December 23, 2013 at 03:10 pm

Effective mental arithmetic or brain exercise

• Mathematics

This article is inspired by the topic and is intended to spread the techniques of S.A. Rachinsky for oral counting.
Rachinsky was a wonderful teacher who taught in rural schools in the 19th century and showed from his own experience that it is possible to develop the skill of rapid mental calculation. For his students, it was not particularly difficult to calculate such an example in their heads:

Using round numbers

One of the most common mental counting techniques is that any number can be represented as a sum or difference of numbers, one or more of which are “round”:

Because on 10 , 100 , 1000 etc. it’s faster to multiply round numbers; in your mind you need to reduce everything to such simple operations as 18 x 100 or 36 x 10. Accordingly, it is easier to add by “splitting off” a round number and then adding a “tail”: 1800 + 200 + 190 .
Another example:
31 x 29 = (30 + 1) x (30 - 1) = 30 x 30 - 1 x 1 = 900 - 1 = 899.

Let's simplify multiplication by division

When counting mentally, it can be more convenient to operate with a dividend and a divisor rather than with a whole number (for example, 5 represent in the form 10:2 , A 50 as 100:2 ):
68 x 50 = (68 x 100) : 2 = 6800: 2 = 3400; 3400: 50 = (3400 x 2) : 100 = 6800: 100 = 68.
Multiplying or dividing by is done in the same way. 25 , after all 25 = 100:4 . For example,
600: 25 = (600: 100) x 4 = 6 x 4 = 24; 24 x 25 = (24 x 100) : 4 = 2400: 4 = 600.
Now it doesn't seem impossible to multiply in your head 625 on 53 :
625 x 53 = 625 x 50 + 625 x 3 = (625 x 100) : 2 + 600 x 3 + 25 x 3 = (625 x 100) : 2 + 1800 + (20 + 5) x 3 = = (60000 + 2500) : 2 + 1800 + 60 + 15 = 30000 + 1250 + 1800 + 50 + 25 = 33000 + 50 + 50 + 25 = 33125.

Squaring a two-digit number

It turns out that in order to simply square any two-digit number, it is enough to remember the squares of all numbers from 1 before 25 . Fortunately, squares up 10 we already know from the multiplication table. The remaining squares can be seen in the table below:

Rachinsky's technique is as follows. In order to find the square of any two-digit number, you need the difference between this number and 25 multiply by 100 and to the resulting product add the square of the complement of the given number to 50 or the square of its excess over 50 -Yu. For example,
37^2 = 12 x 100 + 13^2 = 1200 + 169 = 1369; 84^2 = 59 x 100 + 34^2 = 5900 + 9 x 100 + 16^2 = 6800 + 256 = 7056;
In general ( M- two-digit number):

Let's try to apply this trick when squaring a three-digit number, first breaking it into smaller terms:
195^2 = (100 + 95)^2 = 10000 + 2 x 100 x 95 + 95^2 = 10000 + 9500 x 2 + 70 x 100 + 45^2 = 10000 + (90+5) x 2 x 100 + + 7000 + 20 x 100 + 5^2 = 17000 + 19000 + 2000 + 25 = 38025.
Hmm, I wouldn’t say that it’s much easier than erecting it in a column, but perhaps over time you can get used to it.
And, of course, you should start training by squaring two-digit numbers, and from there you can even get to disassembling in your mind.

Multiplying two-digit numbers

This interesting technique was invented by a 12-year-old student of Rachinsky and is one of the options for adding to a round number.
Let two two-digit numbers be given whose sum of units is 10:
M = 10m + n, K = 10a + 10 - n.
Compiling their product, we get:

For example, let's calculate 77 x 13. The sum of the units of these numbers is equal to 10 , because 7 + 3 = 10 . First we put the smaller number before the larger one: 77 x 13 = 13 x 77.
To get round numbers, we take three units from 13 and add them to 77 . Now let's multiply the new numbers 80 x 10, and to the result we add the product of the selected 3 units by the difference of the old number 77 and a new number 10 :
13 x 77 = 10 x 80 + 3 x (77 - 10) = 800 + 3 x 67 = 800 + 3 x (60 + 7) = 800 + 3 x 60 + 3 x 7 = 800 + 180 + 21 = 800 + 201 = 1001.
This technique has a special case: everything is greatly simplified when two factors have the same number of tens. In this case, the number of tens is multiplied by the number following it and the product of the units of these numbers is added to the resulting result. Let's see how elegant this technique is with an example.
48 x 42. Tens number 4 , next number: 5 ; 4 x 5 = 20 . Product of units: 8 x 2 = 16 . So 48 x 42 = 2016.
99 x 91. Tens number: 9 , next number: 10 ; 9 x 10 = 90 . Product of units: 9 x 1 = 09 . So 99 x 91 = 9009.
Yeah, that is, to multiply 95 x 95, just count 9 x 10 = 90 And 5 x 5 = 25 and the answer is ready:
95 x 95 = 9025.
Then the previous example can be calculated a little simpler:
195^2 = (100 + 95)^2 = 10000 + 2 x 100 x 95 + 95^2 = 10000 + 9500 x 2 + 9025 = 10000 + (90+5) x 2 x 100 + 9000 + 25 = 10000 + 19000 + 1000 + 8000 + 25 = 38025.

