What is arithmetic? When did humanity start using numbers and working with them? Where do the roots of such everyday concepts as numbers, fractions, subtraction, addition and multiplication, which a person has made an inseparable part of his life and worldview, go? Ancient Greek minds admired sciences such as mathematics, arithmetic and geometry as the most beautiful symphonies of human logic.
Maybe arithmetic is not as deep as other sciences, but what would happen to them if a person forget the elementary multiplication table? The logical thinking habitual to us, using numbers, fractions and other tools, was not easy for people and for a long time was inaccessible to our ancestors. In fact, before the development of arithmetic, no area of human knowledge was truly scientific.
Arithmetic is the ABC of mathematics
Arithmetic is the science of numbers, with which any person begins to get acquainted with the fascinating world of mathematics. As M. V. Lomonosov said, arithmetic is the gate of learning, opening the way to world knowledge for us. But he's rightIs the knowledge of the world can be separated from the 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 own laws.
The word "arithmetic" (Greek "arithmos") of Greek origin, means "number". She studies numbers and everything that can be connected with them. This is the world of numbers: various operations on numbers, numerical rules, solving problems that are related to multiplication, subtraction, etc.
It is generally accepted that arithmetic is the initial step of mathematics and a solid foundation for its more complex sections, such as algebra, mathematical analysis, higher mathematics, etc.
Main object of arithmetic
The basis of arithmetic is an integer, the properties and patterns of which are considered in higher arithmetic or number theory. In fact, the strength of the whole building - mathematics depends on how correct the approach is taken in considering such a small block as a natural number.
Therefore, the question of what arithmetic is 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 cannot write good or even the most mediocre poetry without an elementary alphabet, you cannot solve even an elementary problem without arithmetic. That is why all sciences advanced only after the development of arithmetic and mathematics, before that being just a set of assumptions.
Arithmetic is a phantom science
What is arithmetic - natural science or phantom? In fact, as the ancient Greek philosophers argued, neither numbers nor figures exist in reality. This is just a phantom that is created in human thinking when considering the environment with its processes. Indeed, what is a number? Nowhere around 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. Or maybe it is the study of ourselves from the inside? Philosophers have been arguing about this for many centuries in a row, so we do not undertake to give an exhaustive answer. One way or another, arithmetic has managed to take its place so firmly that in the modern world no one can be considered socially adapted without knowing its basics.
How did the natural number appear
Of course, the main object that arithmetic operates on is a natural number, such as 1, 2, 3, 4, …, 152… etc. The arithmetic of natural numbers is the result of counting ordinary objects, such as cows in a meadow. Still, the definition of "a lot" or "little" once ceased to suit people, and they had to invent more advanced counting techniques.
But the real breakthrough happened when human thought reached the point that one and the same number "two" can designate 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 perform some actions with these objects in the form of natural numbers. Thus was born the arithmetic of numbers, whichfurther developed and expanded, occupying ever greater positions in the life of society.
Such in-depth concepts of number as zero and negative number, fractions, designations of numbers by numbers and in other ways, have a rich and interesting history of development.
Arithmetic and practical Egyptians
The two oldest human companions in exploring the world around us 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 Rinda papyrus of Egyptian origin (so named because it belonged to the owner of the same name), dating back to the 20th century. BC, in addition to other valuable data, contains the expansion of one fraction into the sum of fractions with different denominators and a 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 abstract thoughts about numbers, on the contrary, calculations were made only for practical purposes. That is, the Egyptian will engage in 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 a person to sit down behind the papyrus. As you can see, the Egyptian progress in calculations was caused, rather, by mass construction than by love for science.
For this reason, the calculations found on the papyri cannot be called reflections on the topic of fractions. Most likely, this is a practical preparation that helped in the future.solve problems with fractions. The ancient Egyptians, who did not know the multiplication tables, made rather long calculations, decomposed into many subtasks. Perhaps this is one of those subtasks. It is easy to see that calculations with such workpieces are very laborious and unpromising. 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 knowledge of the Ancient East was successfully mastered by the ancient Greeks, famous lovers of abstract, abstract and philosophical reflections. They were no less interested in practice, but it is difficult to find the best theorists and thinkers. This has benefited science, since it is impossible to delve into arithmetic without breaking it off from reality. Of course, you can multiply 10 cows and 100 liters of milk, but you won't get very far.
