Multiplied by their number determines. Game "Mathematical Comparisons"

Let's look at the concept of multiplication using an example:

The tourists were on the road for three days. Every day they walked the same path of 4200 m. How much distance did they cover in three days? Solve the problem in two ways.

Solution:
Let's consider the problem in detail.

On the first day, tourists walked 4200m. On the second day, tourists walked the same path 4200m and on the third day – 4200m. Let's write it in mathematical language:
4200+4200+4200=12600m.
We see a pattern in which the number 4200 is repeated three times, therefore, the sum can be replaced by multiplication:
4200⋅3=12600m.
Answer: tourists walked 12,600 meters in three days.

Let's look at an example:

To avoid writing a long entry, we can write it in the form of multiplication. The number 2 is repeated 11 times, so an example with multiplication would look like this:
2⋅11=22

Let's summarize. What is multiplication?

Multiplication– this is an action that replaces the repetition of the term m n times.

The notation m⋅n and the result of this expression are called product of numbers, and the numbers m and n are called multipliers.

Let's look at this with an example:
7⋅12=84
The expression 7⋅12 and the result 84 are called product of numbers.
The numbers 7 and 12 are called multipliers.

There are several multiplication laws in mathematics. Let's look at them:

Commutative law of multiplication.

Let's consider the problem:

We gave two apples to 5 of our friends. Mathematically, the entry will look like this: 2⋅5.
Or we gave 5 apples to two of our friends. Mathematically, the entry will look like this: 5⋅2.
In the first and second cases, we will distribute the same number of apples equal to 10 pieces.

If we multiply 2⋅5=10 and 5⋅2=10, the result will not change.

Property of the commutative multiplication law:
Changing the places of the factors does not change the product.
m⋅ n=n⋅m

Combinative law of multiplication.

Let's look at an example:

(2⋅3)⋅4=6⋅4=24 or 2⋅(3⋅4)=2⋅12=24 we get,
(2⋅3)⋅4=2⋅(3⋅4)
(a⋅ b) ⋅ c= a⋅(b⋅ c)

Property of the associative multiplication law:
To multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second.

By swapping multiple factors and putting them in parentheses, the result or product will not change.

These laws are true for any natural numbers.

Multiplying any natural number by one.

Let's look at an example:
7⋅1=7 or 1⋅7=7
a⋅1=a or 1⋅a= a
When any natural number is multiplied by one, the product will always be the same number.

Multiplying any natural number by zero.

6⋅0=0 or 0⋅6=0
a⋅0=0 or 0⋅a=0
When any natural number is multiplied by zero, the product will equal zero.

Questions for the topic “Multiplication”:

What is a product of numbers?
Answer: the product of numbers or the multiplication of numbers is the expression m⋅n, where m is a term, and n is the number of repetitions of this term.

What is multiplication used for?
Answer: in order not to write long addition of numbers, but to write abbreviated. For example, 3+3+3+3+3+3=3⋅6=18

What is the result of multiplication?
Answer: the meaning of the work.

What does multiplication 3⋅5 mean?
Answer: 3⋅5=5+5+5=3+3+3+3+3=15

If you multiply a million by zero, what is the product equal to?
Answer: 0

Example #1:
Replace the sum with the product: a) 12+12+12+12+12 b)3+3+3+3+3+3+3+3+3
Answer: a) 12⋅5=60 b) 3⋅9=27

Example #2:
Write it down as a product: a) a+a+a+a b) c+c+c+c+c+c+c
Solution:
a)a+a+a+a=4⋅a
b) s+s+s+s+s+s+s=7⋅s

Task #1:
Mom bought 3 boxes of chocolates. Each box contains 8 candies. How many candies did mom buy?
Solution:
There are 8 candies in one box, and we have 3 such boxes.
8+8+8=8⋅3=24 candies
Answer: 24 candies.

Task #2:
The art teacher told her eight students to prepare seven pencils for each lesson. How many pencils did the children have in total?
Solution:
You can calculate the sum of the task. The first student had 7 pencils, the second student had 7 pencils, etc.
7+7+7+7+7+7+7+7=56
The recording turned out to be inconvenient and long, let’s replace the sum with the product.
7⋅8=56
The answer is 56 pencils.

In this context, the multiplication sign is a binary operator. There is no multiplication sign special name, such as the addition sign called “plus”.

The oldest symbol in use is the diagonal cross (×). It was first used by the English mathematician William Oughtred in his work “Clavis Mathematicae” in 1631. The German mathematician Leibniz preferred the sign in the form of a raised dot (∙) He used this symbol in a letter of 1698. Johann Rahn introduced the asterisk (∗) as a multiplication sign, it appeared in his book Teutsche Algebra of 1659.

IN Russian textbooks In mathematics, the sign in the form of a raised dot (∙) is mainly used. The asterisk (∗) is used in computer notation. The result is written using the equal sign " $=$ ", For example:

$a \cdot b = c$ ; $6\cdot 3 = 18$ (“six times three equals eighteen” or “six times three equals eighteen”).

table for multiplication in decimal number system

*	1	2	3	4	5	6	7	8	9
0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81

This procedure is applicable to multiplying natural and integer (including sign) numbers. For other numbers, more complex algorithms are used.

Multiplying numbers

Natural numbers

Let's use the definition of natural numbers $\mathbb(N)$ as equivalence classes of finite sets. We denote the equivalence classes of finite sets $C, A, B$ generated by bijections, using parentheses: $[C], [A], [B]$ . Then the arithmetic operation “multiplication” is defined as follows:

$[C]=[A] \cdot [B] = ;$

Where: $A \times B=$(a,b) \mid a \in A , b \in B $$ direct product of sets - set $C$ , whose elements are ordered pairs $(a,b)$ for all kinds $a \in A , b \in B$ . This operation on classes is introduced correctly, that is, it does not depend on the choice of class elements, and coincides with the inductive definition.

One-to-one mapping finite set $A$ for a segment $N_a$ can be understood as the numbering of elements of a set $A:\quad A\sim N_a$ . This numbering process is called "COUNTING". Thus, “counting” is the establishment of a one-to-one correspondence between the elements of a set and a segment of a natural series of numbers.

To multiply natural numbers in the positional number notation system, a bitwise multiplication algorithm is used. If given two natural numbers $a$ And $b$ such that:

$a=a_(n-1) a_(n-2)\dots a_0, \quad b=b_(n-1) b_(n-2)\dots b_0, \quad \forall a_(k),b_(k ) \in $P $, \quad \forall a_(n-1), b_(n-1) \ne 0, \quad \exists 0\in \N;$

Where $a_(0 \dots n-1)=a_k P^k, \quad b_(0 \dots n-1)=b_k P^k$ ; $n$ - number of digits in a number $n \in $1, 2, \dots ,n $$ ; $k$ - serial number rank (position), $k \in $0, 1, \dots ,n-1 $$ ; $P$ - base of the number system; $$P$$ many numerical signs (digits), specific system notation: $$P_2$= $0,1$$ , $$P_(10) $= $0,1,2,3,4,5,6,7,8,9$$ , $$P_(16) $= $0,1,2,3,4,5,6,7,8,9,A,B,C,D,F $$ ; Then:

$c=a \sdot b; \quad c_(n-1) c_(n-2)\dots c_0=a_(n-1) a_(n-2)\dots a_0 \sdot b_(n-1) b_(n-2)\dots b_0 ;$

multiplying bitwise we get $n$ intermediate results:

$t_(n-1,~0) = mod(a_(n-1) \cdot b_0 + r_(n-1),P), \quad r_(n)=div(a_(n-1) \cdot b_0 + r_(n-1),P)~,~~ t_0 \sdot~ P^k;$
$t_(n-1,~1) = mod(a_(n-1) \cdot b_1 + r_(n-1),P), \quad r_(n)=div(a_(n-1) \cdot b_1 + r_(n-1),P)~,~~ t_1 \sdot~ P^k;$
$... \qquad \qquad... \qquad \qquad \qquad \qquad \qquad \qquad... \qquad \qquad \qquad \qquad \qquad \qquad...$
$t_(n-1,~k) = mod(a_(n-1) \cdot b_(k) + r_(n-1),P), \quad r_(n)=div(a_(n-1) \cdot b_(k) + r_(n-1),P)~,~~ t_(k) \sdot~ P^k;$

where: r is the carry value, mod() is the function of finding the remainder of division, div() is the function of finding the incomplete quotient.

Then received $n$ We add up the intermediate results: $c=t_0+t_1+...+t_(k).$

Thus, the multiplication operation is reduced to a sequential procedure simple multiplication single digit numbers $a_(k)\sdot b_(k)$ , with the formation of a carry if necessary, which is done either by the tabular method or by sequential addition. And then on to addition.

Arithmetic operations on numbers in any positional number system are performed according to the same rules as in the decimal system, since they are all based on the rules for performing operations on the corresponding polynomials. In this case, you need to use the multiplication table corresponding to the given base. $P$ number systems.

