks like this.

I deliberately did not use the conventions of set theory, since we were looking at a set element in its natural habitat prior to the emergence of set theory. Each pair of letters in brackets denotes a separate value, consisting of the number indicated by the letter " n"and units of measurement indicated by the letter" a". The indices next to the letters indicate that the numbers and units of measurement are different. One element of the set can consist of an infinite number of quantities (as far as we and our descendants have enough imagination). Each bracket is geometrically depicted as a separate segment. In the example with the sea urchin one bracket is one needle.

How do shamans form sets from different elements? In fact, by units or numbers. Without understanding anything in mathematics, they take different sea urchins and carefully examine them in search of that single needle, along which they form a set. If there is such a needle, then this element belongs to the set, if there is no such needle, it is an element not from this set. Shamans tell us fables about thought processes and a single whole.

As you may have guessed, the same element can belong to very different sets. Further I will show you how sets, subsets and other shamanic nonsense are formed. As you can see, "there cannot be two identical elements in a set", but if there are identical elements in a set, such a set is called a "multiset". Such logic of absurdity will never be understood by rational beings. This is the level of talking parrots and trained monkeys, who lack intelligence from the word "completely". Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the incompetent engineer died under the rubble of his creation. If the bridge could withstand the load, a talented engineer would build other bridges.

No matter how mathematicians hide behind the phrase "chur, I'm in the house", or rather "mathematics studies abstract concepts," there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to the mathematicians themselves.

We studied mathematics very well and now we are sitting at the checkout, giving out salaries. Here comes a mathematician to us for his money. We count the entire amount to him and lay out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and hand the mathematician his “mathematical set of salary”. Let us explain the mathematics that he will receive the rest of the bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.

First of all, the logic of the deputies will work: "You can apply it to others, you can not apply it to me!" Further, we will begin to assure us that there are different banknote numbers on bills of the same denomination, which means that they cannot be considered the same elements. Okay, let's count the salary in coins - there are no numbers on the coins. Here the mathematician will start to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms in each coin is unique ...

And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science did not lie anywhere near here.

Look here. We select football stadiums with the same pitch. The area of ​​the fields is the same, which means we have got a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How is it correct? And here the mathematician-shaman-shuller takes a trump ace out of his sleeve and begins to tell us either about the set or about the multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "thinkable as not a single whole" or "not thinkable as a whole."