Who invented addition. The history of the action of addition from ancient times to the present day
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The history of the origin of mathematical signs Prepared by: Cherepanov Ivan, student 5th grade Mathematics teacher: Mosunova O.A. As there is no table in the world without table legs, As there is no goat horns in the world, Cats without mustaches and without crayfish shells, So there are no actions in arithmetic without signs!
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Tasks Consider where mathematical signs came to us from and what they originally meant. Compare mathematical signs of different peoples. Consider the similarities of modern mathematical signs with the signs of our ancestors
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Object: mathematical signs of different nations. Main research methods: literature analysis, comparison, student survey, analysis and generalization of the data obtained during the study.
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Why in our time do we use just such mathematical signs: + “plus”, - “minus”, ∙ “multiplication” and: “division”, and not some others? Problem
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Hypothesis I think that mathematical signs arose simultaneously with the appearance of numbers and numbers
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The origin of mathematical signs The origin of these signs cannot always be precisely established. The symbols for the arithmetic operations of addition (plus "+'') and subtraction (minus "-'') are so common that we almost never think that they did not always exist. In fact, someone had to invent these symbols (or at least others that later evolved into the ones we use today). Certainly also some time passed before these symbols became generally accepted. There is an opinion that the signs "+" and "-" originated in trading practice. The vintner marked with dashes how many measures of wine he sold from the barrel. Pouring new reserves into the barrel, he crossed out as many expendable lines as he restored the measures. So, supposedly, there were signs of addition and subtraction in the 15th century. There is another explanation regarding the origin of the “+” sign. Instead of "a + b" they wrote "a and b", in Latin "a et b". Since the word “et” (“and”) had to be written very often, they began to abbreviate it: first they wrote one letter t, which, in the end, turned into a “+” sign
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The algebraic sign “-” The first use of the modern algebraic sign “+” refers to a German manuscript on algebra from 1481, which was found in the Dresden library. In a Latin manuscript from the same time (also from the Dresden library), there are both symbols: + and - . Johann Widmann is known to have reviewed and commented on both of these manuscripts. In 1489, in Leipzig, he published the first printed book (Mercantile Arithmetic - “Commercial Arithmetic”), in which both + and - signs were present (see figure). The fact that Widman used these symbols as if they were common knowledge points to the possibility of their origin in trade. An anonymous manuscript, apparently written around the same time, also contains the same characters, and this provided two additional books published in 1518 and 1525.
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Some mathematicians such as Record, Harriot and Descartes used the same sign. Others (eg Hume, Huygens, and Fermat) used the Latin cross “†” sometimes placed horizontally, with a crossbar at one end or the other. Finally, some (such as Halley) used the more decorative look of Widman
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The first occurrence of "+" and "-" in English is found in the 1551 algebra book "The Whetstone of Witte" by the Oxford mathematician Robert Record, who also introduced the equals sign, which was much longer than the current sign. In describing the plus and minus signs, Record wrote: “Other two signs are often used, the first of which is written “+” and means more, and the second “-” and means less.
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Subtraction sign The subtraction notation was somewhat less fancy, but perhaps more confusing (for us, at least), as instead of the simple "-" sign, German, Swiss, and Dutch books sometimes used the symbol "÷'', which we now denote division. Several books of the seventeenth century (for example, those of Halley and Mersenne) use two dots "∙ ∙" or three dots "∙ ∙ ∙" to indicate subtraction.
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In ancient Egypt In the famous Egyptian papyrus of Ahmes, a pair of legs going forward indicates addition, and leaving - subtraction
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The ancient Greeks denoted addition by writing side by side, but occasionally used the slash symbol “/'' and the semi-elliptic curve for subtraction. The Hindus, like the Greeks, usually did not denote addition in any way, except that the characters used in Bakhshali's Arithmetic manuscript (probably third or fourth century).