An example of multiplying natural numbers in binary, decimal and hexadecimal number systems; for convenience, the numbers are written under each other according to the digits, the carry is written on top:

$\begin(array)(cccccccccccc)& & & & & & & & \\ & & & &1&1&0&1&1&0 \\ & & &*& & &1&1&0&1 \\ \hline & & & &1&1&0&1&1&0 \\ & & &0&0&0&0&0&0&(\color(Gray)0) \\ & &1&1&0&1&1&0&( \color(Gray)0) &(\color(Gray)0) \\ +&1&1&0&1&1&0&(\color(Gray)0) &(\color(Gray)0) &(\color(Gray)0) \\ \hline 1&0&1&0&1&1&1&1&1&0 \end(array); \quad \quad \begin(array)(cccccccccc) & & & &_2&_2&_3&_3& \\ & & & &_1&_2&_2&_2& \\ & & & &8&4&5&6&7 \\ & & &*& & &5&4&1 \\ \hline & & &0&8&4&5&6&7 \\ & &3&3&8&2&6 &8&(\ color(Gray)0) \\ +&4&2&2&8&3&5&(\color(Gray)0)&(\color(Gray)0) \\ \hline &4&5&7&5&0&7&4&7 \end(array); \quad \quad\begin(array)(cccccccc) &&&&_8&_8&_2 \\ &&&&_D&_D&_3 \\ &&&&6&D&E&4 \\ &&&(*)&&A&1&F \\ \hline &&&6&7&0&5&C \\ &&0&6&D&E&4&(\color(Gray)0) \\ +&4&4& A&E&8&(\color(Gray)0) &(\color(Gray)0) \\ \hline &4&5&8&3&6&9&C\end(array)~~.$

Integers

\alpha = \pm a_0, a_1 a_2 \ldots a_n \ldots = \(a_n\)

\beta = \pm b_0, b_1 b_2 \ldots b_n \ldots = \(b_n\)

defined respectively by fundamental sequences of rational numbers (satisfying the Cauchy condition), denoted as: $\alpha =$ And $\beta =$ , then their product is the number $\gamma =$ , defined by the product of sequences $$a_n$$ And $$b_n$$ :

$\gamma = \alpha \cdot \beta \overset(\text(def))(=) \cdot =$ ;

real number $\gamma = \alpha \cdot \beta$ , satisfies the following condition:

\forall a",a , b", b\in \mathbb(Q); ~~~~ (a" \leqslant \alpha \leqslant a ) \and (b" \leqslant \beta \leqslant b) \Rightarrow (a" \cdot b" \leqslant \alpha \times \beta \leqslant a \cdot b) \Rightarrow (a" \cdot b" \leqslant \gamma \leqslant a \cdot b)

Thus, the product of two real numbers $\alpha$ And $\beta$ is such a real number $\gamma$ which is contained between all products of the form $a" \cdot b"$ on the one hand, and all works of the form $a$ \cdot b on the other side .

In practice, in order to multiply two numbers $\alpha$ And $\beta$ , it is necessary to replace them with the required accuracy with approximate rational numbers $a$ And $b$ . For the approximate value of the product of numbers $\alpha \cdot \beta$ take the product of the indicated rational numbers $a \cdot b$ . In this case, it does not matter from which side (by deficiency or excess) the taken rational numbers bring closer $\alpha$ And $\beta$ . Multiplication is performed using the bitwise multiplication algorithm.

In order to multiply two complex numbers in trigonometric notation, you need to multiply their modules and add their arguments:

$c=a \cdot b=r_1 (Cos \varphi _1+ iSin \varphi _1) ~\cdot~ r_2 (Cos \varphi _2+ iSin\varphi _2) =r_1 \cdot r_2 (Cos (\varphi _1+\varphi _2)+ iSin (\varphi _1+\varphi _2)),$ Where: $r=|z|=|a+ib|=\sqrt(a^2+b^2);~~~\varphi = Arg(z)=arctg \biggl(\frac(b)(a) \biggr) ,$ modulus and argument of a complex number.

Multiplying a complex number $a = r_1 e^ (i\varphi _1)$ in exponential form, for a complex number $b = r_2 e^ (i\varphi _2)$ reduces to rotating the vector corresponding to the number $a$ , to the corner $Arg(b)$ and changing its length in $|b|$ once. For a piece complex numbers in exponential form the equality is true:

$c=re^ (i\varphi)=a \cdot b = r_1 e^ (i\varphi _1) \cdot r_2 e^ (i\varphi _2)= r_1\cdot r_2\cdot e^ (i(\varphi _1+ \varphi _2)),$

Where: $e=2.718281828...$ - number e.

Exponential notation

For example, if you multiply the speed $V=4 ~m/s$ for a while $T=2 ~s$ , corresponding to one physical process, you get a named number ( physical quantity) corresponding to the same physical process, which is called “length” and is measured in meters: $L=8 ~m$ .

$L=V \cdot T = 4~\frac(m)(s) \cdot 2~s =8 ~\frac(m \cdot s)(s)= 8 ~m.$

When describing mathematical means physical processes An important role is played by the concept of homogeneity, which means, for example, that “1 kg of flour” and “1 kg of copper” belong different sets(flour) and (copper) respectively. Also, the concept of homogeneity assumes that the multiplied quantities belong to the same physical process.

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Notes

Literature

Ilyin V.A. etc.. - Moscow State University, 1985. - T. 1. - 662 p.
Enderton G. Elements of Set Theory = Elements of Set Theory. - Gulf Professional Publishing, 1977. - 279 p. - ISBN 0-12-238440-7.
Barsukov A.N.. - Education, 1966. - 296 p.
Gusev V.A., Mordkovich A.G.. - Education, 1988. - 416 p.
Istomina N.B.. - Association XXI century, 2005. - 272 p. - ISBN 5-89308-193-5.
Vygodsky M. Ya. Guide to elementary mathematics. - M.: AST, 2003. - ISBN 5-17-009554-6.
V.I. Igoshin(Russian): article. - Saratov state university named after N.G. Chernyshevsky, 2010.
Kononyuk A.E.. - Osvita Ukraine, 2012. - T. 2. - 548 p. - ISBN 978-966-7599-50-8.
: [August 24, 2011] // Horsemanship / Artemy Lebedev. - January 15, 2003 - § 97.

Excerpt describing Multiplication

“Please change your clothes, please,” he said, walking away.

