Column division. The history of the emergence of arithmetic operations Notation in Germany
Multiplication and division signs played an important role in the development of mathematics. The cross multiplication sign (x) was first introduced by the English mathematician William Outred (1575–1660). Multiplication by a column, familiar to us from the school bench, is an invention of not so distant time! (He was also invented by Outred.) His students were the famous Christopher Wren, the creator of St. Paul's Cathedral in London, and the great mathematician J. Wallis. Another remarkable invention of Outred was also the well-known logarithmic, which was introduced into wide engineering practice by the creator of the universal steam engine at his engineering plant in Soho. Later, in 1698, the German mathematician G. Leibniz introduced the multiplication sign "dot".
People learned to divide numbers much later than to multiply. In division using tables of reciprocals was reduced to multiplication, the Egyptians used a special table of basic fractions. The European mathematician Herbert (born in 950 in Aquitaine) cited rules in his writings. But they were too complicated and were called "iron fission". Later in Europe, the Arabic method of division appeared, which we still use. It was much simpler, and therefore it was called the "golden division". The oldest division sign, most likely looked like this: "/". It was first used by the English mathematician William Outred in his Clavis Mathematicae (1631, London). The German mathematician Johan Rahn introduced the "+" sign for multiplication. It appeared in his book "Deutsche Algebra" (1659). Rahn's sign is often referred to as the "English sign" because the English were the first to use it, although its roots lie in Germany. The German mathematician Leibniz preferred the colon ":" - this character he first used in 1684 in his work "Acta eruditomm". Before Leibniz, this sign was used by the Englishman Johnson in 1633 in one book, but as a fraction sign, and not division in the narrow sense. In most countries, the colon ":" is preferred, in English-speaking countries and on calculator keys, the "+" symbol. For mathematical formulas around the world, the "/" sign is preferred. The signs of multiplication and division did not immediately receive universal recognition. How slowly the most elementary symbols came into use is shown by the following fact. In 1731, Steven Hels published his "Etudes in Statics", a large serious work, addressed by the author primarily to fellow members of the Royal Society of London and signed for publication by the president of the society, Isaac Newton. In the preface to this book, the author writes: “Since complaints are heard that the signs I use are incomprehensible to many (the book was published in the second edition), I will say: the sign “+” means “more” or “add”; so on page 18, line 4: "6 ounces + 240 grains" is the same as saying "to 6 ounces add 240 grains", and on line 16 of the same page the "x" sign means "multiply"; two short parallel lines mean "equals ", so 1820x4 is 7280, it's like 1820 times 4 gives (equal to) 7280".
The multiplication and division signs (÷) and (:) can also be used to denote a range. For example, "5÷10" may indicate a range, i.e. from 5 to 10 inclusive. If there is a table whose rows are denoted by numbers and columns by Latin letters, then the entry of the form "D4:F11" can be used to denote a cell array (two-dimensional range) from D to F and from 4 to 11.
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The Arabs did not erase the numbers, but crossed them out and inscribed a new number over the crossed out one. It was very inconvenient. Then the Arab mathematicians, using the same method of subtraction, began to start the action from the lowest digits, that is, once they worked out a new method of subtraction, similar to the modern one. To denote subtraction in the III century. BC e. in Greece, the inverted Greek letter psi (F) was used. Italian mathematicians used the letter M, initial in the word minus, to denote subtraction. In the 16th century, the - sign began to be used to indicate subtraction. Probably, this sign passed into mathematics from trading. Merchants, pouring wine from barrels for sale, indicated with a dash in chalk the number of measures of wine sold from the barrel.
Multiplication
Multiplication is a special case of adding multiple identical numbers. In ancient times, people learned to multiply already when counting objects. So, counting in order the numbers 17, 18, 19, 20, they were supposed to represent
20 is not only like 10 + 10, but also like two tens, that is, 2 10;
30 - like three tens, that is, repeat the term ten times three times - 3 - 10 - and so on
People began to multiply much later than to add. The Egyptians performed multiplication by repeated addition or successive doubling. In Babylon, when multiplying numbers, they used special multiplication tables - the "ancestors" of modern ones. In ancient India, a method of multiplying numbers was used, which is also quite close to the modern one. The Indians multiplied numbers starting from the highest digits. At the same time, they erased those numbers that had to be replaced during subsequent actions, since they added the number that we now remember when multiplying. Thus, the mathematicians of India immediately wrote down the product, performing intermediate calculations on the sand or in their minds. The Indian method of multiplication passed to the Arabs. But the Arabs did not erase the numbers, but crossed them out and inscribed a new number over the crossed out one. In Europe, for a long time, the product was called the sum of multiplication. The name "multiplier" is mentioned in works of the 6th century, and "multiplier" in the 13th century.