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At the end of the fifteenth century, the French mathematician Chuquet (1484) and the Italian Pacioli (1494) used “p” (denoting “plus”) for addition and “m” (denoting “minus”) for subtraction. Shuke
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In Italy In Italy, the symbols "+" and "-" were adopted by the astronomer Christopher Clavius (a German living in Rome), the mathematicians Gloriosi and Cavalieri in the early seventeenth century Christopher Clavius
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Multiplication sign To denote the action of multiplication, some of the European mathematicians of the 16th century used the letter M, which was the initial in the Latin word for increase, multiplication, - animation (the name "cartoon" comes from this word). In the 17th century, some mathematicians began to denote multiplication with a slash "×", while others used a period for this. In Europe, for a long time, the product was called the sum of multiplication. The name "multiplier" is mentioned in the works of the XI century. For thousands of years, the action of division was not indicated by signs. The Arabs introduced the line "/" to indicate division. It 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 at the end of the 17th century. In Russia, the names “divisible”, “divisor”, “private” were first introduced by L.F. Magnitsky at the beginning of the 18th century. The multiplication sign was introduced in 1631 by William Ootred (England) in the form of an oblique cross. Before him, the letter M was used. Later, Leibniz replaced the cross with a dot (late 17th century) so as not to confuse it with the letter x; before him, such symbolism was found in Regiomontanus (XV century) and the English scientist Thomas Harriot (1560-1621).
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Division marks Oughtred preferred the slash "/". Colon division began to denote Leibniz. Before them, the letter D was also often used. In England and the United States, the ÷ (obelus) symbol, which was proposed by Johann Rahn and John Pell in the middle of the 17th century, became widespread.
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Equality and Inequality Signs The equal sign was designated at different times in different ways: both by words and by various symbols. The “=” sign, so convenient and understandable now, came into general use only in the 18th century. And this sign was proposed to denote the equality of two expressions by the English author of the algebra textbook Robert Ricord in 1557. He explained that there is nothing more equal in the world than two parallel segments of the same length. In continental Europe, the equal sign was introduced by Leibniz. The "not equal" sign is first encountered by Euler. Comparison marks were introduced by Thomas Harriot in his work, published posthumously in 1631. Before him, they wrote in words: more, less.
ADDITION
Meaning:
ADDITION, -i, cf.
2. A mathematical operation, by means of which one obtains a new one from two or more numbers (or values), containing as many units (or values) as there were in all given numbers (values) together. Task on p.
3. A word formed according to the method of word formation (special).
II. ADDITION, -i, cf. Same as body~ . Bogatyrskoye s.
Meaning:
complicated e nie
cf.1) The process of action by value. verb: add up (2*).
2) A mathematical operation by means of which a new one is obtained from two or more numbers - terms - a sum containing as many units as there were in all the named numbers together.
4) One of the layers of canvas, tape, roving, laid parallel to other layers or superimposed on other layers (in spinning).
Modern explanatory dictionary ed. "Great Soviet Encyclopedia"
ADDITION
Meaning:
arithmetic operation. Denoted by a + (plus) sign. In the area of positive integers (natural numbers), as a result of addition by given numbers (terms), a new number (sum) is found, containing as many units as there are in all terms. The action of addition is also defined for the case of arbitrary real or complex numbers, as well as vectors, etc.
Small academic dictionary of the Russian language
addition
Meaning:
I, cf.
Action on verb. add (into 2, 5 and 8 values).
Addition of numbers. Abdication.
The reverse of subtraction is a mathematical operation by which two or more numbers (or values) are obtained from two or more numbers (or values) containing as many units (or values) as there were in all given numbers (values) together.
The beauty of the Grebenskaya woman is especially striking by the combination of the purest type of the Circassian face with the broad and powerful build of a northern woman. L. Tolstoy, Cossacks.
Tsygankov Alexander, student of the 4th grade, secondary school No. 7, Mirny
In mathematics lessons, we constantly work with one of the mathematical operations - addition, and we thought about when people first began to add, who and when gave the names to the components of this action, and what else can be learned about the addition action.
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HISTORY OF THE ACTION OF ADDITION FROM ANCIENT TIMES TO THE PRESENT DAYS.
In mathematics lessons, we constantly work with one of the mathematical operations - addition, and we thought about when people first began to add, who and when gave the names to the components of this action, and what else can be learned about the addition action.
Gradually, we learned that everyone needs mathematics in everyday life. Everyone has to count in life, we often use (without noticing it) knowledge about the quantities of length, time, mass. We realized that mathematics is an important part of human culture.
This paper discusses a number of interesting questions about the operation of addition, as one of the basic arithmetic operations.
Since ancient times, people have counted objects. People have been learning how to do arithmetic for over a thousand years.
Human fingers were not only the first counting instrument, but also the first calculating machine. Nature itself provided man with this universal counting tool. For many peoples, fingers (or their joints) played the role of the first counting device in any trading operations. For most of the everyday needs of people, their help was enough.
However, counting results were recorded in various ways.: notching, counting sticks, knots, etc. For example, the peoples of pre-Columbian America had a highly developed knot count. Moreover, the system of nodules also served as a storage and chronicle, having a rather complex structure. However, its use required a good memory training.