- He's coming! - the makhalny shouted at this time.
The regimental commander, blushing, ran up to the horse, with trembling hands took the stirrup, threw the body over, straightened himself, took out his sword and with a happy, decisive face, his mouth open to the side, prepared to shout. The regiment perked up like a recovering bird and froze.
- Smir r r r na! - the regimental commander shouted in a soul-shaking voice, joyful for himself, strict in relation to the regiment and friendly in relation to the approaching commander.
Along a wide, tree-lined, highwayless road, a tall blue Viennese carriage was moving in a train at a brisk trot, its springs slightly rattling. Behind the carriage galloped a retinue and a convoy of Croats. Next to Kutuzov sat an Austrian general in a strange white uniform among the black Russians. The carriage stopped at the shelf. Kutuzov and the Austrian general were talking quietly about something, and Kutuzov smiled slightly, while, stepping heavily, he lowered his foot from the footrest, as if these 2,000 people were not there, who were looking at him and the regimental commander without breathing .
A shout of command was heard, and again the regiment trembled with a ringing sound, putting itself on guard. In the dead silence I heard weak voice commander-in-chief. The regiment barked: “We wish you good health, yours!” And again everything froze. At first, Kutuzov stood in one place while the regiment moved; then Kutuzov, next to the white general, on foot, accompanied by his retinue, began to walk along the ranks.
By the way the regimental commander saluted the commander-in-chief, glaring at him with his eyes, stretching out and getting closer, how he leaned forward and followed the generals along the ranks, barely maintaining a trembling movement, how he jumped at every word and movement of the commander-in-chief, it was clear that he was fulfilling his duties subordinate with even greater pleasure than the duties of a superior. The regiment, thanks to the rigor and diligence of the regimental commander, was in excellent condition compared to others who came to Braunau at the same time. There were only 217 people who were retarded and sick. And everything was fine, except for the shoes.
Kutuzov walked through the ranks, occasionally stopping and speaking a few kind words to the officers whom he knew from the Turkish war, and sometimes to the soldiers. Looking at the shoes, he sadly shook his head several times and pointed them out to the Austrian general with such an expression that he didn’t seem to blame anyone for it, but he couldn’t help but see how bad it was. Each time the regimental commander ran ahead, afraid to miss the commander-in-chief's word regarding the regiment. Behind Kutuzov, at such a distance that any faintly spoken word could be heard, walked about 20 people in his retinue. The gentlemen of the retinue talked among themselves and sometimes laughed. The handsome adjutant walked closest to the commander-in-chief. It was Prince Bolkonsky. Next to him walked his comrade Nesvitsky, a tall staff officer, extremely fat, with a kind and smiling beautiful face and wet eyes; Nesvitsky could hardly restrain himself from laughing, excited by the blackish hussar officer walking next to him. The hussar officer, without smiling, without changing the expression of his fixed eyes, looked with a serious face at the back of the regimental commander and imitated his every movement. Every time the regimental commander flinched and bent forward, in exactly the same way, in exactly the same way, the hussar officer flinched and bent forward. Nesvitsky laughed and pushed others to look at the funny man.
Kutuzov walked slowly and sluggishly past thousands of eyes that rolled out of their sockets, watching their boss. Having caught up with the 3rd company, he suddenly stopped. The retinue, not anticipating this stop, involuntarily moved towards him.
- Ah, Timokhin! - said the commander-in-chief, recognizing the captain with the red nose, who suffered for his blue overcoat.
It seemed impossible to stretch out moreover, as Timokhin stretched out, while the regimental commander reprimanded him. But at that moment the commander-in-chief addressed him, the captain stood up straight so that it seemed that if the commander-in-chief had looked at him for a little longer, the captain would not have been able to stand it; and therefore Kutuzov, apparently understanding his position and wishing, on the contrary, all the best for the captain, hastily turned away. A barely noticeable smile ran across Kutuzov’s plump, wound-disfigured face.
“Another Izmailovo comrade,” he said. - Brave officer! Are you happy with it? – Kutuzov asked the regimental commander.
And the regimental commander, reflected as in a mirror, invisible to himself, in a hussar officer, shuddered, came forward and answered:
- I am very pleased, Your Excellency.
“We are all not without weaknesses,” said Kutuzov, smiling and moving away from him. “He had a devotion to Bacchus.
The regimental commander was afraid that he was to blame for this, and did not answer anything. The officer at that moment noticed the captain’s face with a red nose and a tucked belly and imitated his face and pose so closely that Nesvitsky could not stop laughing.
Kutuzov turned around. It was clear that the officer could control his face as he wanted: the minute Kutuzov turned around, the officer managed to make a grimace, and after that take on the most serious, respectful and innocent expression.
The third company was the last, and Kutuzov thought about it, apparently remembering something. Prince Andrei stepped out from his retinue and said quietly in French:
“You ordered a reminder of the demoted Dolokhov in this regiment.
-Where is Dolokhov? – asked Kutuzov.
Dolokhov, already dressed in a soldier’s gray overcoat, did not wait to be called. Slender figure blond with clear hair blue eyes the soldier stepped out from the front. He approached the commander-in-chief and put him on guard.
- Claim? – Kutuzov asked, frowning slightly.
“This is Dolokhov,” said Prince Andrei.
- A! - said Kutuzov. “I hope this lesson will correct you, serve well.” The Lord is merciful. And I will not forget you if you deserve it.
Blue, clear eyes looked at the commander-in-chief as defiantly as at the regimental commander, as if with their expression they were tearing apart the veil of convention that so far separated the commander-in-chief from the soldier.
“I ask one thing, Your Excellency,” he said in his sonorous, firm, unhurried voice. “Please give me a chance to make amends for my guilt and prove my devotion to the Emperor and Russia.”
Kutuzov turned away. The same smile in his eyes flashed across his face as when he turned away from Captain Timokhin. He turned away and winced, as if he wanted to express that everything that Dolokhov told him, and everything that he could tell him, he had known for a long, long time, that all this had already bored him and that all this was not at all what he needed . He turned away and headed towards the stroller.
The regiment disbanded in companies and headed to assigned quarters not far from Braunau, where they hoped to put on shoes, dress and rest after difficult marches.
– You don’t lay claim to me, Prokhor Ignatyich? - said the regimental commander, driving around the 3rd company moving towards the place and approaching Captain Timokhin, who was walking in front of it. The regimental commander’s face expressed uncontrollable joy after a happily completed review. - The royal service... it’s impossible... another time you’ll end it at the front... I’ll apologize first, you know me... I thanked you very much! - And he extended his hand to the company commander.
- For mercy's sake, general, do I dare! - answered the captain, turning red with his nose, smiling and revealing with a smile the lack of two front teeth, knocked out by the butt under Ishmael.
- Yes, tell Mr. Dolokhov that I will not forget him, so that he can be calm. Yes, please tell me, I kept wanting to ask how he is, how he is behaving? And that's all...
“He is very serviceable in his service, Your Excellency... but the charterer...” said Timokhin.
- What, what character? – asked the regimental commander.
“Your Excellency finds, for days,” said the captain, “that he is smart, and learned, and kind.” It's a beast. He killed a Jew in Poland, if you please...
“Well, yes, well,” said the regimental commander, “we still need to feel sorry for the young man in misfortune.” After all, great connections... So you...
“I’m listening, Your Excellency,” Timokhin said, smiling, making it feel like he understood the boss’s wishes.
- Well, yes, well, yes.
The regimental commander found Dolokhov in the ranks and reined in his horse.
“Before the first task, epaulets,” he told him.
Dolokhov looked around, said nothing and did not change the expression of his mockingly smiling mouth.
“Well, that’s good,” continued the regimental commander. “The people each have a glass of vodka from me,” he added so that the soldiers could hear. – Thank you everyone! God bless! - And he, overtaking the company, drove up to another.
- Well, he really good man; “You can serve with him,” said subaltern Timokhin to the officer walking next to him.
“One word, the red one!... (the regimental commander was nicknamed the king of reds),” the subaltern officer said, laughing.
The happy mood of the authorities after the review spread to the soldiers. The company walked cheerfully. Soldiers' voices were talking from all sides.
- What did they say, crooked Kutuzov, about one eye?
- Otherwise, no! Totally crooked.
- No... brother, he has bigger eyes than you. Boots and tucks - I looked at everything...
- How can he, my brother, look at my feet... well! Think…
- And the other Austrian, with him, was as if smeared with chalk. Like flour, white. I tea, how they clean ammunition!
- What, Fedeshow!... did he say that when the fighting began, you stood closer? They all said that Bunaparte himself stands in Brunovo.
- Bunaparte is worth it! he's lying, you fool! What he doesn’t know! Now the Prussian is rebelling. The Austrian, therefore, pacifies him. As soon as he makes peace, then war will open with Bunaparte. Otherwise, he says, Bunaparte is standing in Brunow! That's what shows that he's a fool. Listen more.
- Look, damn the lodgers! The fifth company, look, is already turning into the village, they will cook porridge, and we still won’t reach the place.
- Give me some crackers, damn it.
- Did you give me tobacco yesterday? That's it, brother. Well, here you go, God be with you.
“At least they made a stop, otherwise we won’t eat for another five miles.”
– It was nice how the Germans gave us strollers. When you go, know: it’s important!
“And here, brother, the people have gone completely rabid.” Everything there seemed to be a Pole, everything was from the Russian crown; and now, brother, he’s gone completely German.
– Songwriters forward! – the captain’s cry was heard.
And twenty people ran out from different rows in front of the company. The drummer began to sing and turned to face the songwriters, and, waving his hand, began a drawn-out soldier’s song, which began: “Isn’t it dawn, the sun has broken...” and ended with the words: “So, brothers, there will be glory for us and Kamensky’s father...” This song was composed in Turkey and was now sung in Austria, only with the change that in place of “Kamensky’s father” the words were inserted: “Kutuzov’s father.”
Having torn off these last words like a soldier and waving his hands, as if he was throwing something to the ground, the drummer, a dry and handsome soldier of about forty, looked sternly at the soldier songwriters and closed his eyes. Then, making sure that all eyes were fixed on him, he seemed to carefully lift with both hands some invisible, precious thing above his head, held it like that for several seconds and suddenly desperately threw it:
Oh, you, my canopy, my canopy!
“My new canopy...”, twenty voices echoed, and the spoon holder, despite the weight of his ammunition, quickly jumped forward and walked backwards in front of the company, moving his shoulders and threatening someone with his spoons. The soldiers, waving their arms to the beat of the song, walked with long strides, involuntarily hitting their feet. From behind the company the sounds of wheels, the crunching of springs and the trampling of horses were heard.
Kutuzov and his retinue were returning to the city. The commander-in-chief gave a sign for the people to continue walking freely, and pleasure was expressed on his face and on all the faces of his retinue at the sounds of the song, at the sight of the dancing soldier and the soldiers of the company walking cheerfully and briskly. In the second row, from the right flank, from which the carriage overtook the companies, one involuntarily caught the eye of a blue-eyed soldier, Dolokhov, who especially briskly and gracefully walked to the beat of the song and looked at the faces of those passing with such an expression, as if he felt sorry for everyone who did not go at this time with the company. A hussar cornet from Kutuzov's retinue, imitating the regimental commander, fell behind the carriage and drove up to Dolokhov.
The hussar cornet Zherkov at one time in St. Petersburg belonged to that violent society led by Dolokhov. Abroad, Zherkov met Dolokhov as a soldier, but did not consider it necessary to recognize him. Now, after Kutuzov’s conversation with the demoted man, he turned to him with the joy of an old friend:
- Dear friend, how are you? - he said at the sound of the song, matching the step of his horse with the step of the company.
- How am I? - Dolokhov answered coldly, - as you see.
The lively song gave particular significance to the tone of cheeky gaiety with which Zherkov spoke and the deliberate coldness of Dolokhov’s answers.
- Well, how do you get along with your boss? – asked Zherkov.
- Nothing, good people. How did you get into the headquarters?
- Seconded, on duty.
They were silent.
“She released a falcon from her right sleeve,” said the song, involuntarily arousing a cheerful, cheerful feeling. Their conversation would probably have been different if they had not spoken to the sound of a song.
– Is it true that the Austrians were beaten? – asked Dolokhov.
“The devil knows them,” they say.
“I’m glad,” Dolokhov answered briefly and clearly, as the song required.
“Well, come to us in the evening, you’ll pawn the Pharaoh,” said Zherkov.
– Or do you have a lot of money?
- Come.
- It is forbidden. I made a vow. I don’t drink or gamble until they make it.
- Well, on to the first thing...
- We'll see there.
Again they were silent.
“You come in if you need anything, everyone at headquarters will help...” said Zherkov.
Dolokhov grinned.
- You better not worry. I won’t ask for anything I need, I’ll take it myself.
- Well, I’m so...
- Well, so am I.
- Goodbye.
- Be healthy...
... and high and far,
On the home side...
Zherkov touched his spurs to the horse, which, getting excited, kicked three times, not knowing which one to start with, managed and galloped off, overtaking the company and catching up with the carriage, also to the beat of the song.