Instead of a conclusion

It would seem, why be able to count in your head in the 21st century, when you can simply give a voice command to your smartphone? But if you think about it, what will happen to humanity if it puts on machines not only physical work, but also any mental work? Isn't it degrading? Even if you do not consider mental arithmetic as an end in itself, it is quite suitable for training the mind.

References:
“1001 problems for mental arithmetic at the school of S.A. Rachinsky".

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Multiplication table (numbers from 1 to 20)

×	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36	38	40
3	3	6	9	12	15	18	21	24	27	30	33	36	39	42	45	48	51	54	57	60
4	4	8	12	16	20	24	28	32	36	40	44	48	52	56	60	64	68	72	76	80
5	5	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80	85	90	95	100
6	6	12	18	24	30	36	42	48	54	60	66	72	78	84	90	96	102	108	114	120
7	7	14	21	28	35	42	49	56	63	70	77	84	91	98	105	112	119	126	133	140
8	8	16	24	32	40	48	56	64	72	80	88	96	104	112	120	128	136	144	152	160
9	9	18	27	36	45	54	63	72	81	90	99	108	117	126	135	144	153	162	171	180
10	10	20	30	40	50	60	70	80	90	100	110	120	130	140	150	160	170	180	190	200
11	11	22	33	44	55	66	77	88	99	110	121	132	143	154	165	176	187	198	209	220
12	12	24	36	48	60	72	84	96	108	120	132	144	156	168	180	192	204	216	228	240
13	13	26	39	52	65	78	91	104	117	130	143	156	169	182	195	208	221	234	247	260
14	14	28	42	56	70	84	98	112	126	140	154	168	182	196	210	224	238	252	266	280
15	15	30	45	60	75	90	105	120	135	150	165	180	195	210	225	240	255	270	285	300
16	16	32	48	64	80	96	112	128	144	160	176	192	208	224	240	256	272	288	304	320
17	17	34	51	68	85	102	119	136	153	170	187	204	221	238	255	272	289	306	323	340
18	18	36	54	72	90	108	126	144	162	180	198	216	234	252	270	288	306	324	342	360
19	19	38	57	76	95	114	133	152	171	190	209	228	247	266	285	304	323	342	361	380
20	20	40	60	80	100	120	140	160	180	200	220	240	260	280	300	320	340	360	380	400

How to multiply numbers in a column (mathematics video)

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

Verbal counting- an activity that fewer and fewer people bother with these days. It’s much easier to take out a calculator on your phone and calculate any example.

But is this really so? In this article, we will present math hacks that will help you learn how to quickly add, subtract, multiply and divide numbers in your head. Moreover, operating not with units and tens, but with at least two-digit and three-digit numbers.

After mastering the methods in this article, the idea of ​​reaching into your phone for a calculator will no longer seem so good. After all, you can not waste time and calculate everything in your head much faster, and at the same time stretch your brains and impress others (of the opposite sex).

We warn you! If you are an ordinary person and not a child prodigy, then developing mental arithmetic skills will require training and practice, concentration and patience. At first everything may be slow, but then things will get better and you will be able to quickly count any numbers in your head.

Gauss and mental arithmetic

One of the mathematicians with phenomenal mental arithmetic speed was the famous Carl Friedrich Gauss (1777-1855). Yes, yes, the same Gauss who invented the normal distribution.

In his own words, he learned to count before he spoke. When Gauss was 3 years old, the boy looked at his father's payroll and declared, "The calculations are wrong." After the adults double-checked everything, it turned out that little Gauss was right.

Subsequently, this mathematician reached considerable heights, and his works are still actively used in theoretical and applied sciences. Until his death, Gauss performed most of his calculations in his head.

