What is arithmetic? The main theorem of arithmetic. Binary Arithmetic

What is arithmetic? When did humanity begin to use numbers and work with them? Where do the roots of such commonplace concepts, as numbers, fractions, subtraction, addition and multiplication, which man has made an inseparable part of his life and worldview, go? Ancient Greek minds admired such sciences as mathematics, arithmetic and geometry, as the most beautiful symphonies of human logic.

Perhaps arithmetic is not as deep as other sciences, but what would have happened to them, forget the elementary table of multiplication? Usual logical thinking, using numbers, fractions and other tools, was not easily given to people and for a long time was not available for our ancestors. In fact, before the development of arithmetic, no area of human knowledge was truly scientific.

Arithmetic is the alphabet of mathematics

Arithmetic is the science of numbers with which any person begins to get acquainted with the fascinating world of mathematics. As M. Lomonosov said, arithmetic is the gateway of scholarship, which opens us the way to world-knowledge. But he is right, can knowledge of the world be separated from knowledge of numbers and letters, mathematics and speech? Perhaps in the old days, but not in the modern world, where the rapid development of science and technology dictates its laws.

The word "arithmetic" (Greek "arithmos") of Greek origin, means "number". She studies the number and everything that can be connected with them. This is a world of numbers: different actions on numbers, numerical rules, solving problems that involve multiplication, subtraction, and so on.

It is generally accepted that arithmetic is the initial step of mathematics and a solid foundation for more complex sections of it, such as algebra, matanalysis, higher mathematics, and so on.

The main object of arithmetic

The basis of arithmetic is an integer whose properties and regularities are considered in higher arithmetic or number theory. In fact, the strength of the whole building - the mathematics depends on how well the correct approach is taken in considering such a small block as a natural number.

Therefore, the question of what is arithmetic can be answered simply: it is the science of numbers. Yes, about the usual seven, nine and all this diverse community. And just as you can not write good and mediocre poems without elementary alphabet, without arithmetic you can not solve even an elementary problem. That is why all sciences have advanced only after the development of arithmetic and mathematics, being before all just a set of assumptions.

Arithmetic - phantom science

What is arithmetic - natural science or phantom? In fact, as the ancient Greek philosophers argued, there are no numbers or figures in reality. This is just a phantom that is created in human thinking when considering the environment with its processes. In fact, what is a number? Nowhere around do we see anything like that, which could be called a number, rather, a number is a way of the human mind to study the world. And maybe this is a study of ourselves from within? Philosophers have argued about this for many centuries in a row, so we do not undertake an exhaustive answer. One way or another, arithmetic managed to take its positions so firmly that in today's world no one can be considered socially adapted without knowledge of its foundations.

How did a natural number appear

Of course, the main object operated by arithmetic is a natural number, such as 1, 2, 3, 4, ..., 152 ... etc. The arithmetic of natural numbers is the result of counting ordinary objects, for example, cows in a meadow. Still, the definition of "a lot" or "little" once ceased to suit people, and I had to invent better techniques of counting.

But a real breakthrough happened when the human thought reached the point that it is possible to designate the same number "two" and 2 kilograms, and 2 bricks, and 2 parts. The fact is that you need to abstract from the forms, properties and meaning of objects, then you can make some actions with these objects in the form of natural numbers. This was how the arithmetic of numbers was born, which further developed and expanded, occupying ever greater positions in the life of society.

Such in-depth notions of numbers, as zero and a negative number, fractions, the notation of numbers in numbers and in other ways, have the richest and most interesting history of development.

Arithmetic and practical Egyptians

Two of the oldest human companions in the study of the surrounding world and solving everyday problems are arithmetic and geometry.

It is believed that the history of arithmetic originates in the Ancient East: in India, Egypt, Babylon and China. Thus, the papyrus of Rinda of Egyptian origin (named so, because it belonged to the owner of the same name), dated to the XX century. BC, except for other valuable data contains the decomposition of one fraction by the sum of fractions with different denominators and numerator equal to one.

For example: 2/73 = 1/60 + 1/219 + 1/292 + 1/365.

But what is the point of such a complex decomposition? The fact is that the Egyptian approach did not tolerate abstracted thinking about numbers, on the contrary, calculations were made only for practical purposes. That is, the Egyptian will deal with such a thing as calculations, solely in order to build a tomb, for example. It was necessary to calculate the length of the edge of the structure, and this forced the person to sit down for papyrus. Apparently, the Egyptian progress in the calculations was caused, rather by massive, construction, rather than love of science.

For this reason, the calculations found on papyri can not be called reflections on fractions. Most likely, this is a practical procurement, which helped in the future to solve problems with fractions. Ancient Egyptians, who did not know the multiplication tables, produced rather long calculations, decomposed into many sub-tasks. Perhaps this is one of those subtasks. It is not difficult to see that the calculations with such preparations are very laborious and of little prospect. Perhaps, for this reason, we do not see the great contribution of Ancient Egypt to the development of mathematics.