The deep-thinking Greeks left a significant mark on history, and their writings have come down to us:
- Euclid and the Elements.
- Pythagoras.
- Archimedes.
- Eratosthenes.
- Zeno.
- Anaxagoras.
And, of course, the Greeks, who turned everything into philosophy, and especially the successors of the work of Pythagoras, were so fascinated by numbers that they considered them the mystery of the harmony of the world. Numbers have been studied and researched to such an extent that some of them and their pairs have been assigned 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 those numbers, one of whichis equal to the sum of all divisors of the second, and vice versa (the Pythagoreans knew only one such pair: 220 and 284).
The Greeks, who believed that science should be loved, and not be with it for the sake of profit, achieved great success by exploring, playing and adding numbers. It should be noted that not all of their research was widely used, some of them remained only "for beauty".
Eastern thinkers of the Middle Ages
In the same way, in the Middle Ages, arithmetic owes its development to Eastern contemporaries. The Indians gave us the numbers that we actively use, such a concept as "zero", and the positional version of the calculus, familiar to modern perception. From Al-Kashi, who worked in Samarkand in the 15th century, we inherited decimal fractions, 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 work "The Book of the Abacus", introducing Eastern innovations. It became the cornerstone of the development of algebra and arithmetic, research and scientific activities in Europe.
Russian arithmetic
And, finally, arithmetic, which found its place and took root in Europe, began to spread to Russian lands. The first Russian arithmetic was published in 1703 - it was a book about arithmetic by Leonty Magnitsky. For a long time it remained the only textbook in mathematics. It contains the initial moments of algebra and geometry. The numbers used in the examples by the first arithmetic textbook in Russia are Arabic. Although Arabic numerals have been seen before, on engravings dating back to 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 a throne, under her is written in Hebrew a word denoting the name of God, and on the steps that lead to the throne, the words “division”, “multiplication”, “addition”, etc. are inscribed. truths that are now considered commonplace.
A 600-page textbook covers both basics like the addition and multiplication tables and applications to navigational 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 the numerator, there is art honest, unenviable …". This approach to arithmetic is quite justified, because it is its widespread introduction that can be considered the beginning of the rapid development of scientific thought in Russia and general education.
Unprime primes
A prime number is a natural number that has only 2 positive divisors: 1 and itself. All other numbers, except for 1, are called composite. Examples of prime numbers: 2, 3, 5, 7, 11, and all others that have no divisors other than 1 and itself.
As for the number 1, it is on a special account - there is an agreement that it should be considered neither simple nor composite. Simple at first glance, a simple number hides many unsolved mysteries inside.
Euclid's theorem says that there are an infinite number of prime numbers, and Eratosthenes invented a special arithmetic "sieve" that eliminates non-prime numbers, leaving only simple ones.
Its essence is to underline the first non-crossed out number, and subsequently to cross out those that are multiples of it. We repeat this procedure many times - and we get a table of prime numbers.
The Fundamental Theorem of Arithmetic
Among the observations about prime numbers, the fundamental theorem of arithmetic should be mentioned in a special way.
The fundamental theorem of arithmetic says that any integer greater than 1 is either prime, or it can be decomposed into a product of prime numbers up to the order of the factors, and in a unique way.
The main theorem of arithmetic is proved rather cumbersome, and understanding it no longer looks like the simplest basics.
At first glance, prime numbers are an elementary concept, but they are not. Physics also once considered the atom to be elementary, until it found the whole universe inside it. A beautiful story by mathematician Don Tzagir "The First Fifty Million Primes" is dedicated to prime numbers.
From "three apples" to deductive laws
What can truly be called the reinforced foundation of all science is the laws of arithmetic. Even in childhood, everyone is faced with arithmetic, studying the number of legs and arms of dolls,the number of cubes, apples, etc. This is how we study arithmetic, which then goes into more complex rules.
All our 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. We know most of them, although we may not know their exact wording.
The law of addition and multiplication
Two any natural numbers a and b can be expressed as a sum a+b, which will also be a natural number. The following laws apply to addition:
- Commutative, which says that the sum does not change from the rearrangement of terms, or a+b=b+a.
- Associative, which says that the sum does not depend on the way the terms are grouped in places, or a+(b+c)=(a+ b)+ c.