Here we will not engage in complex calculations, but will start with the simplest.

Adding numbers in your head

To learn how to add large numbers in your head, you need to be able to accurately add numbers up to 10 . Ultimately, any complex task comes down to performing a few trivial actions.

Most often, problems and errors arise when adding numbers with “passing through 10 " When adding (and even when subtracting), it is convenient to use the “support by ten” technique. What is this? First, we mentally ask ourselves how much one of the terms is missing to 10 , and then add to 10 the difference remaining until the second term.

For example, let's add the numbers 8 And 6 . To from 8 get 10 , lacks 2 . Then to 10 all that remains is to add 4=6-2 . As a result we get: 8+6=(8+2)+4=10+4=14

The main trick to adding large numbers is to break them down into place value parts, and then add those parts together.

Suppose we need to add two numbers: 356 And 728 . Number 356 can be represented as 300+50+6 . Likewise, 728 will look like 700+20+8 . Now we add:

356+728=(300+700)+(50+20)+(8+6)=1000+70+14=1084

Subtracting numbers in your head

Subtracting numbers will also be easy. But unlike addition, where each number is broken down into place value parts, when subtracting we only need to “break down” the number we are subtracting.

For example, how much will 528-321 ? Breaking down the number 321 into bit parts and we get: 321=300+20+1 .

Now we count: 528-300-20-1=228-20-1=208-1=207

Try to visualize the processes of addition and subtraction. At school everyone was taught to count in a column, that is, from top to bottom. One way to restructure your thinking and speed up counting is to count not from top to bottom, but from left to right, breaking numbers into place parts.

Multiplying numbers in your head

Multiplication is the repetition of a number over and over again. If you need to multiply 8 on 4 , this means that the number 8 need to repeat 4 times.

8*4=8+8+8+8=32

Since all complex problems are reduced to simpler ones, you need to be able to multiply all single-digit numbers. There is a great tool for this - multiplication table . If you do not know this table by heart, then we strongly recommend that you learn it first and only then start practicing mental counting. Besides, there is essentially nothing to learn there.

Multiplying multi-digit numbers by single-digit numbers

First, practice multiplying multi-digit numbers by single-digit numbers. Let it be necessary to multiply 528 on 6 . Breaking down the number 528 into ranks and go from senior to junior. First we multiply and then add the results.

528=500+20+8

528*6=500*6+20*6+8*6=3000+120+48=3168

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Multiplying two-digit numbers

There is nothing complicated here either, only the load on short-term memory is a little greater.

Let's multiply 28 And 32 . To do this, we reduce the entire operation to multiplication by single-digit numbers. Let's imagine 32 How 30+2

28*32=28*30+28*2=20*30+8*30+20*2+8*2=600+240+40+16=896

One more example. Let's multiply 79 on 57 . This means that you need to take the number " 79 » 57 once. Let's break the whole operation into stages. Let's multiply first 79 on 50 , and then - 79 on 7 .

• 79*50=(70+9)*50=3500+450=3950
• 79*7=(70+9)*7=490+63=553
• 3950+553=4503

Multiplying by 11

Here's a quick mental math trick to multiply any two-digit number by 11 at phenomenal speed.

To multiply a two-digit number by 11 , we add the two digits of the number to each other, and enter the resulting amount between the digits of the original number. The resulting three-digit number is the result of multiplying the original number by 11 .

Let's check and multiply 54 on 11 .

• 5+4=9
• 54*11=594

Take any two-digit number and multiply it by 11 and see for yourself - this trick works!

Squaring

Using another interesting mental counting technique, you can quickly and easily square two-digit numbers. This is especially easy to do with numbers that end in 5 .

The result begins with the product of the first digit of a number by the next one in the hierarchy. That is, if this figure is denoted by n , then the next number in the hierarchy will be n+1 . The result ends with the square of the last digit, that is, the square 5 .

Let's check! Let's square the number 75 .

• 7*8=56
• 5*5=25
• 75*75=5625

Dividing numbers in your head

It remains to deal with division. Essentially, this is the inverse operation of multiplication. With division of numbers up to 100 There shouldn’t be any problems at all - after all, there is a multiplication table that you know by heart.

Division by a single digit number

When dividing multi-digit numbers by single-digit numbers, it is necessary to select the largest possible part that can be divided using the multiplication table.