Returning from the review, Kutuzov, accompanied by the Austrian general, went into his office and, calling the adjutant, ordered to be given some papers relating to the state of the arriving troops, and letters received from Archduke Ferdinand, who commanded the advanced army. Prince Andrei Bolkonsky entered the commander-in-chief's office with the required papers. Kutuzov and an Austrian member of the Gofkriegsrat sat in front of the plan laid out on the table.
“Ah...” said Kutuzov, looking back at Bolkonsky, as if with this word he was inviting the adjutant to wait, and continued the conversation he had started in French.
“I’m only saying one thing, General,” Kutuzov said with a pleasant grace of expression and intonation, which forced you to listen carefully to every leisurely spoken word. It was clear that Kutuzov himself enjoyed listening to himself. “I only say one thing, General, that if the matter depended on my personal desire, then the will of His Majesty Emperor Franz would have been fulfilled long ago.” I would have joined the Archduke long ago. And believe my honor, it would be a joy for me personally to hand over the highest command of the army to a more knowledgeable and skilled general than I am, of which Austria is so abundant, and to relinquish all this heavy responsibility. But circumstances are stronger than us, General.
And Kutuzov smiled with an expression as if he was saying: “You have every right not to believe me, and even I don’t care at all whether you believe me or not, but you have no reason to tell me this. And that’s the whole point.”
The Austrian general looked dissatisfied, but could not help but respond to Kutuzov in the same tone.
“On the contrary,” he said in a grumpy and angry tone, so contrary to the flattering meaning of the words he was saying, “on the contrary, your Excellency’s participation in common cause highly valued by His Majesty; but we believe that the present slowdown deprives the glorious Russian troops and their commanders-in-chief of the laurels that they are accustomed to reaping in battles,” he finished his apparently prepared phrase.
Kutuzov bowed without changing his smile.
“And I am so convinced and, based on the last letter with which His Highness Archduke Ferdinand honored me, I assume that the Austrian troops, under the command of such a skillful assistant as General Mack, have now won a decisive victory and no longer need our help,” said Kutuzov.
The general frowned. Although there was no positive news about the defeat of the Austrians, there were too many circumstances that confirmed the general unfavorable rumors; and therefore Kutuzov’s assumption about the victory of the Austrians was very similar to ridicule. But Kutuzov smiled meekly, still with the same expression, which said that he had the right to assume this. Really, last letter, which he received from Mack's army, informed him of victory and the most profitable strategic position army.
“Give me this letter here,” said Kutuzov, turning to Prince Andrei. - If you please see. - And Kutuzov, with a mocking smile at the ends of his lips, read in German to the Austrian general the following passage from a letter from Archduke Ferdinand: “Wir haben vollkommen zusammengehaltene Krafte, nahe an 70,000 Mann, um den Feind, wenn er den Lech passirte, angreifen und schlagen zu konnen. Wir konnen, da wir Meister von Ulm sind, den Vortheil, auch von beiden Uferien der Donau Meister zu bleiben, nicht verlieren; mithin auch jeden Augenblick, wenn der Feind den Lech nicht passirte, die Donau ubersetzen, uns auf seine Communikations Linie werfen, die Donau unterhalb repassiren und dem Feinde, wenn er sich gegen unsere treue Allirte mit ganzer Macht wenden wollte, seine Absicht alabald vereitelien. Wir werden auf solche Weise den Zeitpunkt, wo die Kaiserlich Ruseische Armee ausgerustet sein wird, muthig entgegenharren, und sodann leicht gemeinschaftlich die Moglichkeit finden, dem Feinde das Schicksal zuzubereiten, so er verdient.” [We have quite concentrated forces, about 70,000 people, so that we can attack and defeat the enemy if he crosses Lech. Since we already own Ulm, we can retain the benefit of command of both banks of the Danube, therefore, every minute, if the enemy does not cross the Lech, cross the Danube, rush to his communication line, and below cross the Danube back to the enemy, if he decides to turn all his power on our faithful allies, prevent his intention from being fulfilled. In this way we will cheerfully await the time when the imperial Russian army will be completely prepared, and then together we will easily find the opportunity to prepare for the enemy the fate he deserves.”]
Kutuzov sighed heavily, ending this period, and looked attentively and affectionately at the member of the Gofkriegsrat.
- But you know, Your Excellency, wise rule“, which instructs us to assume the worst,” said the Austrian general, apparently wanting to end the jokes and get down to business.
He involuntarily looked back at the adjutant.
“Excuse me, General,” Kutuzov interrupted him and also turned to Prince Andrei. - That's it, my dear, take all the reports from our spies from Kozlovsky. Here are two letters from Count Nostitz, here is a letter from His Highness Archduke Ferdinand, here is another,” he said, handing him several papers. - And from all this, purely, on French, draw up a memorandum, a note, for the visibility of all the news that we have about the actions Austrian army had. Well, then, introduce him to his Excellency.
Prince Andrei bowed his head as a sign that he understood from the first words not only what was said, but also what Kutuzov wanted to tell him. He collected the papers, and, making a general bow, quietly walking along the carpet, went out into the reception room.
Despite the fact that not much time has passed since Prince Andrei left Russia, he has changed a lot during this time. In the expression of his face, in his movements, in his gait, the former pretense, fatigue and laziness were almost not noticeable; he had the appearance of a man who does not have time to think about the impression he makes on others, and is busy doing something pleasant and interesting. His face expressed more satisfaction with himself and those around him; his smile and gaze were more cheerful and attractive.
Kutuzov, whom he caught up with in Poland, received him very kindly, promised him not to forget him, distinguished him from other adjutants, took him with him to Vienna and gave him more serious assignments. From Vienna, Kutuzov wrote to his old comrade, the father of Prince Andrei:
“Your son,” he wrote, “shows hope of becoming an officer, out of the ordinary in his studies, firmness and diligence. I consider myself lucky to have such a subordinate at hand.”
At Kutuzov's headquarters, among his comrades and colleagues, and in the army in general, Prince Andrei, as well as in St. Petersburg society, had two completely opposite reputations.
Some, a minority, recognized Prince Andrei as something special from themselves and from all other people, they expected from him great success, listened to him, admired him and imitated him; and with these people Prince Andrei was simple and pleasant. Others, the majority, did not like Prince Andrei, considered him pompous, cold and unpleasant person. But with these people, Prince Andrei knew how to position himself in such a way that he was respected and even feared.
Coming out of Kutuzov’s office into the reception area, Prince Andrei with papers approached his comrade, the adjutant on duty Kozlovsky, who was sitting by the window with a book.
- Well, what, prince? – asked Kozlovsky.
“We were ordered to write a note explaining why we shouldn’t go ahead.”
- Why?
Prince Andrey shrugged his shoulders.
- No news from Mac? – asked Kozlovsky.
- No.
“If it were true that he was defeated, then the news would come.”
“Probably,” said Prince Andrei and headed towards the exit door; but at the same time, a tall, obviously visiting, Austrian general in a frock coat, with a black scarf tied around his head and with the Order of Maria Theresa around his neck, quickly entered the reception room, slamming the door. Prince Andrei stopped.
- General Chief Kutuzov? – the visiting general quickly said with a sharp German accent, looking around on both sides and walking without stopping to the office door.
“The general in chief is busy,” said Kozlovsky, hastily approaching the unknown general and blocking his path from the door. - How would you like to report?
The unknown general looked contemptuously down at the short Kozlovsky, as if surprised that he might not be known.
“The general in chief is busy,” Kozlovsky repeated calmly.
The general's face frowned, his lips twitched and trembled. He took out notebook, quickly drew something with a pencil, tore out the piece of paper, gave it to him, walked quickly to the window, threw his body on a chair and looked around at those in the room, as if asking: why are they looking at him? Then the general raised his head, craned his neck, as if intending to say something, but immediately, as if casually starting to hum to himself, he made strange sound, which immediately stopped. The door to the office opened, and Kutuzov appeared on the threshold. The general with his head bandaged, as if running away from danger, bent down and approached Kutuzov with large, fast steps of his thin legs.
“Vous voyez le malheureux Mack, [You see the unfortunate Mack.],” he said in a broken voice.
The face of Kutuzov, standing in the doorway of the office, remained completely motionless for several moments. Then, like a wave, a wrinkle ran across his face, his forehead smoothed out; He bowed his head respectfully, closed his eyes, silently let Mack pass by him and closed the door behind himself.
The rumor, already spread before, about the defeat of the Austrians and the surrender of the entire army at Ulm, turned out to be true. Half an hour later, adjutants were sent in different directions with orders proving that soon the Russian troops, which had hitherto been inactive, would have to meet the enemy.
Prince Andrei was one of those rare officers at the headquarters who believed his main interest in general progress military affairs. Having seen Mack and heard the details of his death, he realized that half of the campaign was lost, understood the difficulty of the position of the Russian troops and vividly imagined what awaited the army, and the role that he would have to play in it.
Involuntarily, he experienced an exciting, joyful feeling at the thought of disgracing arrogant Austria and the fact that in a week he might have to see and take part in a clash between the Russians and the French, for the first time since Suvorov.
But he was afraid of the genius of Bonaparte, who could be stronger than all the courage of the Russian troops, and at the same time could not allow shame for his hero.
Excited and irritated by these thoughts, Prince Andrei went to his room to write to his father, to whom he wrote every day. He met in the corridor with his roommate Nesvitsky and the joker Zherkov; They, as always, laughed at something.

And multiplication. The multiplication operation will be discussed in this article.

Multiplying numbers

Multiplication of numbers is mastered by children in the second grade, and there is nothing complicated about it. Now we will look at multiplication with examples.

Example 2*5. This means either 2+2+2+2+2 or 5+5. Take 5 twice or 2 five times. The answer, accordingly, is 10.

Example 4*3. Likewise, 4+4+4 or 3+3+3+3. Three times 4 or four times 3. Answer 12.

Example 5*3. We do the same as the previous examples. 5+5+5 or 3+3+3+3+3. Answer 15.

Multiplication formulas

Multiplication is the sum of identical numbers, for example, 2 * 5 = 2 + 2 + 2 + 2 + 2 or 2 * 5 = 5 + 5. Multiplication formula:

Where, a is any number, n is the number of terms of a. Let's say a=2, then 2+2+2=6, then n=3 multiplying 3 by 2, we get 6. Consider in reverse order. For example, given: 3 * 3, that is. 3 multiplied by 3 means that three must be taken 3 times: 3 + 3 + 3 = 9. 3 * 3=9.

Abbreviated multiplication

Abbreviated multiplication is a shortening of the multiplication operation in certain cases, and abbreviated multiplication formulas have been derived specifically for this purpose. Which will help make calculations the most rational and fastest:

Abbreviated multiplication formulas

Let a, b belong to R, then:

The square of the sum of two expressions is equal to the square of the first expression plus twice the product of the first expression and the second plus the square of the second expression. Formula: (a+b)^2 = a^2 + 2ab + b^2

The square of the difference of two expressions is equal to the square of the first expression minus twice the product of the first expression and the second plus the square of the second expression. Formula: (a-b)^2 = a^2 - 2ab + b^2

Difference of squares two expressions is equal to the product of the difference of these expressions and their sum. Formula: a^2 - b^2 = (a - b)(a + b)

Cube of sum two expressions equal to cube the first expression plus triple the product of the square of the first expression and the second plus triple the product of the first expression and the square of the second plus the cube of the second expression. Formula: (a + b)^3 = a^3 + 3a(^2)b + 3ab^2 + b^3

Difference cube two expressions is equal to the cube of the first expression minus triple the product of the square of the first expression and the second plus triple the product of the first expression and the square of the second minus the cube of the second expression. Formula: (a-b)^3 = a^3 - 3a(^2)b + 3ab^2 - b^3

Sum of cubes a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Difference of cubes two expressions is equal to the product of the sum of the first and second expressions and the incomplete square of the difference of these expressions. Formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Sign up for the course "Speed up mental arithmetic, NOT mental arithmetic" to learn how to quickly and correctly add, subtract, multiply, divide, square numbers and even extract roots. In 30 days, you'll learn how to use easy tricks to simplify arithmetic operations. Each lesson contains new techniques, clear examples and useful tasks.