In the 17th century, some of the mathematicians began to denote multiplication with an oblique cross - x, while others used a dot for this. In the 16th and 17th centuries, various symbols were used to denote actions - there was no uniformity in their use. Only at the end of the 18th century, most mathematicians began to use a point as a multiplication sign, but they also allowed the use of an oblique cross. The multiplication signs ( , x) and the equal sign (=) became universally recognized thanks to the authority of the famous German mathematician Gottfried Wilhelm Leibniz (1646-1716).
Division
Any two natural numbers can always be added and also multiplied. A subtraction from a natural number can be performed only when the subtrahend is less than the minuend. Division without a remainder is feasible only for some numbers, and it is difficult to find out whether one number is divisible by another. In addition, there are numbers that cannot be divided at all by any number other than one. You can't divide by zero. These features of the action greatly complicated the path to understanding the methods of division. In ancient Egypt, the division of numbers was performed by the method of doubling and mediation, that is, dividing by two, followed by the addition of the selected numbers. The mathematicians of India invented the "division up" method. They wrote the divisor below the dividend, and all intermediate calculations - above the dividend. Moreover, those figures that were subject to change during intermediate calculations were erased by the Indians and new ones were written in their place. Having borrowed this method, the Arabs in intermediate calculations began to cross out the numbers and inscribe others above them. This innovation greatly complicated the "division up". The method of division, close to the modern one, first appeared in Italy in the 15th century.
For thousands of years, the action of division was not denoted by any sign - it was simply called and written down as a word. Indian mathematicians were the first to designate division with the initial letter from the name of this action. The Arabs introduced a line to indicate division. In the 13th century, the Italian mathematician Fibonacci adopted the line to indicate division from the Arabs. He was the first to use the term private. The colon sign (:) to indicate division came into use in the late 17th century.
The equal sign (=) was first introduced by the English mathematics teacher R. Rikorrd in the 16th century. He explained: "No two objects can be more equal to each other than two parallel lines." But even in the Egyptian papyri there is a sign that denoted the equality of two numbers, although this sign is completely different from the sign =.
Column division- a standard procedure in arithmetic designed to divide simple or complex multi-digit numbers by breaking the division into a number of simpler steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. This method allows division of arbitrarily large numbers by breaking the process into a series of successive simple steps.
Designation in Russia, Kazakhstan, Kyrgyzstan, France, Belgium, Spain, Ukraine, Belarus, Moldova, Georgia, Tajikistan, Uzbekistan, Mongolia
In Russia, the divisor is located to the right of the dividend, separated from it by a vertical bar. Division also occurs in a column, but the quotient (result) is written below the divider and separated from it by a horizontal line.
8420│4 500│4 -8 │2105 -4 │125 4 10 - 4 - 8 20 20 - 20 -20 0 0
Designation in Germany
- In some European countries, a different designation is used. The calculation is exactly the same, but written differently, as shown in the example:
127 ÷ 4 \u003d 31.75 (12 - 12 \u003d 0 which is written on the next line) 07 (seven is carried over from the dividend 127) 4 2 8 20 (5 × 4 = 20) 0
Designation in the Netherlands
The calculation is exactly the same, but written differently (the divisor is to the left of the dividend), as shown in the example of dividing 135 by 11 (with a result of 12 and a remainder of 3):
11 / 135 \ 12 11 -- 25 22 -- 3
Designation in America and Great Britain
Division on paper does not use forward slashes ( / ) or obelus ( ÷ ) . Instead, the dividend, divisor, and quotient (in the process of being found) are placed in a table. An example of dividing 500 by 4 (resulting in 125):
1 2 5 (Explanation) 4|500 4 (4 × 1 = 4) 1 0 (5 - 4 = 1 ) 8 (4 × 2 = 8) 2 0 (10 - 8 = 2) 20 (4 × 5 = 20) 0 (20 - 20 = 0)An example of division with a remainder:
31.75 4|127 12 (12 - 12 = 0 which is written on the next line) 07 (seven carried over from dividend 127) 4 3.0 (3 is the remainder divided by 4 to get 0.75) 2 8 (7 × 4 = 28) 20 (extra zero carried over) 20 (5 × 4 = 20) 0- First, look at the dividend (127) to determine if the divisor (4) can be subtracted from it (in our case, it can't, since we have one as the first digit and we can't use negative numbers, so we can't write − 3)
- If the first digit is not large enough, we take the next digit along with it. Thus, we will now have the number 12 as the first number.
