The history of the emergence of arithmetic operations. Column division Designation in America and Great Britain
School-lyceum No. __
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“The history of arithmetic operations”
Completed: __ 5th _ grade exercises
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Karaganda, 2015
The Arabs did not erase numbers, but crossed them out and wrote new figure above the crossed one. It was very inconvenient. Then the Arab mathematicians, using the same method of subtraction, began to begin the action from the lowest ranks, i.e., once they worked on a new method of subtraction, similar to the modern one. To indicate subtraction in the 3rd century. BC e. in Greece they used the inverted Greek letter psi (F). Italian mathematicians used the letter M, the initial letter in the word minus, to denote subtraction. In the 16th century, the sign - began to be used to indicate subtraction. This sign probably passed into mathematics from trade. Merchants, pouring wine from barrels for sale, used a chalk line to mark the number of measures of wine sold from the barrel.
Multiplication
Multiplication is a special case of addition of several identical numbers. In ancient times, people learned to multiply when counting objects. So, counting the numbers 17, 18, 19, 20 in order, they were supposed to represent
20 is not only like 10+10, but also like two tens, that is, 2 10;
30 is like three tens, that is, repeat the ten term three times - 3 - 10 - and so on
People started multiplying much later than adding. The Egyptians performed multiplication by repeated addition or successive doublings. In Babylon, when multiplying numbers, they used special multiplication tables - the “ancestors” of modern ones. IN Ancient India They used a method of multiplying numbers, also quite close to the modern one. The Indians multiplied numbers starting from the highest ranks. At the same time, they erased those numbers that had to be replaced during subsequent actions, since they added to them the number that we now remember when multiplying. Thus, Indian mathematicians immediately wrote down the product, performing intermediate calculations in the sand or in their heads. The Indian method of multiplication was passed on to the Arabs. But the Arabs did not erase the numbers, but crossed them out and wrote a new number above 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 "multiplicand" in the 13th century.
In the 17th century, some 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 indicate actions; there was no uniformity in their use. Only at the end of the 18th century did most mathematicians begin to use a dot as a multiplication sign, but they also allowed the use of an oblique cross. Multiplication signs ( , x) and the equal sign (=) became generally accepted 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. Subtraction from natural number can only be performed 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 by any number other than one. You cannot divide by zero. These features of the action significantly complicated the path to understanding division techniques. IN Ancient Egypt The division of numbers was carried out by the method of doubling and mediation, that is, division by two and subsequent addition of the selected numbers. Indian mathematicians invented the "up division" method. They wrote the divisor below the dividend, and all intermediate calculations above the dividend. Moreover, those numbers 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 began to cross out numbers in intermediate calculations and write others over them. This innovation made “up division” much more difficult. A 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 indicated by any sign - it was simply called and written down as a word. Indian mathematicians were the first to denote division with the initial letter from the name of this action. The Arabs introduced a line to denote the division. The line for marking division was adopted from the Arabs in the 13th century by the Italian mathematician Fibonacci. 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. Ricorrd in the 16th century. He explained: “No two objects can be more equal to each other than two parallel lines" But even in Egyptian papyri there is a sign that denoted the equality of two numbers, although this sign is completely different from the = sign.
Multiplication and division signs played a huge role in the development of mathematics. The multiplication sign "slash" (x) was first introduced by the English mathematician William Oughtred (1575–1660). Column multiplication, familiar to us from school, is an invention of not so distant times! (He was also invented by Oughtred.) 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 Oughtred was the well-known logarithmic one, which was introduced into widespread 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. While division using tables of reciprocal numbers was reduced to multiplication, the Egyptians used a special table of basic fractions. The European mathematician Herbert (born in 950 in Aquitaine) gave rules in his writings. But they were too complex and were called “iron fission”. Later, the Arabic method of division appeared in Europe, which we still use today. It was much simpler, and that's why it was called the "golden division". The oldest division sign, most likely looked like this: "/". It was first used by the English mathematician William Oughtred in his work "Clavis Mathematicae" (1631, London). German mathematician Johan Rahn introduced the "+" sign for multiplication. It appeared in his book "Deutsche Algebra" (1659). The Rana sign is often called 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 ":" - he first used this symbol 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 sign for a fraction, not a division in in the narrow sense. In most countries, the colon ":" is preferred; in English-speaking countries and on the keys of microcalculators, the "+" symbol is preferred. For mathematical formulas, the "/" sign is preferred throughout the world. The signs of multiplication and division did not immediately gain universal recognition. How slowly the most elementary symbols came into use is shown by the following fact. In 1731, Stephen Hels published his “Etudes on 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 society’s president, 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 its second edition), I will say: the sign “+” means “more” or “add”; so on page 18, line 4: "6 ounces + 240 grains" means the same as saying "to 6 ounces add 240 grains", and on line 16 of the same page the sign "x" means "multiply" two short parallel lines mean "equals"; “so 1820x4 is 7280, it’s the same as 1820 multiplied by 4 gives (equal to) 7280.”
The multiplication and division signs (÷) and (:) can also be used to indicate a range. For example, "5÷10" can indicate a range, that is, from 5 to 10 inclusive. If you have a table whose rows are designated by numbers and columns by Latin letters, then an entry like "D4:F11" can be used to designate an array of cells (a two-dimensional range) from D to F and from 4 to 11.
division sign, division sign mathematicsDivision sign- a mathematical symbol in the form of a colon (:), obelus (÷), or slash (/) used to represent the division operator.