Multiplying fractions

Considering the addition and subtraction of fractions, the rule was voiced for bringing fractions to common denominator to perform the calculation. When multiplying this do No need! When multiplying two fractions, the denominator is multiplied by the denominator, and the numerator by the numerator.

For example, (2/5) * (3 * 4). Let's multiply two thirds by one quarter. We multiply the denominator by the denominator, and the numerator by the numerator: (2 * 3)/(5 * 4), then 6/20, make a reduction, we get 3/10.

Multiplication 2nd grade

Second grade is just the beginning of learning multiplication, so second graders solve simple problems to replace addition with multiplication, multiply numbers, and learn the multiplication table. Let's look at multiplication problems at the second grade level:

Oleg lives in a five-story building, on the top floor. The height of one floor is 2 meters. What is the height of the house?

The box contains 10 packages of cookies. There are 7 of them in each package. How many cookies are in the box?

Misha arranged his toy cars in a row. There are 7 of them in each row, but there are only 8 rows. How many cars does Misha have?

There are 6 tables in the dining room, and 5 chairs are pushed behind each table. How many chairs are there in the dining room?

Mom brought 3 bags of oranges from the store. The bags contain 22 oranges. How many oranges did mom bring?

There are 9 strawberry bushes in the garden, and each bush has 11 berries. How many berries grow on all the bushes?

Roma placed 8 pipe parts one after another, same size 2 meters each. What is the length of the complete pipe?

Parents brought their children to school on September 1st. 12 cars arrived, each with 2 children. How many children did their parents bring in these cars?

Multiplication 3rd grade

In third grade, more serious tasks are given. In addition to multiplication, Division will also be covered.

Multiplication tasks will include: multiplying two-digit numbers, multiplying by columns, replacing addition with multiplication and vice versa.

Column multiplication:

Column multiplication is the easiest way to multiply large numbers. Let's consider this method using the example of two numbers 427 * 36.

1 step. Let's write the numbers one below the other, so that 427 is at the top and 36 at the bottom, that is, 6 under 7, 3 under 2.

Step 2. We begin multiplication with the rightmost digit of the bottom number. That is, the order of multiplication is: 6 * 7, 6 * 2, 6 * 4, then the same with three: 3 * 7, 3 * 2, 3 * 4.

So, first we multiply 6 by 7, answer: 42. We write it this way: since it turned out to be 42, then 4 are tens, and 2 are units, the recording is similar to addition, which means we write 2 under the six, and 4 we add the number 427 to the two.

Step 3. Then we do the same with 6 * 2. Answer: 12. The first ten, which is added to the four of the number 427, and the second - ones. We add the resulting two with the four from the previous multiplication.

Step 4. Multiply 6 by 4. The answer is 24 and add 1 from the previous multiplication. We get 25.

So, multiplying 427 by 6, the answer is 2562

REMEMBER! The result of the second multiplication should begin to be written down SECOND number of the first result!

Step 5. We perform similar actions with the number 3. We get the multiplication answer 427 * 3 = 1281

Step 6. Then we add up the obtained answers during multiplication and get the final multiplication answer 427 * 36. Answer: 15372.

Multiplication 4th grade

The fourth class is already the multiplication of large numbers only. The calculation is performed using the column multiplication method. The method is described above in accessible language.

For example, find the product of the following pairs of numbers:

988 * 98 =
99 * 114 =
17 * 174 =
164 * 19 =

Presentation on multiplication

Download a presentation on multiplication with simple tasks for second graders. The presentation will help children better navigate this operation, because it is written colorfully and in a playful style - in the best option for teaching a child!

Multiplication table

Every student in the second grade learns the multiplication table. Everyone should know it!

Examples for multiplication

Multiplying by one digit

9 * 5 =
9 * 8 =
8 * 4 =
3 * 9 =
7 * 4 =
9 * 5 =
8 * 8 =
6 * 9 =
6 * 7 =
9 * 2 =
8 * 5 =
3 * 6 =

Multiplying by two digits

4 * 16 =
11 * 6 =
24 * 3 =
9 * 19 =
16 * 8 =
27 * 5 =
4 * 31 =
17 * 5 =
28 * 2 =
12 * 9 =

Multiplying two-digit by two-digit

24 * 16 =
14 * 17 =
19 * 31 =
18 * 18 =
10 * 15 =
15 * 40 =
31 * 27 =
23 * 25 =
17 * 13 =

Multiplying three-digit numbers

630 * 50 =
123 * 8 =
201 * 18 =
282 * 72 =
96 * 660 =
910 * 7 =
428 * 37 =
920 * 14 =

Games for developing mental arithmetic

Special educational games developed with the participation of Russian scientists from Skolkovo will help improve skills oral counting in an interesting playful way.

Game "Quick Count"

The game "quick count" will help you improve your thinking. The essence of the game is that in the picture presented to you, you will need to choose the answer “yes” or “no” to the question “are there 5 identical fruits?” Follow your goal, and this game will help you with this.

Game "Mathematical matrices"

"Mathematical Matrices" is great brain exercise for kids which will help you develop his mental work, mental calculation, quick search necessary components, care. The essence of the game is that the player has to find a pair from the proposed 16 numbers that will add up to a given number, for example in the picture below the given number is “29”, and the desired pair is “5” and “24”.

Game "Number Span"

The number span game will challenge your memory while practicing this exercise.

The essence of the game is to remember the number, which takes about three seconds to remember. Then you need to play it back. As you progress through the stages of the game, the number of numbers increases, starting with two and further.

Game "Guess the operation"

The game “Guess the Operation” develops thinking and memory. The main point games need to be selected mathematical sign so that the equality is true. Examples are given on the screen, look carefully and put the required “+” or “-” sign so that the equality is true. The “+” and “-” signs are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you score points and continue playing.

Game "Simplification"

The game “Simplification” develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and a mathematical operation is given; the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need using the mouse. If you answered correctly, you score points and continue playing.

Game "Quick addition"

Game " Quick addition» develops thinking and memory. The main essence of the game is to choose numbers whose sum is equal to a given number. In this game, a matrix from one to sixteen is given. Above the matrix it is written given number, you need to select the numbers in the matrix so that the sum of these numbers is equal to the given number. If you answered correctly, you score points and continue playing.

Visual Geometry Game

Game " Visual geometry» develops thinking and memory. The main essence of the game is to quickly count the number of shaded objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, you need to quickly count them, then they close. There are four numbers written below the table, you need to choose one correct number and click on it with the mouse. If you answered correctly, you score points and continue playing.

Game "Mathematical Comparisons"

Game " Mathematical comparisons» develops thinking and memory. The main essence of the game is to compare numbers and mathematical operations. In this game you need to compare two numbers. At the top there is a question written, read it and answer the question correctly. You can answer using the buttons below. There are three buttons “left”, “equal” and “right”. If you answered correctly, you score points and continue playing.

Development of phenomenal mental arithmetic

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In essence, the whole difficulty lies in how to correctly place intermediate results multiplications (partial products. In an effort to make calculations easier, people have come up with many ways to multiply numbers. Over the centuries-old history of mathematics, there have been several dozen of them.

The heritage of the Hindus is the lattice method.

Hindus, who have known since ancient times decimal system notation, preferred mental counting written. They invented several ways to do fast multiplication. Later they were borrowed by the Arabs, and from them these methods were passed on to the Europeans. Those, however, did not limit themselves to them and developed new ones, in particular the one that is studied at school - multiplication by column. This method has been known since the beginning of the 15th century; in the next century it firmly came into use among mathematicians, and today it is used everywhere. But is column multiplication the best way to do this arithmetic operation? In fact, there are other, now forgotten, methods of multiplication that are no worse, for example, the lattice method.

This method was used in ancient times, in the Middle Ages it became widespread in the East, and in the Renaissance - in Europe. The grid method was also called Indian, Muslim or “Cellular Multiplication”. And in Italy it was called “Gelosia”, or “lattice multiplication” (Gelosia translated from Italian means “blinds”, “lattice shutters”. Indeed, the resulting figures from numbers were similar to shutters - blinds that covered windows from the sun Venetian houses.

We will explain the essence of this simple method of multiplication with an example: we will calculate the product 296 x 73. Let's start by drawing a table with square cells, in which there will be three columns and two rows, according to the number of digits in the factors. Divide the cells in half diagonally. Above the table we write the number 296, and with right side vertically - the number 73. Multiply each digit of the first number with each digit of the second and write the products in the corresponding cells, placing the tens above the diagonal and the ones below it. We obtain the digits of the desired product by adding the digits in the oblique stripes. In this case, we will move clockwise, starting from the lower right cell: 8, 2 1 7, etc. We will write the results under the table, as well as to the left of it. In the event that the addition results in a two-digit sum, we indicate only the ones, and add the tens to the sum of the digits from the next strip. Answer: 21,608. So, 296 x 73 = 21,608.

The lattice method is in no way inferior to column multiplication. It is even simpler and more reliable, despite the fact that the number of actions performed in both cases is the same. Firstly, you only have to work with single and double digit numbers, and they are easy to operate with in your head. Secondly, there is no need to remember intermediate results and keep track of the order in which they are written down. Memory is unloaded and attention is retained, so the likelihood of error is reduced. In addition, the lattice method allows you to get results faster. Once you master it, you can see for yourself.

Why does the lattice method lead to the correct answer? What is its “Mechanism”? Let's figure this out using a table constructed similarly to the first one, only in this case the factors are presented as the sums of 200 90 6 and 70 3.

As you can see, in the first oblique stripe there are units, in the second - tens, in the third - hundreds, etc. when added, they give the answer, respectively, the number of units, tens, hundreds, etc. the rest is obvious:

10 10 1500. 100. 8 _ 21608.

In other words, in accordance with the laws of arithmetic, the product of the numbers 296 and 73 is calculated as follows:

296 x 73 = (200 90 6) x (70 3) = 14,000 6300 420 600 270 18 = 10,000 (4000 6000) (300 400 600 200) (70 20 10) 8 = 21,608.

Nepera sticks.

Multiplication using the lattice method is the basis of a simple and original calculating device - neper sticks.

Its inventor, John Napier, a Scottish baron and a lover of mathematics, worked with professionals to improve the means and methods of calculation. In the history of science, he is known primarily as one of the creators of logarithms.