- Take the maximum number of fours that can be subtracted from the first number. In our case, 3 fours can be subtracted from 12
- In private (above the second digit of the dividend, since this is the last digit that is used), write the resulting triple, and under the dividend, the number 12
- Subtract the 12 you wrote from the corresponding number above it (the result will be 0 of course)
- Repeat the first step
- Since 0 is not a good number for the dividend, move the next digit from the dividend (7). The result will be 07
- Repeat steps 3, 4 and 7
- You will have the number 31 in the quotient, 3 as the remainder, and no more numbers in the dividend
- You can continue dividing by getting a decimal in the quotient: add a dot to the quotient on the right, and zero to the remainder (3) on the right and continue the division, adding zero whenever the dividend is less than the divisor (4)
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Notes
Links
- Alternative Division Algorithms: , (unavailable link from 23-05-2013 (2432 days) - story , copy) ,
An excerpt characterizing the Division by a column
- Quel beau regne aurait pu etre celui de l "Empereur Alexandre! [He would owe all this to my friendship ... Oh, what a wonderful reign, what a wonderful reign! Oh, what a wonderful reign the reign of Emperor Alexander could be!]He glanced at Balashev with regret, and Balashev had just wanted to notice something, as he again hastily interrupted him.
“What could he desire and look for that he would not find in my friendship?” Napoleon said, shrugging his shoulders in bewilderment. - No, he found it best to surround himself with my enemies, and with whom? he continued. - He called the Steins, Armfelds, Wintzingerode, Benigsen, Stein - a traitor expelled from his fatherland, Armfeld - a libertine and intriguer, Wintzingerode - a fugitive subject of France, Benigsen is somewhat more military than others, but still incapable, who could not do anything done in 1807 and which should arouse terrible memories in Emperor Alexander ... Suppose, if they were capable, we could use them, ”continued Napoleon, barely managing to keep up with the incessantly arising considerations showing him his rightness or strength (which in his concept was one and the same) - but even that is not: they are not suitable either for war or for peace. Barclay, they say, is more efficient than all of them; but I won't say that, judging by his first movements. What are they doing? What are all these courtiers doing! Pfuel proposes, Armfeld argues, Bennigsen considers, and Barclay, called to act, does not know what to decide on, and time passes. One Bagration is a military man. He is stupid, but he has experience, eye and determination ... And what role does your young sovereign play in this ugly crowd. They compromise him and blame everything that happens on him. Un souverain ne doit etre a l "armee que quand il est general, [The sovereign should be with the army only when he is a commander,] - he said, obviously sending these words directly as a challenge to the sovereign's face. Napoleon knew how the emperor wanted Alexander to be a commander.
“It's been a week since the campaign started and you haven't been able to defend Vilna. You are cut in two and driven out of the Polish provinces. Your army murmurs...
“On the contrary, Your Majesty,” said Balashev, who barely had time to memorize what was said to him, and with difficulty following this firework of words, “the troops are burning with desire ...
“I know everything,” Napoleon interrupted him, “I know everything, and I know the number of your battalions as surely as mine. You do not have two hundred thousand troops, but I have three times as many. I give you my word of honor, ”said Napoleon, forgetting that his word of honor could not matter in any way,“ I give you ma parole d "honneur que j" ai cinq cent trente mille hommes de ce cote de la Vistule. [on my word that I have five hundred and thirty thousand people on this side of the Vistula.] The Turks are no help to you: they are no good and have proved it by making peace with you. The Swedes are predestined to be ruled by crazy kings. Their king was mad; they changed him and took another - Bernadotte, who immediately went mad, because a madman only, being a Swede, can make alliances with Russia. Napoleon grinned wickedly and raised the snuffbox to his nose again.
To each of Napoleon's phrases, Balashev wanted and had something to object to; he incessantly made the gesture of a man who wanted to say something, but Napoleon interrupted him. For example, about the madness of the Swedes, Balashev wanted to say that Sweden is an island when Russia is for it; but Napoleon cried out angrily to drown out his voice. Napoleon was in that state of irritation in which one must speak, speak, and speak, only in order to prove his justice to himself. It became hard for Balashev: he, as an ambassador, was afraid to drop his dignity and felt the need to object; but, like a man, he shrunk morally before forgetting the unreasonable anger in which, obviously, Napoleon was. He knew that all the words now spoken by Napoleon were of no importance, that he himself, when he came to his senses, would be ashamed of them. Balashev stood with lowered eyes, looking at Napoleon's moving thick legs, and tried to avoid his gaze.
“What are these allies of yours to me?” Napoleon said. - My allies are the Poles: there are eighty thousand of them, they fight like lions. And there will be two hundred thousand.