In most countries, the colon (:) is preferred; in English-speaking countries and on the keys of microcalculators, the symbol (÷) is preferred. For mathematical formulas, the sign (⁄) is preferred throughout the world.
- 1 History of the symbol
- 2 Other uses of the symbols (÷) and (:)
- 3 Encoding
- 4 Literature
- 5 See also
History of the symbol
The oldest division sign is most likely the (/) sign. It was first used by the English mathematician William Oughtred in his work Clavis Mathematicae (1631, London).
The German mathematician Leibniz preferred the colon (:). He first used this symbol in 1684 in his work Acta eruditorum. Before Leibniz, this sign was used by the Englishman Johnson in 1633 in one book, but as a sign for a fraction, and not for division in the narrow sense.
The German mathematician Johann Rahn introduced the symbol (÷) to denote division. Together with the asterisk (∗) multiplication sign, it appeared in his book Teutsche Algebra in 1659. Due to its distribution in England, the Rana sign is often called the "English division sign", but its roots lie in Germany.
Other uses of the symbols (÷) and (:)
The symbols (÷) and (:) can also be used to indicate a range. For example, "5÷10" can indicate a range, that is, from 5 to 10 inclusive. If you have a table whose rows are designated by numbers and columns by Latin letters, then an entry like “D4:F11” can be used to designate an array of cells (two-dimensional range) from D to F and from 4 to 11. This is how the Japanese use the sign (-
Encoding
Sign | Unicode | Name | HTML/XML | LaTeX | |||
---|---|---|---|---|---|---|---|
code | Name | hexadecimal | decimal | named | |||
(:) | U+003A | Colon | colon | : | : | absent | : |
(÷) | U+00F7 | Division sign | ÷ | ÷ | ÷ | \div | |
(∕) | U+2215 | Division slash | ∕ | ∕ | absent | / | |
(⁄) | U+2044 | Fraction slash | fraction sign | ⁄ | ⁄ | ⁄ | / |
Literature
- Florian Cajori: A History of Mathematical Notations. Dover Publications 1993
see also
Fraction (mathematics)
Mathematical signs | |
---|---|
Plus ( + ) Minus ( − ) Multiplication sign ( · or × ) Division sign (: or / ) Root sign ( √ ) Equal sign ( = , ≈ , ≡ etc.) Inequality signs ( ≠ , > , < etc.) Infinity sign ( ∞ ) Integral sign ( ∫ ) Factorial ( ! ) Vertical bar ( | ) Degree sign ( ° ) Minute degree ( ′ ) |
Most countries prefer a colon ( : ) , in English-speaking countries and on the keys of microcalculators - the symbol ( ÷ ) . For mathematical formulas all over the world, preference is given to the sign ( ⁄ ) .
History of the symbol
The oldest division sign is most likely the sign ( / ) . It was first used by an English mathematician William Oughtred in his work Clavis Mathematicae ( , London).
Other uses of symbols ( ÷ ) And ( : )
Characters ( ÷ ) And ( : ) can also be used to indicate a range. For example, "5÷10" can indicate a range, that is, from 5 to 10 inclusive. If you have a table whose rows are designated by numbers and columns by Latin letters, then an entry like “D4:F11” can be used to designate 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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Excerpt characterizing the division sign
But this happiness on one side of her soul not only did not prevent her from feeling grief for her brother with all her might, but, on the contrary, this peace of mind in one respect gave her a greater opportunity to fully surrender to her feelings for her brother. This feeling was so strong in the first minute of leaving Voronezh that those accompanying her were sure, looking at her exhausted, desperate face, that she would certainly get sick on the way; but it was precisely the difficulties and worries of the journey, which Princess Marya took on with such activity, that saved her for a time from her grief and gave her strength.As always happens during a trip, Princess Marya thought only about one journey, forgetting what was its goal. But, approaching Yaroslavl, when what could lie ahead of her was revealed again, and not many days later, but this evening, Princess Marya’s excitement reached its extreme limits.
When the guide sent ahead to find out in Yaroslavl where the Rostovs were standing and in what position Prince Andrei was, met a large carriage entering at the gate, he was horrified when he saw the terribly pale face of the princess, which leaned out of the window.
“I found out everything, your Excellency: the Rostov men are standing on the square, in the house of the merchant Bronnikov.” “Not far away, just above the Volga,” said the hayduk.
Princess Marya looked fearfully and questioningly at his face, not understanding what he was telling her, not understanding why he did not answer the main question: what about brother? M lle Bourienne asked this question for Princess Marya.
- What about the prince? – she asked.
“Their Lordships are standing with them in the same house.”
“So he is alive,” thought the princess and quietly asked: what is he?
“People said they were all in the same situation.”
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 steps rattled.
The doors opened. On the left there was water - a large river, on the right there was a porch; on the porch there were people, servants and some kind of ruddy girl with a large black braid who was smiling unpleasantly, as it seemed to Princess Marya (it was Sonya). The princess ran up the stairs, the girl feigning a smile said: “Here, here!” - and the princess found herself in the hallway in front of an old woman with oriental type face, who quickly walked towards her with a touched expression. It was the Countess. She hugged Princess Marya and began to kiss her.