The device consists of ten rulers on which the multiplication table is placed. In each cell, divided by a diagonal, the product of two single-digit numbers from 1 to 9 is written: the number of tens is indicated in the upper part, the number of units is indicated in the lower part. One ruler (the left one) is stationary, the rest can be rearranged from place to place, laying out the desired number combination. Using neper sticks, it is easy to multiply multi-digit numbers, reducing this operation to addition.

For example, to calculate the product of the numbers 296 and 73, you need to multiply 296 by 3 and 70 (first by 7, then by 10) and add the resulting numbers. Let's attach three others to the fixed ruler - with the numbers 2, 9 and 6 at the top (they should form the number 296. Now let's look at the third line (the line numbers are indicated on the outer ruler. The numbers in it form a set that is already familiar to us.

Adding them, as in the lattice method, we get 296 x 3 = 888. Likewise, ra? 6

At school they study the multiplication table, and then teach children to multiply numbers in a column. Of course, this is not the only way to multiply. In fact, there were several dozen ways to multiply and divide multi-digit numbers. I will present here, perhaps, an even simpler “lattice method” (see the book by I.Ya. Depman, N.Ya. Vilenkin “Beyond the Pages of a Textbook”). Let's look at this method with an example.

Let's say we need to multiply 347 by 29. Let's draw a table, as in Figure a), write above it the number 347 from left to right, and to the right of it - the number 29 from top to bottom. In each cell we write the product of the numbers above this cell and to the right of it. In this case, we will write the tens digit of the product above the slash, and the units digit below it. Now we will add the numbers in each oblique stripe shown in the figure, performing this operation from right to left. If the amount is less than 10, then it is written under the bottom number of the strip. If it turns out to be greater than 10, then only the units digit of the amount is written, and the tens digit is added to the next amount. As a result, we obtain the desired product, which is equal to 10063.

This method of multiplication was previously common in the East and Italy. To understand its meaning, let's look at figure b). We see that in the first stripe there are units, in the second – tens, in the third – hundreds, etc. In other words, the product 347\cdot29 is calculated as follows:

There are some other rules to help quick counting. So, to square a two-digit number ending in 5, you need to add 1 to the first digit and multiply the resulting number by this digit, and then add 25 to the result. For example, let's square 35. The first digit of this number is 3, add 1: 3+1=4. Let's multiply 3 by 4, we get 12, then we just add 25. So the answer is: 1225.

This rule follows immediately from the fact that

Of course, this can also be used to square three-digit numbers ending in 5 and numbers that have even more digits. However, in these cases, you will have to calculate the product a\cdot(a+1) , where the number a already has several decimal places, and this also has to be done, say, in a column, that is, this is more complicated!

And now the video shows a method of multiplication, widely viewed and discussed on the Internet, which is called the Chinese method. Funny and interesting. By the way, some generalizations of this method have already been posted, because drawing 9 straight lines when multiplying by 9 is somehow long and uninteresting, and then counting the intersection points... In general, you still need to know the multiplication table! I think you can explain why the method works. Attention, question: why?

What is multiplication?

Multiplication is an arithmetic operation in which the first number is repeated as a term as many times as the second number shows.

A number that repeats as a term is called multiplyable(it is multiplied), the number that shows how many times to repeat the term is called multiplier. The number resulting from multiplication is called work.

For example, multiplying the natural number 2 by the natural number 5 means finding the sum of five terms, each of which is equal to 2:

2 + 2 + 2 + 2 + 2 = 10

In this example, we find the sum by ordinary addition. But when the number of equal terms is large, finding the sum by adding all the terms becomes too tedious.

Multiplication is indicated by the sign × (slash) or the sign · (dot) and is read: multiply by. The multiplication sign is placed between the multiplicand and the multiplier. The multiplicand is written to the left of the multiplication sign, and the multiplier is written to the right:

This entry reads like this: the product of 2 and 5 equals 10 or 2 times 5 equals 10.

So we see that multiplication is simply short form records of addition of identical terms.

Multiplication check

To check multiplication, you can divide the product by the factor. If the result of division is a number equal to the multiplicand, then the multiplication is done correctly:

Now let's check the multiplication:

Multiplication can also be checked by dividing the product by the multiplicand. If the result of division is a number equal to the multiplier, then the multiplication is done correctly:

Let's check:

Multiplying one and by one

a the following equalities are true:

1 · a = a
a· 1 = a

If the multiplicand is the number 1, then the product is equal to the multiplier. For example, 1 · 3 = 3 because the sum of 1 + 1 + 1 is three.
If the factor is one, then the product will be equal to the multiplicand. For example, 5 · 1 = 5. If we take the number 5 once, we get 5.

Number 0 in multiplication

For any natural number a the following equalities are true:

a· 0 = 0
0 · a = 0

These equalities mean the following:

If the factor is zero, then the product is zero. For example, 5 · 0 = 0 (if we don’t take 5 even once, then naturally we won’t get anything).
If the multiplicand is zero, then the product is zero. For example, 0 · 3 = 0 because the sum of 0 + 0 + 0 is zero.

Matching property multiplication indicates the equality of two products a·(b·c) and (a·b)·c, where a, b And c– any natural numbers. Thus, the result of multiplying three numbers a, b And c does not depend on the way the brackets are placed. Because of this, in the products a·(b·c) and (a·b)·c, parentheses are often not placed, and the products are written in the form a·b·c. Expression a·b·c called the product of three numbers a, b And c, numbers a, b And c all are also called multipliers.

Similarly, the combinatory property of multiplication allows us to state that the products (a·b)·(c·d) , (a·(b·c))·d , ((a·b)·c)·d , a·(b ·(c·d)) and a·((b·c)·d) are equal. That is, the result of multiplying four numbers also does not depend on the distribution of brackets. Product of four numbers a, b, c And d write it down as a b c d.

In general, the result of multiplying two, three, four, and so on numbers does not depend on the method of placing parentheses, and in writing such products, parentheses are usually omitted.

Now let's figure out how to calculate the product of several numbers, the notation of which does not contain parentheses. In this case multiplying three or more numbers is reduced to sequentially replacing two adjacent factors with their product until we get the required result. In other words, in writing the product, we place the brackets ourselves in any acceptable way, after which we sequentially multiply the two numbers.

Consider an example of calculating the product of five natural numbers 2 , 1 , 3 , 1 And 8 . Let's write down the work: 2 1 3 1 8. We will show two methods of solution (there are more than two methods of solution).

First way. We will successively replace the two factors on the left with their product. Since the result of multiplying numbers 2 And 1 is the number 2 , That 2·1·3·1·8=2·3·1·8. Because 2·3=6, That 2·3·1·8=6·1·8. Further, because 6 1=6, That 6·1·8=6·8. Finally, 6·8=48. So, the product of five numbers 2 , 1 , 3 , 1 And 8 equals 48 . This solution corresponds to the following method of arranging brackets: (((2 1) 3) 1) 8.

Second way. Let's arrange the brackets in the product like this: ((2 1) 3) (1 8) . Because 2 1=2 And 1·8=8, then ((2·1)·3)·(1·8)=(2·3)·8 . Two times three is six, then (2·3)·8=6·8. Finally, 6·8=48. So, 2·1·3·1·8=48.

Note that the result of multiplying three or more numbers is not affected by the order of the factors. In other words, the factors in the product can be written in any order, and can also be swapped. This statement follows from the properties of multiplication of natural numbers.

Let's look at an example.

Multiply four numbers 3 , 9 , 2 And 1 . Let's write down their product: 3·9·2·1. If we replace the factors 3 And 9 their product or factors 9 And 2 their product, then at the next stage we will have to multiply by two-digit numbers 27 or 18 (which we don’t know how to do yet). You can do without this by swapping the terms and arranging the brackets in a certain way. We have 3·9·2·1=3·2·9·1=(3·2)·(9·1)=6·9=54.

Thus, by swapping the factors, we can calculate the products in the most convenient way.

To complete the picture, consider a problem whose solution boils down to multiplying several numbers.

Each box contains 3 subject. Each box contains 2 boxes. How many items are contained in 4 boxes?

Since in one box there are 2 boxes, each of which 3 item, then in one box there is 3·2=6 items. Then in four drawers there is 6·4=24 subject.

One can argue differently. Since in one box there are 2 boxes, then in four boxes there are 2·4=8 boxes Since each box contains 3 subject, then in 8 boxes are 3·8=24 subject.

The announced solutions can be briefly written as (3·2)·4=6·4=24 or 3·(2·4)=3·8=24.

Thus, the required number of objects is equal to the product of the numbers 3 , 2 And 4 , that is, 3·2·4=24.

Let's summarize the information in this paragraph.

Multiplying three or more natural numbers is a sequential multiplication of two numbers. In addition, due to the commutative and combinative properties of multiplication, the factors can be swapped and any two of the multiplied numbers can be replaced with their product.

Multiplying a sum by a natural number and a natural number by a sum.

Addition and multiplication of numbers are related distributive property multiplication. This property allows you to study addition and multiplication together, which opens up much more opportunities than studying these actions separately.

We formulated the distribution property of multiplication relative to addition for two terms: (a+b) c=a c+b c , a, b, c– arbitrary natural numbers. Starting from this equality, we can prove the validity of the equalities (a+b+c) d=a d+b d+c d , (a+b+c+d) h=a h+b h+c h+d h etc., a, b, c, d, h– some natural numbers.

Thus, the product of the sum of several numbers and a given number is equal to the sum of the products of each of the terms and the given number. This rule can be used when multiplying a sum by a given number.

For example, let's multiply the sum of five numbers 7 , 2 , 3 , 8 , 8 per number 3 . Let's use the resulting rule: (7+2+3+8+8) 3=7 3+2 3+3 3+8 3+8 3. Because 7·3=21, 2·3=6, 3·3=9, 8·3=24, That 7·3+2·3+3·3+8·3+8·3=21+6+9+24+24. It remains to calculate the sum of five numbers 21+6+9+24+24=84 .