Many number systems go back to counting on the fingers, for example, five-fold (one hand), decimal (two hands), vigesimal (fingers and toes), forty-fold (the total number of fingers and toes of the buyer and seller). For many peoples, the fingers of the hands remained for a long time a counting tool even at the highest levels of development.
Well-known medieval mathematicians recommended finger counting as an auxiliary tool, which allows quite effective counting systems.
However, in different countries and at different times they thought differently.
Despite the fact that for many peoples the hand is a synonym and the actual basis of the numeral "five", for different peoples with a finger count from one to five, the index and thumb can have different meanings.
For Italians, when counting on the fingers, the thumb indicates the number 1, and the index finger indicates the number 2; when the Americans and the British count, the index finger means the number 1, and the middle finger means 2, in this case the thumb represents the number 5. And the Russians start counting on the fingers, bending the little finger first, and end with the thumb indicating the number 5, while the index the finger was compared with the number 4. But when they show the number, they put up the index finger, then the middle and ring fingers.
Each nation had its own arithmetic operations. And they were all used to perform operations on numbers. For a long time, people performed addition of numbers only verbally with the help of any objects - fingers, pebbles, shells, beans, sticks.
In ancient India, they found a way to add numbers in writing. When calculating, they wrote down the numbers with a stick on the sand, poured on a special board.
Indian sages suggested writing numbers in a column - one under the other; the answer is written below.
In ancient China, addition was done on the board with the help of special sticks. They were made from bamboo or ivory.
In ancient Egypt, the hieroglyph in the form of walking legs was used for addition. The direction of the legs coincided with the direction of the letter, which means that addition must be performed.
In ancient Russia, Russian people in their calculations used only two arithmetic operations - addition and subtraction, and called them doubling and bifurcation.
Some signs for addition appeared in antiquity, but until the 15th century there was almost no generally accepted sign. There are several points of view on how the sign for addition appeared.
In the 15th and 16th centuries, the Latin letter "P", the initial letter of the word plus, was used for the addition sign. Gradually, this letter began to be written with two lines. For addition, the Latin word " et" (floor) , denoting "And", which means "greater than". Since the word “et” had to be written very often, they began to shorten it: first they wrote one letter “t”, which gradually turned into the sign “+ ». There is a third opinion: the “+” sign originated in trading practice.
For the first time, the “+” sign appears in print in the book “A Quick and Beautiful Account for Merchants”. It was written by the Czech mathematician Jan Widman in 1489.
Man has always sought to simplify and speed up the solution of expressions, and this has led to the creation of computing devices. The ancient peoples used the abacus counting device in calculations.
An abacus is a counting board used for arithmetic calculations in ancient Greece and Rome. The abacus board was divided by lines into stripes, the count was carried out with the help of 5 stones and bones placed on the strips. In China and Japan, oriental abacuses of 7 bones were common: Chinese suan-pan and Japanese - soroban.
Russian abacus - abacus, appeared at the end of the 15th century. They have horizontal knitting needles with underwire and are based on the decimal system. Russian abacus was widely used for calculations. They are easy and quick to add and subtract.
For almost three centuries, talented scientists, engineers and designers have been creating mechanical calculating machines that make it easier to perform the four mathematical operations.
At the beginning of the 19th century, the French inventor Karl Thomas, took advantage of the ideas of the famous German scientist Leibniz and invented a calculating machine for performing 4 arithmetic operations and called it an adding machine. Adding machines until the early 1970s remained good helpers of calculators of all countries.
And 20 years ago, small devices were made that perform complex calculations in a matter of seconds - calculators. A calculator is an electronic computing device. Calculators can be desktop or (pocket) calculators, calculators built into computers, cell phones, and even wristwatches. But even faster than a calculator, a computer performs various mathematical operations. All these are assistants to a person in counting. Despite all the advantages of the computer age, there is the fact that many adults have forgotten how to count without a calculator. And many children even count on their fingers - this is very inconvenient. Therefore, I propose to learn how to count "in an adult way", using mathematical tricks - ways to memorize the addition table within 20 and quickly count without a calculator and fingers. Cunning mathematical tricks will allow you to instantly add in your mind. At first glance, these techniques seem confusing and incomprehensible. But having understood them and bringing the execution to automaticity, you will understand how simple, convenient and easy these techniques are. Count faster, count better!
From interviews with subject teachers, we learned that the action of addition is actively used in other sciences.
Russian language . Topic: "Word formation" (primary school teacher)
As a result of addition, a complex word-word with several roots is formed: snowfall, cinema, forest park.