Ancient Greece and Philosophical Arithmetic

Many of the knowledge of the Ancient East has been successfully mastered by the ancient Greeks, known lovers of abstract, abstract and philosophical reflections. Practice of them was of no less interest, but it is difficult to find the best theorists and thinkers. This has gone to the benefit of science, since it is impossible to delve into arithmetic without breaking it with reality. Of course, you can multiply 10 cows and 100 liters of milk, but it will not be possible to go far.

Thinking deeply the Greeks left a significant mark in history, and their writings have reached us:

• Euclid and the "Beginning."
• Pythagoras.
• Archimedes.
• Eratosthenes.
• Zeno.
• Anaxagoras.

And, of course, the Greeks turning everything into philosophy, and especially the continuers of the Pythagorean case, were so keen on numbers that they considered them to be the mystery of the harmony of the world. The numbers have been so studied and studied that some of them and their pairs have been attributed special properties. For example:

• Perfect numbers are those that are equal to the sum of all their divisors, except for the number itself (6 = 1 + 2 + 3).
• Friendly numbers are numbers, one of which is equal to the sum of all the divisors of the second, and vice versa (the Pythagoreans only knew one such pair: 220 and 284).

The Greeks, who believed that science needed to be loved, and not to be with her for the sake of gain, achieved great success, exploring, playing and adding numbers. It should be noted that not all of their findings have found wide application, some of them remained only "for beauty."

Eastern thinkers of the Middle Ages

Similarly, in the Middle Ages, arithmetic owes its development to Eastern contemporaries. The Indians gave us figures that we are actively using, such a notion as "zero", and a positional version of the calculus system, familiar to modern perception. From al-kasha, who worked in Samarkand in the 15th century, we inherited decimals, without which it is difficult to imagine modern arithmetic.

In many ways, Europe's acquaintance with the achievements of the East became possible thanks to the work of the Italian scientist Leonardo Fibonacci, who wrote the book "The Abacus Book", introducing the eastern innovations. It became the cornerstone of the development of algebra and arithmetic, research and scientific activity in Europe.

Russian arithmetic

And, finally, arithmetic, which found its place and rooted in Europe, began to spread to the Russian lands. The first Russian arithmetic was published in 1703 - it was a book about the arithmetic of Leonty Magnitsky. For a long time it remained the only teaching manual on mathematics. It contains the initial moments of algebra and geometry. Figures, which used in the examples the first in Russia textbook of arithmetic, Arabic. Although the Arabic numerals were encountered earlier, on engravings dating from the 17th century.

The book itself is decorated with images of Archimedes and Pythagoras, and on the first sheet - the image of arithmetic in the form of a woman. She sits on the throne, under it is written in Hebrew the word denoting the name of God, and on the steps that lead to the throne, the words "division", "multiplication", "addition", etc. are inscribed, etc. One can only imagine the significance betrayed Such truths, which are now considered commonplace.

A textbook of 600 pages describes both the basics like the addition and multiplication table, and the applications to the navigation sciences.

It is not surprising that the author chose images of Greek thinkers for his book, because he himself was captivated by the beauty of arithmetic, saying: "Arithmetic is a numerator, there is an honest, ungrounded art ...". This approach to arithmetic is fully justified, because it is its widespread introduction that can be considered the beginning of rapid development of scientific thought in Russia and general education.

Uneasy prime numbers

A prime number is a natural number that has only 2 positive divisors: 1 and itself. All other numbers, not counting 1, are called composite. Examples of prime numbers: 2, 3, 5, 7, 11, and all others that do not have other divisors, except the number 1 and yourself.

As for the number 1, it is on a special account - there is a conviction that it must be considered neither simple nor complex. A simple at first sight simple number conceals a lot of unsolved mysteries within yourself.

Euclid's theorem says that prime numbers are an infinite set, and Eratosthenes came up with a special arithmetic "sieve" that sifts out uneasy numbers, leaving only simple ones.

Its essence is to emphasize the first non-underscored number, and in the future to delete those that are multiple to it. We repeat this procedure many times and get a table of prime numbers.

The main theorem of arithmetic

Among the observations on prime numbers, we need to mention in a special way the basic theorem of arithmetic.

The basic theorem of arithmetic says that any integer greater than 1 is either simple or it can be decomposed into a product of primes to within the order of the factors, in a unique way.

The main theorem of arithmetic is proved to be rather cumbersome, and its understanding is no longer similar to the simplest foundations.

At first glance, prime numbers are an elementary concept, but this is not so. Physics also once considered the atom elementary, until it found a whole universe inside it. The beautiful story of the mathematician Don Tsagir "The first fifty million prime numbers" is devoted to prime numbers.

From "three apples" to deductive laws

What truly can be called the reinforced foundation of the whole of science is the laws of arithmetic. As a child, everyone is faced with arithmetic, studying the number of legs and pens in dolls, the number of cubes, apples, etc. So we study arithmetic, which goes on to more complex rules.