Of course, it was possible to first calculate the sum of the five given numbers, and then carry out the multiplication. But in this case we would have to multiply a two-digit number 7+2+3+8+8=28 per number 3 , which we don’t know how to do yet (we’ll talk about multiplying such numbers later in the section).

The commutative property of multiplication allows us to reformulate the rule for multiplying the sum of numbers by a given number as follows: the product of a given number and the sum of several numbers is equal to the sum of the products of a given number and each of the terms. This is the rule for multiplying a given number by a sum.

Here is an example of using the rule for multiplying a number by a sum: 2·(6+1+3)=2·6+2·1+2·3=12+2+6=20.

Let's look at a problem whose solution boils down to multiplying the sum of numbers by a given number.

Each box contains 3 red, 7 green and 2 blue items. How many items are in the four boxes?

One box contains 3+7+2 items. Then there are (3+7+2)·4 items in four boxes. Let's calculate the product of the sum and the number using the resulting rule: (3+7+2) 4=3 4+7 4+2 4=12+28+8=48.

48 items.

Multiplying a natural number by 10 , 100 , 1 000 and so on.

First, let's get the rule for multiplying an arbitrary natural number by 10 .

Natural numbers 20 , 30 , …, 90 inherently correspond 2 dozens, 3 dozens... 9 dozens, that is, 20=10+10 , 30=10+10+10 , ... Since we gave the multiplication of two natural numbers the meaning of the sum of identical terms, we have
2·10=20, 3·10=30, ..., 9·10=90.

Reasoning similarly, we arrive at the following equalities:
2·100=200, 3·100=300, ..., 9·100=900;
2·1 000=2 000, 3·1 000=3 000, ..., 9·1 000=9 000;
2·10,000=20,000, 3·10,000=30,000, ..., 9·10,000=90,000; ...

Since ten tens is a hundred, then 10·10=100;
since ten hundreds is a thousand, then 100·10=1,000;
since ten thousand is ten thousand, then 1,000·10=10,000.
Continuing these arguments, we have 10,000·10=100,000, 100,000·10=1,000,000, …

Let's now look at an example that will allow us to formulate a rule for multiplying an arbitrary natural number by ten.

Multiply a natural number 7 032 on 10 .

For this number 7 032 let's represent it as a sum bit terms, after which we will use the rule for multiplying the sum by the number that we obtained in the previous paragraph of this article: 7,032·10=(7,000+30+2)·10= 7,000·10+30·10+2·10.

Because 7 000=7 1 000 And 30=3·10, then the resulting amount 7 000 10+30 10+2 10 equal to the sum (7 1 000) 10+(3 10) 10+2 10, and the associative property of multiplication allows us to write the following equality:
(7 1 000) 10+(3 10) 10+2 10= 7·(1 000·10)+3·(10·10)+2·10.

By virtue of the results written before this example, we have 7·(1,000·10)+3·(10·10)+2·10= 7·10,000+3·100+2·10= 70,000+300+20.

Amount received 70 000+300+20 represents the expansion into digits of a number 70 320 .

7,032·10=70,320.

By performing similar actions, we can multiply any natural number by ten. At the same time, it is not difficult to notice that as a result we will receive numbers, the writing of which will differ from the writing of the number being multiplied only by the digit 0 , located on the right.

All the above considerations allow us to voice rule for multiplying an arbitrary natural number by ten: if in the notation of a given natural number, add a digit to the right 0 , then the resulting entry will correspond to the number that is the result of multiplying this natural number by 10 .

For example, 4·10=40, 43·10=430, 501·10=5 010, 79,020·10=790,200 etc.

And now, based on the rule of multiplying a natural number by 10 , we can obtain rules for multiplying an arbitrary natural number by 100 , on 1 000 etc.

Because 100=10·10, then multiplying any natural number by 100 boils down to multiplying this number by 10 10 . For example,
17·100=17·10·10=170·10=1,700;
504·100=504·10·10=5,040·10=50,400;
100 497 100=100 497 10 10= 1 004 970 10=10 049 700.

That is, if you add two digits to the right of the number being multiplied 0 , then we get the result of multiplying this number by 100 . This is it rule for multiplying a natural number by 100 .

Because 1 000=100·10, then multiplying any natural number by a thousand is reduced to multiplying this number by 100 and then multiplying the result by 10 . From these reasoning it follows rule for multiplying an arbitrary natural number by 1 000 : if you add three digits to the right of a number 0 , then we get the result of multiplying this number by a thousand.

Similarly, when multiplying a natural number by 10 000 , 100 000 and so on, you need to add four numbers to the right, respectively 0 , five digits 0 and so on.

For example,
58·1 000=58 000;
6,032·1,000,000=6,032,000,000;
777·10 000=7 770 000.

Multiplication of multi-valued and single-valued natural numbers.

Now we have all the skills necessary to perform multi-digit and single-digit multiplication of natural numbers.

What needs to be done for this?

Let's immediately understand it with an example.

Multiply a three-digit number 763 to a single digit number 5 , that is, we calculate the product 763·5.

First you need to represent a multi-digit number as a sum of digit terms. In our example 763=700+60+3 , then we have 763·5=(700+60+3)·5.

Now we apply: (700+60+3) 5=700 5+60 5+3 5.

Because 700=7·100 And 60=6·10(we talked about this in the previous paragraph), then the amount 700·5+60·5+3·5 can be written as (7 100) 5+(6 10) 5+3 5.

Due to the commutative and combinative properties of multiplication, the following equality is true: (7 100) 5+(6 10) 5+3 5= (5 7) 100+(5 6) 10+3 5 .

Because 5·7=35, 5·6=30 And 3·5=15, then (5·7)·100+(5·6)·10+3·5= 35·100+30·10+15.

All that remains is to multiply by 100 and on 10 , then add the three terms:
35 100+30 10+15= 3 500+300+15=3 815

Work 763 And 5 equals 3 815 .

It is clear that multiplying a single-digit number by multi-digit number carried out in a similar way.

To consolidate the material, we will give the solution to another example, but this time we will do without explanations.

3 And 104 558 .

3 104 558= 3·(100,000+4,000+500+50+8)=
=3·100,000+3·4,000+ 3·500+3·50+3·8=
=3·100,000+3·(4·1,000)+ 3·(5·100)+3·(5·10)+3·8=
=3·100,000+(3·4)·1,000+ (3·5)·100+(3·5)·10+3·8=
=3·100,000+12·1,000+ 15 100+15 10+3 8=
=300 000+12 000+ 1 500+150+24=313 674

The result of multiplying numbers 3 And 104 558 is the number 313 674 .

Multiplying two multi-digit natural numbers.

Now we have come to the culmination - the multiplication of two multi-digit natural numbers. First of all, you need to expand one of the factors into digits (usually the number whose record consists of a larger number of characters is expanded), then use the rule for multiplying a number by a sum (or a sum by a number). Further calculations will not cause difficulties if you have thoroughly mastered the information in the previous sections of this article.

Let's look at all the stages of multiplying two multi-digit natural numbers using an example.

Calculate product of numbers 41 And 3 806 .

Natural number expansion 3 806 by digits has the form 3 000+800+6 , therefore, 41·3 806=41·(3 000+800+6) .

Let's apply the rule for multiplying a number by a sum: 41·(3,000+800+6)= 41·3,000+41·800+41·6.

Because 3,000=3·1,000 And 800=8·100, then the equality 41·3 000+41·800+41·6= is true 41·(3·1 000)+41·(8·100)+41·6.

The combinational property of multiplication allows us to rewrite the last sum in the following form (41·3)·1,000+(41·8)·100+41·6.

To multiply one integer by another means to repeat one number as many times as the other contains units. To repeat a number means to take it as an addend several times and determine the sum.

Definition of multiplication

Multiplication of integers is an operation in which you need to take one number as addends as many times as another number contains units, and find the sum of these addends.

Multiplying 7 by 3 means taking the number 7 as its addend three times and finding the sum. The required amount is 21.

Multiplication is the addition of equal terms.

The data in multiplication is called multiplicand and multiplier, and the required - work.

In the proposed example, the data will be the multiplicand 7, the multiplier 3, and the desired product 21.

Multiplicand. A multiplicand is a number that is multiplied or repeated by a addend. A multiplicand expresses the magnitude of equal terms.

Factor. The multiplier shows how many times the multiplicand is repeated by the addend. The multiplier shows the number of equal terms.

Work. A product is a number that is obtained from multiplication. It is the sum of equal terms.

The multiplicand and the multiplier together are called manufacturers.

When multiplying integers, one number increases by as many times as the other number contains units.

Multiplication sign. The action of multiplication is denoted by the sign × (indirect cross) or. (dot). The multiplication sign is placed between the multiplicand and the multiplier.

Repeating the number 7 three times as a addend and finding the sum means 7 multiplied by 3. Instead of writing

write using the multiplication sign in short:

7 × 3 or 7 3

Multiplication is a shortened addition of equal terms.

Sign ( × ) was introduced by Oughtred (1631), and the sign. Christian Wolf (1752).

The relationship between the data and the desired number is expressed in multiplication

in writing:

7 × 3 = 21 or 7 3 = 21

verbally:

seven multiplied by three is 21.

To make a product of 21, you need to repeat 7 three times

To make a factor of 3, you need to repeat the unit three times

From here we have another definition of multiplication: Multiplication is an action in which a product is made up of the multiplicand in the same way as a factor is made up of a unit.

The main property of the work

The product does not change due to a change in the order of producers.

Proof. Multiplying 7 by 3 means repeating 7 three times. Replacing 7 with the sum of 7 units and inserting them in vertical order, we have:

Thus, when multiplying two numbers, we can consider either of the two producers to be the multiplier. On this basis, manufacturers are called factors or just multipliers.

The most common method of multiplication is to add equal terms; but if the producers are large, this technique leads to long calculations, so the calculation itself is arranged differently.

Multiplying single digit numbers. Pythagorean table

To multiply two single-digit numbers, you need to repeat one number as a addend as many times as the other number contains units, and find their sum. Since multiplying integers leads to multiplying single-digit numbers, they create a table of products of all single-digit numbers in pairs. Such a table of all products of single-digit numbers in pairs is called multiplication table.

Its invention is attributed to the Greek philosopher Pythagoras, after whom it is called Pythagorean table. (Pythagoras was born around 569 BC).