Biology . Topic: "Human nutrition" (biology teacher)
Addition of calories is performed to determine the energy value of the product (proteins, fats, carbohydrates)
Geography . Topic: "Climate" (geography teacher)
Temperatures are added up for a certain period to find the average daily, average monthly, average annual temperature.
Physics . Topic "Interference" (physics teacher)
The addition in space of two (or several) waves, in which at different points an increase or decrease in the amplitude of the wave is obtained - wave interference.
We can see the action of addition everywhere: in the construction of houses, in the design and construction of a rocket, a car, in tailoring, for cooking, for raising animals, for making medicines and in many other areas of activity.
Findings :
- Addition has been used for a long time to count various objects.
- addition action is used in many sciences
- most often in life, both adults and children use addition
- the easiest way to add numbers on a calculator
- there are "easy" ways of mental counting when adding
There is an action by which the set of given numbers is reduced to the form a010n + a110n-1+ a210n-2 +.. . + an+an+110-1 + an+210-2 +.. . where all coefficients are less than ten. Everyone knows how to perform this transformation, and therefore we do not consider it necessary to go into details. D.S. Encyclopedic Dictionary of Brockhaus and Efron
addition
additions, cf.
only ed. action on verb. add in 2 5 and 7 digits. - fold - fold. Addition of forces (replacement of several forces by one that produces an equivalent action; physical). Addition of values. Addition of responsibilities.
only ed. One of the four arithmetic operations, by means of which a new (sum) is obtained from two or more numbers (summands), containing as many units as there were in all these numbers together. Addition rule. Addition task. Perform addition.
Same as physique; general physical condition of the body. A heroic addition, hefty was a kid. Nekrasov. I do not brag about my constitution, but I am cheerful and fresh, and lived to gray hair. Griboyedov.
The structure of matter (spec.). Nasal fold.
Explanatory dictionary of the Russian language. S.I. Ozhegov, N.Yu. Shvedova.
addition
A mathematical action by which two or more numbers - terms - get a new one - a sum containing as many units as there were in all the named numbers together.
One of the layers of canvas, tape, roving, laid parallel to other layers or superimposed on other layers (in spinning).
Encyclopedic Dictionary, 1998
addition
arithmetic operation. Denoted by a + (plus) sign. In the area of positive integers (natural numbers), as a result of addition by given numbers (terms), a new number (sum) is found, containing as many units as there are in all terms. The action of addition is also defined for the case of arbitrary real or complex numbers, as well as vectors, etc.
Addition
arithmetic operation. The result of the S. numbers a and b is a number called the sum of the numbers a and b (terms) and denoted by a + b. With S., the commutative (commutative) law is fulfilled: a + b \u003d b + a and the associative (associative) law: (a + b) + c \u003d a + (b + c). In addition to the scaling of numbers, mathematics considers actions, also called scaling, on various other mathematical objects (scaling of polynomials, vectors, matrices, and so on). For operations that do not obey the commutative and combinational laws, the term "S." do not apply.
Wikipedia
Addition (disambiguation)
Addition- a fundamental term, in different areas, almost always meaning that something whole is made up of some parts. It is most often used in a mathematical sense: addition is an arithmetic operation. As well as:
- Addition- the process of building walls from blocks, bricks.
- Addition- making syllables from letters, adding words from syllables.
- Addition- synonym figures .
Addition
Addition(often denoted by a plus sign "+") - an arithmetic operation. The result of adding numbers a and b is the number called the sum of the numbers a and b and denoted a + b. It is one of the four mathematical operations of arithmetic, along with subtraction, multiplication, and division. The addition of two natural numbers is the total sum of these quantities. For example, a combination of three and two apples gives a total of 5 apples. This observation is equivalent to the algebraic expression "3 + 2 = 5", i.e. "3 plus 2 equals 5."
Using systematic generalizations, addition can be defined for abstract quantities such as integers, rational numbers, real numbers, and complex numbers, and for other abstract objects such as vectors and matrices.
That is, each pair of elements ( a, b) from the set A c = a + b, called the sum a and b.
Addition has several important properties (for example, for A- sets of real numbers) (see Sum):
Commutativity: a + b = b + a, ∀a, b ∈ A Associativity: ( a + b) + c = a + (b + c), ∀a, b, c ∈ A Distributivity: x ⋅ (a + b) = (x ⋅ a) + (x ⋅ b), ∀a, b ∈ A. Adding 0 gives a number equal to the original: x + 0 = 0 + x = x, ∀x ∈ A, ∃0 ∈ A.