Our whole life acquaints us with the rules of arithmetic, which have become for the common man the most useful of all that science gives. The study of numbers is "arithmetic-baby", which introduces a person to the world of numbers in the form of numbers in early childhood.

Higher arithmetic is a deductive science that studies the laws of arithmetic. Most of them we know, although, perhaps, we do not know their exact formulations.

The law of addition and multiplication

Any two natural numbers a and b can be expressed as a + b, which is also a natural number. Concerning the addition, the following laws apply:

• A commutative one that says that the sum does not change from the permutation of the summands in places, or a + b = b + a.
• Associative , which says that the sum does not depend on the way of grouping the summands in places, or a + (b + c) = (a + b) + c.

The rules of arithmetic, such as addition, are some of the elementary, but they are used by all sciences, not to mention everyday life.

Any two natural numbers a and b can be expressed in the product a * b or a * b, which is also a natural number. The same commutative and associative laws apply to the product as to the addition:

• A * b = b * a;
• A * (b * c) = (a * b) * c.

It is interesting that there is a law that combines addition and multiplication, also called distributive, or distributive law:

A (b + c) = ab + ac

This law actually teaches us to work with brackets, revealing them, thus we can work with more complex formulas. These are exactly the laws that will guide us through the bizarre and complex world of algebra.

The law of arithmetic order

The law of order uses human logic every day, comparing clocks and counting bills. And, nevertheless, and it needs to be formalized in the form of concrete formulations.

If we have two natural numbers a and b, then the following options are possible:

• A is b, or a = b;
• A is less than b, or a
• A is greater than b, or a> b.

Of the three options, only one can be fair. The basic law that governs order says: if a

There are also laws that link order with the actions of multiplication and addition: if a

The laws of arithmetic teach us to work with numbers, signs and brackets, turning everything into a harmonious symphony of numbers.

Positional and nonpositional systems of calculation

We can say that numbers are a mathematical language, from the convenience of which much depends. There are many systems of calculus, which, like the alphabets of different languages, differ from each other.

Consider the number system from the point of view of the influence of the position on the quantitative value of the digit at this position. For example, the Roman system is non-positional, where each number is coded by a specific set of special symbols: I / V / X / L / C / D / M. They are equal to 1/5/10/50/100/500 / 1000. In such a system, the figure does not change its quantitative definition, depending on what it stands for position: first, second, etc. To get other numbers, you need to add the base ones. For example:

• DCC = 700.
• CCM = 800.

More familiar to us the number system using Arabic numerals is positional. In such a system, the digit number determines the number of digits, for example, three-digit numbers: 333, 567, etc. The weight of any digit depends on the position on which this or that digit is located, for example the number 8 in the second position has a value of 80. This is characteristic of the decimal system, there are other positioning systems, for example, binary.

Binary Arithmetic

We are familiar with the decimal system of calculation, consisting of single-digit numbers and multi-digit ones. The digit on the left in a multi-digit number is ten times larger in importance than the one on the right. So, we used to read 2, 17, 467, etc. A completely different logic and approach for the section, which is called "binary arithmetic." This is not surprising, because binary arithmetic is not created for human logic, but for the computer. If the arithmetic of numbers has occurred from the account of objects, which later abstracted from the properties of the object to "bare" arithmetic, then this will not work with the computer. To be able to share their knowledge with computers, a person had to invent such a model of calculation.

Binary arithmetic works with a binary alphabet, which consists of only 0 and 1. And the use of this alphabet is called the binary system of the calculus.

The difference between binary arithmetic and decimal is that the importance of the position on the left is no longer 10, but 2 times. Binary numbers have the form 111, 1001, etc. How to understand such numbers? So, consider the number 1100:

1. The first digit on the left is 1 * 8 = 8, remembering that the fourth digit, and therefore, it must be multiplied by 2, we get position 8.
2. The second digit is 1 * 4 = 4 (position 4).
3. The third digit is 0 * 2 = 0 (position 2).
4. The fourth digit is 0 * 1 = 0 (position 1).
5. So, our number is 1100 = 8 + 4 + 0 + 0 = 12.

That is, when you switch to a new digit on the left, its significance in the binary system is multiplied by 2, and in the decimal one by 10. This system has one minus: it is too large a growth of digits, which are necessary for writing numbers. Examples of representing decimal numbers in the form of two-digit numbers can be found in the following table.

Decimal numbers in binary form are shown below.

Also used are the octal and hexadecimal systems of the calculus.

This mysterious arithmetic

What is arithmetic, "twice two" or unknown mysteries of numbers? As you can see, arithmetic may seem simple at first sight, but its unobvious ease is deceptive. It can be studied and children together with Aunt Sova from the cartoon "Arithmetic-baby", and you can immerse yourself in deeply scientific research of almost philosophical order. In history, she went from the counting of objects to the worship of the beauty of numbers. Only one thing is certain: with the establishment of the basic postulates of arithmetic, all science can rely on its strong shoulder.