And, probably even more indignant that, having said this, he had told an obvious lie and that Balashev, in the same pose of submission to his fate, silently stood in front of him, he abruptly turned back, went up to Balashev’s very face and, making energetic and quick gestures with his white hands, almost shouted:
“Know that if you shake Prussia against me, know that I will erase her from the map of Europe,” he said with a pale face distorted by anger, striking with an energetic gesture of one small hand on the other. - Yes, I will throw you beyond the Dvina, beyond the Dnieper and restore against you that barrier that Europe was criminal and blind, which allowed it to be destroyed. Yes, that’s what will happen to you, that’s what you won by moving away from me, ”he said and silently walked several times around the room, shaking his thick shoulders. He put a snuff-box in his waistcoat pocket, took it out again, put it to his nose several times, and stopped in front of Balashev. He paused, looked mockingly straight into Balashev's eyes, and said in a low voice: "Et cependant quel beau regne aurait pu avoir votre maitre!"
Most countries prefer the colon ( : ) , in English-speaking countries and on the keys of calculators - the symbol ( ÷ ) . For mathematical formulas around the world, the sign ( ⁄ ) .
Symbol history
The oldest division sign is most likely the sign ( / ) . It was first used by the English mathematician William Oughtred in his work Clavis Mathematicae( , London).
Other uses of the characters ( ÷ ) And ( : )
Symbols ( ÷ ) And ( : ) can also be used to indicate a range. For example, "5÷10" may indicate a range, i.e. from 5 to 10 inclusive. If there is a table, the rows of which are denoted by numbers, and the columns by Latin letters, then the record of the form "D4:F11" can be used to denote a cell array (two-dimensional range) from D before F and from 4 to 11.
Encoding
| Sign | Unicode | Name | HTML/XML | LaTeX | |||
|---|---|---|---|---|---|---|---|
| Code | Name | Hexadecimal | Decimal | Mnemonics | |||
| : | U+003A | COLON | colon | : | : | - | : |
| ÷ | U+00F7 | DIVISION SIGN | ÷ | ÷ | ÷ | \div | |
| ∕ | U+2215 | DIVISION SLASH | ∕ | ∕ | - | / | |
| ⁄ | U+2044 | FRACTION SLASH | fraction sign | ⁄ | ⁄ | ⁄ | / |
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Literature
- Florian Cajori: A History of Mathematical Notations. Dover Publications 1993
see also
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An excerpt characterizing the sign of division
But this happiness of one side of her soul not only did not prevent her from feeling sorrow for her brother with all her strength, but, on the contrary, this peace of mind in one respect gave her a great opportunity to give herself completely to her feelings for her brother. This feeling was so strong in the first minute of leaving Voronezh that those who saw her off were sure, looking at her exhausted, desperate face, that she would certainly fall ill on the way; but it was precisely the difficulties and worries of the journey, which Princess Marya undertook with such activity, saved her for a while from her grief and gave her strength.As always happens during a trip, Princess Marya thought about only one trip, forgetting what was his goal. But, approaching Yaroslavl, when something that could lie ahead of her again opened up, and not many days later, but this evening, Princess Mary's excitement reached its extreme limits.
When a haiduk sent ahead to find out in Yaroslavl where the Rostovs were and in what position Prince Andrei was, he met a large carriage driving in at the outpost, he was horrified to see the terribly pale face of the princess, which stuck out to him from the window.
- I found out everything, Your Excellency: the Rostov people are standing on the square, in the house of the merchant Bronnikov. Not far, above the Volga itself, - said the haiduk.
Princess Mary looked at his face in a frightened questioning way, not understanding what he was saying to her, not understanding why he did not answer the main question: what is a brother? M lle Bourienne made this question for Princess Mary.
- What is the prince? she asked.
“Their excellencies are in the same house with them.
“So he is alive,” thought the princess, and quietly asked: what is he?
“People said they were all in the same position.
What did "everything in the same position" mean, the princess did not ask, and only briefly, glancing imperceptibly at the seven-year-old Nikolushka, who was sitting in front of her and rejoicing at the city, lowered her head and did not raise it until the heavy carriage, rattling, shaking and swaying, did not stop somewhere. The folding footboards rattled.
The doors opened. On the left was water - a big river, on the right was a porch; there were people on the porch, servants, and some sort of ruddy-faced girl with a big black plait, who smiled unpleasantly feignedly, as it seemed to Princess Marya (it was Sonya). The princess ran up the stairs, the smiling girl said: “Here, here!” - and the princess found herself in the hall in front of an old woman with an oriental type of face, who, with a touched expression, quickly walked towards her. It was the Countess. She embraced Princess Mary and began to kiss her.