Addition is one of the simplest number operations. The addition of very small numbers is understandable even to children; the simplest problem, 1 + 1, can be solved by a five-month-old baby and even by some animals. Elementary school teaches to count in decimal notation, starting with adding simple numbers and gradually moving on to more complex problems.
Various devices for addition are known: from ancient abacus to modern computers,
Addition (mathematics)
Addition- one of the basic binary mathematical operations (arithmetic operations) of two arguments, the result of which is a new number (sum), obtained by increasing the value of the first argument by the value of the second argument. On a letter, it is usually indicated with a plus sign: a + b = c.
In general terms, one can write: S(a, b) = c, where a ∈ A and b ∈ A. That is, each pair of elements ( a, b) from the set A element is assigned c = a + b, called the sum a and b.
Addition is only possible if both arguments belong to the same set of elements (have the same type).
On the set of real numbers, the graph of the addition function has the form of a plane passing through the origin and inclined to the axes by 45° of angular degrees.
Addition has several important properties (for example, for A= R):
Commutativity: a + b = b + a, ∀a, b ∈ A. Associativity (see Sum): ( a + b) + c = a + (b + c), ∀a, b, c ∈ A. Distributivity: x ⋅ (a + b) = (x ⋅ a) + (x ⋅ b), ∀a, b ∈ A. Adding 0 (zero element) gives a number equal to the original: x + 0 = 0 + x = x, ∀x ∈ A, ∃0 ∈ A. Adding with the opposite element gives 0: a + ( − a) = 0, ∀a ∈ A, ∃ − a ∈ A.
As an example, in the picture on the right, 3 + 2 means three apples and two apples together, for a total of five apples. Note that you cannot add, for example, 3 apples and 2 pears. Thus, 3 + 2 = 5 In addition to counting apples, addition can also represent the union of other physical and abstract quantities, such as: negative numbers, fractional numbers, vectors, functions, and others.
Various addition devices are known: from ancient abacus to modern computers, the task of implementing the most efficient addition for the latter is relevant to this day.
Examples of the use of the word addition in the literature.
State Councilor Dorofeev - short-legged, square, apoplectic additions- opened the piano, played a few chords, then pulled up the sleeves of a dark green business card and played one of Grieg's sad melodies.
Next to Avramy was a young crossbowman, a heroic additions a boy with a scarred face, in whose mighty hands the heavy legion crossbow looked like a child's toy.
Lord Dono was a vigorous man of medium height with a short, broad black beard, wearing a Vor-style mourning suit, black with gray trim, to accentuate his athletic addition.
Este Ronde was tall, like all the strikers, but had unusual power for his middle age. addition.
young, strong additions a boy and a tall, dark-eyed girl in a long sleeveless fur robe, trimmed with white fur along the hem, boldly approached the counter where Thure Hund was standing.
tall, strong additions radiating energy, a kind of bon vivant, he grew into a major figure more due to his appearance than to oratory, which was owned by Hitler.
The captain is a heavy man about the same additions, as Mark Brehm, but physically more resilient - approached Stephen.
The black man himself, a hefty fellow of Hercules, seemed especially terrible to him. additions, and the Spaniard Cesare, small, overgrown with hair, black as a beetle, with a sly look of an evil and cunning animal.
But - only on condition that the glide path is in the center, which means that the plane moves along the hypotenuse, and all laws additions vectors are valid.
When he returned to the beach, a glider came close to the shore, and an athletic guy additions, who was driving, looked at those sitting and lying on the shore, looking for someone.
This does not contradict the existence of sorcery through the evil eye, leading to the bewitchment of a tender child. additions, or through other methods that cause a change in the state of bodies in people and animals, the transition of one element to another, resulting in hail, etc.
Recall that the increment and decrement operations of a pointer are equivalent addition 1 with a pointer, or subtracting 1 from a pointer, and the calculation takes place in the elements of the array to which the pointer is set.
He quickly learned them and mastered the simplest examples. additions and subtraction, although the decimal system, invented by beings with ten fingers on their hands, and different from the octal system of the Tendu, who had eight fingers, made matters difficult.
The complication of these appeals occurred through duplication and multiplication, additions two different bases, and differentiation also through intonations.
Meaning comes from additions numbers indicated by capital letters of this verse.
Two heads and six legs; four walk, and two lie still
Self-esteem - what is it: concept, structure, types and levels
Cassandra's Path, or Pasta Adventures War on Earth and Underground