The rules of arithmetic, such as addition, are among the most elementary, but they are used by all sciences, not to mention everyday life.
Two any natural numbers a and b can be expressed as a product ab or ab, which is also a natural number. The same commutative and associative laws apply to the product as to addition:
- ab=b a;
- a(bc)=(a b) c.
I wonderthat there is a law that unites addition and multiplication, also called a distributive or distributive law:
a(b+c)=ab+ac
This law actually teaches us to work with brackets by expanding them, thus we can work with more complex formulas. These are the laws that will guide us through the bizarre and complex world of algebra.
The law of arithmetic order
The law of order is used by human logic every day, comparing watches and counting banknotes. And, nevertheless, it needs to be formalized in the form of specific formulations.
If we have two natural numbers a and b, then the following options are possible:
- a equals b, or a=b;
- a is less than b, or a < b;
- a is greater than b, or a > b.
Out of three options, only one can be fair. The basic law that governs the order says: if a < b and b < c, then a< c.
There are also laws relating order to multiplication and addition: if a< is b, then a + c < b+c and ac< bc.
The laws of arithmetic teach us to work with numbers, signs and brackets, turning everything into a harmonious symphony of numbers.
Positional and non-positional calculus
It can be said that numbers are a mathematical language, on the convenience of which a lot depends. There are many number systems, which, like the alphabets of different languages, differ from each other.
Let's consider the number systems from the point of view of the influence of the position on the quantitative valuenumbers in this position. So, for example, the Roman system is non-positional, where each number is encoded by a certain set of special characters: I/ V/ X/L/ C/ D/ M. They are equal, respectively, to the numbers 1/ 5/10/50/100/500/ 1000. In such a system, the number does not change its quantitative definition depending on what position it is in: first, second, etc. To get other numbers, you need to add the base ones. For example:
- DCC=700.
- CCM=800.
The number system more familiar to us using Arabic numerals is positional. In such a system, the digit of a number determines the number of digits, for example, three-digit numbers: 333, 567, etc. The weight of any digit depends on the position in which this or that digit is located, for example, the number 8 in the second position has a value of 80. This is typical for the decimal system, there are other positional systems, for example, binary.
Binary arithmetic
We are familiar with the decimal system, consisting of single-digit numbers and multi-digit ones. The number on the left of a multi-digit number is ten times more significant than the one on the right. So, we are used to reading 2, 17, 467, etc. The section called "binary arithmetic" has a completely different logic and approach. This is not surprising, because binary arithmetic was created not for human logic, but for computer logic. If the arithmetic of numbers originated from the counting of objects, which was further abstracted from the properties of the object to "bare" arithmetic, then this will not work with a computer. To be able to sharewith his knowledge of a computer, a person had to invent such a model of calculus.
Binary arithmetic works with the binary alphabet, which consists of only 0 and 1. And the use of this alphabet is called the binary system.
The difference between binary arithmetic and decimal arithmetic is that the significance of the position on the left is no longer 10, but 2 times. Binary numbers are of the form 111, 1001, etc. How to understand such numbers? So, consider the number 1100:
- The first digit on the left is 18=8, remembering that the fourth digit, which means it needs to be multiplied by 2, we get position 8.
- Second digit 14=4 (position 4).
- Third digit 02=0 (position 2).
- Fourth digit 01=0 (position 1).
- So our number is 1100=8+4+0+0=12.
That is, when moving to a new digit on the left, its significance in the binary system is multiplied by 2, and in decimal - by 10. Such a system has one minus: it is too large an increase in digits that are needed to write numbers. Examples of representing decimal numbers as binary numbers can be found in the following table.
Decimal numbers in binary form are shown below.
Both octal and hexadecimal systems are also used.
This mysterious arithmetic
What is arithmetic, "twice two" or unexplored mysteries of numbers? As you can see, arithmetic may seem simple at first glance, but its unobvious ease is deceptive. It can also be studied by children along with Aunt Owl fromcartoon "Arithmetic-baby", and you can immerse yourself in deeply scientific research of an almost philosophical order. In history, she has gone from counting objects to worshiping the beauty of numbers. Only one thing is known for sure: with the establishment of the basic postulates of arithmetic, all science can rely on its strong shoulder.