For example, there is a number 6144 , which must be divided by 8 . We recall the multiplication table and understand that 8 the number will be divided 5600 . Let's present an example in the form:

6144:8=(5600+544):8=700+544:8

544:8=(480+64):8=60+64:8

It remains to divide 64 on 8 and get the result by adding all the division results

64:8=8

6144:8=700+60+8=768

Division by two digits

When dividing by a two-digit number, you must use the rule of the last digit of the result when multiplying two numbers.

When multiplying two multi-digit numbers, the last digit of the multiplication result is always the same as the last digit of the result of multiplying the last digits of those numbers.

For example, let's multiply 1325 on 656 . According to the rule, the last digit in the resulting number will be 0 , because 5*6=30 . Really, 1325*656=869200 .

Now, armed with this valuable information, let's look at division by a two-digit number.

How much will 4424:56 ?

Initially, we will use the “fitting” method and find the limits within which the result lies. We need to find a number that, when multiplied by 56 will give 4424 . Intuitively let's try the number 80.

56*80=4480

This means that the required number is less 80 and obviously more 70 . Let's determine its last digit. Her work on 6 must end with a number 4 . According to the multiplication table, the results suit us 4 And 9 . It is logical to assume that the result of division can be either a number 74 , or 79 . We check:

79*56=4424

Done, solution found! If the number didn't fit 79 , the second option would definitely be correct.

In conclusion, here are some useful tips that will help you quickly learn mental arithmetic:

• Don't forget to exercise every day;
• do not quit training if the results do not come as quickly as you would like;
• download a mobile application for mental calculation: this way you don’t have to come up with examples for yourself;
• Read books on fast mental counting techniques. There are different mental counting techniques, and you can master the one that best suits you.

The benefits of mental counting are undeniable. Practice and every day you will count faster and faster. And if you need help in solving more complex and multi-level problems, contact student service specialists for quick and qualified help!

Some quick ways oral multiplication We’ve already figured it out, now let’s take a closer look at how to quickly multiply numbers in your head using various auxiliary methods. You may already know, and some of them are quite exotic, such as the ancient Chinese way of multiplying numbers.

Layout by ranks

It is the simplest technique for quickly multiplying two-digit numbers. Both factors need to be divided into tens and ones, and then all these new numbers must be multiplied by each other.

This method requires the ability to hold up to four numbers in memory at the same time, and to do calculations with these numbers.

For example, you need to multiply numbers 38 And 56 . We do it this way:

38 * 56 = (30 + 8) * (50 + 6) = 30 * 50 + 8 * 50 + 30 * 6 + 8 * 6 = 1500 + 400 + 180 + 48 = 2128 It will be even easier to do oral multiplication of two-digit numbers in three operations. First you need to multiply the tens, then add two products of ones by tens, and then add the product of ones by ones. It looks like this: 38 * 56 = (30 + 8) * (50 + 6) = 30 * 50 + (8 * 50 + 30 * 6) + 8 * 6 = 1500 + 580 + 48 = 2128 In order to successfully use this method, you need to know the multiplication table well, be able to quickly add two-digit and three-digit numbers, and switch between mathematical operations without forgetting intermediate results. The last skill is achieved through help and visualization.

This method is not the fastest and most effective, so it is worth exploring other methods of oral multiplication.

Fitting the numbers

You can try to bring the arithmetic calculation to a more convenient form. For example, the product of numbers 35 And 49 can be imagined this way: 35 * 49 = (35 * 100) / 2 — 35 = 1715
This method may be more effective than the previous one, but it is not universal and is not suitable for all cases. It is not always possible to find a suitable algorithm to simplify the problem.

On this topic, I remembered an anecdote about how a mathematician sailed along the river past a farm and told his interlocutors that he was able to quickly count the number of sheep in the pen, 1358 sheep. When asked how he did it, he said it was simple - you need to count the number of legs and divide by 4.

Visualization of columnar multiplication

This is one of the most universal ways of oral multiplication of numbers, developing spatial imagination and memory. First, you should learn to multiply two-digit numbers by single-digit numbers in a column in your head. After this, you can easily multiply two-digit numbers in three steps. First, a two-digit number must be multiplied by the tens of another number, then multiplied by the units of another number, and then sum the resulting numbers.

It looks like this: 38 * 56 = (38 * 5) * 10 + 38 * 6 = 1900 + 228 = 2128

Visualization with number arrangement

A very interesting way to multiply two-digit numbers is as follows. You need to sequentially multiply the digits in numbers to get hundreds, ones and tens.

Let's say you need to multiply 35 on 49 .

First you multiply 3 on 4 , you get 12 , then 5 And 9 , you get 45 . Recording 12 And 5 , with a space between them, and 4 remember.