To create this table, you need to write the first 9 numbers in a horizontal row:

1, 2, 3, 4, 5, 6, 7, 8, 9.

Then under this line you need to sign a series of numbers expressing the product of these numbers by 2. This series of numbers will be obtained when in the first line we add each number to itself. From the second line of numbers we move sequentially to 3, 4, etc. Each subsequent line is obtained from the previous one by adding the numbers of the first line to it.

Continuing to do this until line 9, we get the Pythagorean table in the following form

To find the product of two single-digit numbers using this table, you need to find one manufacturer in the first horizontal row, and the other in the first vertical column; then the required product will be at the intersection of the corresponding column and row. Thus, the product 6 × 7 = 42 is at the intersection of the 6th row and 7th column. The product of zero and a number and a number and zero always produces zero.

Since multiplying a number by 1 gives the number itself and changing the order of the factors does not change the product, all the different products of two single-digit numbers that you should pay attention to are contained in the following table:

Products of single-digit numbers not contained in this table are obtained from the data if only the order of the factor in them is changed; thus 9 × 4 = 4 × 9 = 36.

Multiplying a multi-digit number by a single-digit number

Multiplying the number 8094 by 3 is indicated by signing the multiplier under the multiplicand, placing a multiplication sign on the left and drawing a line to separate the product.

Multiplying the multi-digit number 8094 by 3 means finding the sum of three equal terms

therefore, to multiply, you need to repeat all orders of a multi-digit number three times, that is, multiply by 3 units, tens, hundreds, etc. Addition begins with one, therefore, multiplication must begin with one, and then move from the right hand to left to higher order units.

In this case, the progress of calculations is expressed verbally:

We start multiplication with units: 3 × 4 equals 12, we sign 2 under the units, and apply the unit (1 ten) to the product of the next order by the factor (or remember it in our minds).

Multiplying tens: 3 × 9 equals 27, but 1 in your head equals 28; We sign the tens 8 and 2 in our heads.

Multiplying hundreds: Zero multiplied by 3 gives zero, but 2 in your head equals 2, we sign 2 under the hundreds.

Multiplying thousands: 3 × 8 = 24, we sign completely 24, because we do not have the following orders.

This action will be expressed in writing:

From the previous example we deduce next rule. To multiply a multi-digit number by a single-digit number, you need:

Sign the multiplier under the units of the multiplicand, put a multiplication sign on the left and draw a line.

Start multiplication with simple units, then, moving from the right hand to the left, sequentially multiply tens, hundreds, thousands, etc.

If, during multiplication, the product is expressed as a single-digit number, then it is signed under the multiplied digit of the multiplicand.

If the product is expressed as a two-digit number, then the units digit is signed under the same column, and the tens digit is added to the product of the next order by the factor.

Multiplication continues until the full product is obtained.

Multiplying numbers by 10, 100, 1000...

Multiplying numbers by 10 means turning simple units into tens, tens into hundreds, etc., that is, increasing the order of all numbers by one. This is achieved by adding one zero to the right. Multiplying by 100 means increasing all orders of magnitude of what is being multiplied by two units, that is, turning units into hundreds, tens into thousands, etc.

This is achieved by adding two zeros to the number.

From here we conclude:

To multiply an integer by 10, 100, 1000, and generally by 1 with zeros, you need to add as many zeros to the right as there are in the factor.

Multiplying the number 6035 by 1000 can be expressed in writing:

When the multiplier is a number ending in zeros, only the significant digits are signed under the multiplicand, and the zeros of the multiplier are added to the right.

To multiply 2039 by 300, you need to take the number 2029 by adding it 300 times. Taking 300 terms is the same as taking three times 100 terms or 100 times three terms. To do this, we multiply the number by 3, and then by 100, or multiply first by 3, and then add two zeros to the right.

The progress of the calculation will be expressed in writing:

Rule. To multiply one number by another, represented by a digit with zeros, you must first multiply the multiplicand by the number expressed by the significant digit, and then add as many zeros as there are in the multiplier.

Multiplying a multi-digit number by a multi-digit number

To multiply a multi-digit number 3029 by a multi-digit 429, or find the product 3029 * 429, you need to repeat the 3029 addend 429 times and find the sum. Repeating 3029 with terms 429 times means repeating it with terms first 9, then 20 and finally 400 times. Therefore, to multiply 3029 by 429, you need to multiply 3029 first by 9, then by 20 and finally by 400 and find the sum of these three products.

Three works

are called private works.

The total product 3029 × 429 is equal to the sum of three quotients:

3029 × 429 = 3029 × 9 + 3029 × 20 + 3029 × 400.

Let us find the values of these three partial products.

Multiplying 3029 by 9, we find:

3029 ×9 27261 first private work

Multiplying 3029 by 20, we find:

3029 × 20 60580 second particular work

Multiplying 3026 by 400, we find:

3029 × 400 1211600 third partial work

Adding these partial products, we get the product 3029 × 429:

It is not difficult to notice that all these partial products are products of the number 3029 by single digit numbers 9, 2, 4, and one zero is added to the second product, resulting from multiplication by tens, and two zeros to the third.

Zeros assigned to partial products are omitted during multiplication and the progress of the calculation is expressed in writing:

In this case, when multiplying by 2 (the tens digit of the multiplier), sign 8 under the tens, or move to the left by one digit; when multiplying by the hundreds digit 4, sign 6 in the third column, or move to the left by 2 digits. In general, each particular work begins to be signed from the right hand to the left, according to the order to which the multiplier digit belongs.

Looking for the product of 3247 by 209, we have:

Here we begin to sign the second quotient product under the third column, since it expresses the product of 3247 by 2, the third digit of the multiplier.

Here we have omitted only two zeros, which should have appeared in the second partial product, as it expresses the product of a number by 2 hundreds or by 200.

From all that has been said, we derive a rule. To multiply a multi-digit number by a multi-digit number,

you need to sign the multiplier under the multiplicand so that the numbers of the same orders are in the same vertical column, put a multiplication sign on the left and draw a line.

Multiplication begins with simple units, then moves from the right hand to the left, multiplying the sequential multiplicand by the digit of tens, hundreds, etc. and creating as many partial products as there are significant digits in the multiplier.

The units of each partial product are signed under the column to which the digit of the multiplier belongs.

All partial products found in this way are added together and the total product is obtained.

To multiply a multi-digit number by a factor ending in zeros, you need to discard the zeros in the factor, multiply by the remaining number, and then add as many zeros to the product as there are in the factor.

Example. Find the product of 342 by 2700.

If the multiplicand and the multiplier both end in zeros, during multiplication they are discarded and then as many zeros are added to the product as are contained in both producers.

Example. Calculating the product of 2700 by 35000, we multiply 27 by 35

By adding five zeros to 945, we get the desired product:

2700 × 35000 = 94500000.

Number of digits of the product. The number of digits of the product 3728 × 496 can be determined as follows. This product is more than 3728 × 100 and less than 3728 × 1000. The number of digits of the first product 6 is equal to the number of digits in the multiplicand 3728 and in the multiplier 496 without one. The number of digits of the second product 7 is equal to the number of digits in the multiplicand and in the multiplier. A given product of 3728 × 496 cannot have digits less than 6 (the number of digits of the product is 3728 × 100, and more than 7 (the number of digits of the product is 3728 × 1000).

Where we conclude: the number of digits of any product is either equal to the number of digits in the multiplicand and in the factor, or equal to this number without a unit.

Our product may contain either 7 or 6 digits.

Degrees

Among different works, those in which the producers are equal deserve special attention. So, for example:

2 × 2 = 4, 3 × 3 = 9.

Squares. The product of two equal factors is called the square of a number.

In our examples, 4 is square 2, 9 is square 3.

cubes. The product of three equal factors is called the cube of a number.

So, in the examples 2 × 2 × 2 = 8, 3 × 3 × 3 = 27, the number 8 is the cube of 2, 27 is the cube of 3.

At all the product of several equal factors is calledpower of number . The powers get their names from the number of equal factors.

Products of two equal factors or squares are called second degrees.

Products of three equal factors or cubes are called third degrees, etc.

*	1	2	3	4	5	6	7	8	9
0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81

*	1	2	3	4	5	6	7	8	9
0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81

Multiplied by their number determines. Game "Mathematical Comparisons"

Commutative law of multiplication.

Combinative law of multiplication.

Multiplying any natural number by one.

Multiplying any natural number by zero.

Multiplying numbers

Natural numbers

Integers

Exponential notation

See also

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Notes

Literature

Excerpt describing Multiplication

Multiplying numbers

Multiplication formulas

Abbreviated multiplication

Abbreviated multiplication formulas

Multiplying fractions

Multiplication 2nd grade

Multiplication 3rd grade

Column multiplication:

Multiplication 4th grade

Presentation on multiplication

Multiplication table

Examples for multiplication

Multiplying by one digit

Multiplying by two digits

Multiplying two-digit by two-digit

Multiplying three-digit numbers

Games for developing mental arithmetic

Game "Quick Count"

Game "Mathematical matrices"

Game "Number Span"

Game "Guess the operation"

Game "Simplification"

Game "Quick addition"

Visual Geometry Game

Game "Mathematical Comparisons"

Development of phenomenal mental arithmetic

Speed ​​reading in 30 days

Development of memory and attention in a child 5-10 years old

Super memory in 30 days

Secrets of brain fitness, training memory, attention, thinking, counting

Money and the Millionaire Mindset

What is multiplication?

Multiplication check

Multiplying one and by one

Number 0 in multiplication

Multiplying a sum by a natural number and a natural number by a sum.

Multiplying a natural number by 10 , 100 , 1 000 and so on.

Multiplication of multi-valued and single-valued natural numbers.

Multiplying two multi-digit natural numbers.

Definition of multiplication

The main property of the work

Multiplying single digit numbers. Pythagorean table

Multiplying a multi-digit number by a single-digit number

Multiplying numbers by 10, 100, 1000...

Multiplying a multi-digit number by a multi-digit number

Degrees

Speed reading in 30 days

*	1	2	3	4	5	6	7	8	9
0	0	0	0	0	0	0	0	0	0
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81