The history of the emergence of arithmetic operations. Project "History of the Origin of Mathematical Signs" What is addition
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 mathematics is needed by everyone 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.
In this paper, we consider a number of interesting questions about the action of addition, as one of the main 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, fivefold (one hand), decimal (two hands), vigesimal (fingers and toes), forty (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 an instrument of counting and for the most high steps 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 considered 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.
In 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, forefinger means the number 1, and the middle one - 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 finger was compared with the number 4. But when they show number, put out 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.
AT ancient China addition was made on the board with the help of special sticks. They were made from bamboo or ivory.
AT Ancient Egypt for addition, a hieroglyph in the form of walking legs was used. The direction of the legs coincided with the direction of the letter, which means that addition must be performed.
AT 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.
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 bones and are based on 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 benefits 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, compound word with several roots: 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 accumulate over 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, in cooking, in raising animals, in making medicines, and in many other areas of activity.
Conclusions :
- 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
Addition is an operation in which, from two or more numbers, a number is found that is equal to all of them taken together.
Addition is the combination of two or more numbers into one.
These numbers in addition are called terms, and the desired - sum.
The sum contains as many units as there are in all terms.
When adding two numbers, one number increases by as many units as there are in the other number. Adding one number to another means add one number to another.
Addition sign. The action of addition is indicated by the + (plus) sign.
Single digit addition
To indicate that you need to add the numbers 2, 7, 8, 9, 6, write these numbers side by side, placing the addition sign + between them:
2 + 7 + 8 + 9 + 6.
For addition, the second number is added to the first number, then the third number is added to the result, and so on, up to the last number.
The very course of the calculation is expressed in writing:
2 + 7 + 8 + 9 + 6 = 32,
verbally:
2 and 7 are 9, 9 and 8 are seventeen, 17 and 9 are twenty-six, 26 and 6 are thirty-two.
The numbers 2, 7, 8, 9, 6 are terms, and the number 32 is the sum.
The main property of the sum. The sum will not change if we add the same numbers in a different order, since in this case the sum will contain the same units, therefore, the sum does not change from changing the order of the terms.
All addition rules are based on this sum property.
Multi-digit addition
To indicate that you need to add several multi-digit numbers (2302, 495, 30) they usually write:
2302 + 495 + 30.
We can consider each number as consisting of units, tens, hundreds, etc. Knowing that the sum does not change from a change in the order of the terms, we can separately add units with units, tens with tens, hundreds with hundreds, etc.
To facilitate addition, the terms of the numbers are signed one under the other so that the ones are under the ones, the tens under the tens, etc., that is, so that the numbers of the same order are in the same vertical column. Then we draw a line to separate the terms from the sum.
In our example, the numbers should be written like this:
2302 495 30
The course of calculation is expressed verbally:
Starting addition from units: 2 yes 5 make seven; We sign under units 7.
Adding tens: 9 yes 3 make 12; 12 tens make one hundred and 2 tens; we sign the number 2 under the tens, and we add the unit to the hundreds, we write it over the hundreds, or, as they usually say: we notice it in the mind.
Adding hundreds: 1 (in mind) yes 3 is 4, 4 yes 4 is 8; sign under hundreds 8.
Adding up thousands, we get 2.
The action itself will be expressed in writing:
Example. Adding the numbers 3275 + 41297 + 135 + 97, we have:
From the previous examples, we deduce addition rules:
To add whole numbers, you need to sign the terms one under the other so that units of the same order stand in one vertical column, that is, units under units, tens under tens, hundreds under hundreds, etc., draw a line and thus separate the terms from amounts.
Addition must begin with simple units, that is, from the first column, and then, moving from the right hand to the left to the next columns, add tens to tens, hundreds to hundreds, etc.
If, when adding simple units, you get a total of 9 or a number less than 9, you need to sign it under the unit column. If the sum turns out to be a number greater than 9, the number of units is signed under the column of units, and the number expressing tens is added to the next column.
When adding a column of tens, you need to do the same and continue adding until you get the full amount.
Explanatory Dictionary of the Living Great Russian Language by Vladimir Dahl
Addition, add up, complex, etc., see add up.
Explanatory dictionary of Ozhegov
Addition, -i, cf.
see fold.
A mathematical operation, by means of which from two or more numbers (or values) a new one is obtained, containing as many units (or values) as there were in all the given numbers (values) together. Task on p.
A word formed according to the method of conditional addition (special). , -i, cf. Same as physique. Bogatyrskoye s.
Explanatory dictionary of the Russian language Ushakov
ADDITION, additions, cf.
Only ed. action on verb. add up to 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 state organism. 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.
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 whole positive numbers (natural numbers) as a result of addition, according to the 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 algebraic expression"3 + 2 = 5", i.e. "3 a 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 task, 1 + 1, can be solved by a five-month-old baby and even by some animals. AT primary school learn to count in decimal notation, starting with addition prime numbers and gradually move on to more complex tasks.
Various devices for addition are known: from ancient abacus to modern computers,
Addition (mathematics)
Addition- one of the main 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 oratory 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 means, causing change the state of bodies in people and animals, the transition of some elements into others, leading to hail, etc.
Recall that the increment and decrement operations of a pointer are equivalent addition 1 with the pointer, or subtracting 1 from the 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 creatures with ten fingers on hands and different from the octal system of the Tendu, which 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.
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