You receive: 12 __ 5 (remember 4 ).

Now you multiply 3 on 9 , And 5 on 4 , and sum up: 3 * 9 + 5 * 4 = 27 + 20 = 47 .

Now we need to 47 add 4 which we remember. We get 51 .

We write 1 in the middle and 5 add to 12 , we get 17 .

In total, the number we were looking for is 1715 , it is the answer:

35 * 49 = 1715
Try multiplying in your head in the same way: 18 * 34, 45 * 91, 31 * 52 .

Chinese or Japanese multiplication

In Asian countries, it is customary to multiply numbers not in a column, but by drawing lines. For Eastern cultures, the desire for contemplation and visualization is important, which is probably why they came up with such a beautiful method that allows you to multiply any numbers. This method is complicated only at first glance. In fact, greater clarity allows you to use this method much more effectively than multiplying by column.

In addition, knowledge of this ancient oriental method increases your erudition. Agree, not everyone can boast that they know the ancient multiplication system that the Chinese used 3000 years ago.

Video about how the Chinese multiply numbers

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The advantage of the three methods of multiplying two-digit numbers for mental calculation, described in, is that they are universal for any numbers and with good mental calculation skills, they can allow you to quickly come to the correct answer. However, the efficiency of multiplying some two-digit numbers in the mind can be higher due to fewer steps when using special algorithms. In this lesson, you will learn how to quickly multiply any number up to 30. Special techniques are presented here, including an introduction to using a reference number.

To multiply any two-digit number by 11, you need to enter the sum of the first and second digits between the first and second digits of the number being multiplied. For example: 23*11, write 2 and 3, and between them put the sum (2+3). Or in short, that 23*11= 2 (2+3) 3 = 253.

If the sum of the numbers in the center gives a result greater than 10, then add one to the first digit, and instead of the second digit we write the sum of the digits of the number being multiplied minus 10. For example: 29*11 = 2 (2+9) 9 = 2 (11) 9 = 319 .

Any two-digit numbers can be multiplied by 11 in this way. For clarity, examples are given:

81 * 11 = 8 (8+1) 1 = 891

68 * 11 = 6 (6+8) 8 = 748

Squared sum, squared difference

To square a two-digit number, you can use the squared sum or squared difference formulas. For example:

23 2 = (20+3) 2 = 20 2 + 2*3*20 + 3 2 = 400+120+9 = 529

69 2 = (70-1) 2 = 70 2 - 70*2*1 + 1 2 = 4 900-140+1 = 4 761

Squaring numbers ending in 5

To square numbers ending in 5. The algorithm is simple. The number up to the last five, multiply by the same number plus one. Add 25 to the remaining number.

15 2 = (1*(1+1)) 25 = 225

25 2 = (2*(2+1)) 25 = 625

85 2 = (8*(8+1)) 25 = 7 225

This is also true for more complex examples:

155 2 = (15*(15+1)) 25 = (15*16)25 = 24 025

Multiplying numbers up to 20

1 step. For example, let's take two numbers - 16 and 18. To one of the numbers we add the number of units of the second - 16+8=24

Step 2. We multiply the resulting number by 10 - 24*10=240

The technique for multiplying numbers up to 20 is very simple:

To write it down briefly:

16*18 = (16+8)*10+6*8 = 288

Proving the correctness of this method is simple: 16*18 = (10+6)*(10+8) = 10*10+10*6+10*8+6*8 = 10*(10+6+8) +6* 8. The last expression is a demonstration of the method described above.

Essentially, this method is a special way of using reference numbers (which will be discussed in). In this case, the reference number is 10. In the last expression of the proof, we can see that it is by 10 that we multiply the bracket. But any other numbers can be used as a reference number, the most convenient of which are 20, 25, 50, 100... Read more about the method of using a reference number in the next lesson.

Reference number

Look at the essence of this method using the example of multiplying 15 and 18. Here it is convenient to use the reference number 10. 15 is more than ten by 5, and 18 is more than ten by 8. In order to find out their product, you need to perform the following operations:

1. To any of the factors add the number by which the second factor is greater than the reference one. That is, add 8 to 15, or 5 to 18. In the first and second cases, the result is the same: 23.
2. Then we multiply 23 by the reference number, that is, by 10. Answer: 230
3. To 230 we add the product 5*8. Answer: 270.

Training

If you want to improve your skills on the topic of this lesson, you can use the following game. The points you receive are affected by the correctness of your answers and the time spent on completion. Please note that the numbers are different each time.