What is arithmetic? When did humanity start using numbers and working with them? Where do the roots of such ordinary concepts as numbers, fractions, subtraction, addition and multiplication go, which a person has made an integral part of his life and worldview? Ancient Greek minds admired such sciences as mathematics, arithmetic and geometry, as the most beautiful symphonies of human logic.
Arithmetic may not be as deep as other sciences, but what would happen to them if the person had forgotten the elementary multiplication table? Our usual logical thinking, 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, not one area of human knowledge was truly scientific.
Arithmetic is the ABC of Mathematics
Arithmetic is the science of numbers, with which anyone begins to get acquainted with the fascinating world of mathematics. As M.V. Lomonosov said, arithmetic is the gateway to scholarship, which opens the way for us to learn about the world. But he is right, is it possible to separate the knowledge of the world 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 the number and everything that can be connected with them. This is the world of numbers: various actions on numbers, numerical rules, solving problems that are related to multiplication, subtraction, etc.
It is generally accepted that arithmetic is the initial step in mathematics and a solid foundation for its more complex sections, such as algebra, matanalysis, higher mathematics, etc.
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 entire building — mathematics — depends on how the correct approach is taken in considering such a small block as a natural number.
Therefore, the question of what arithmetic can be answered simply: this is the science of numbers. Yes, about the familiar seven, nine and all this diverse community. And just like good and most mediocre verses you cannot write without an elementary alphabet, without arithmetic you cannot solve even an elementary problem. That is why all sciences have advanced only after the development of arithmetic and mathematics, being before that just a set of assumptions.
Arithmetic - Phantom Science
What is arithmetic - natural science or phantom? In fact, as the ancient Greek philosophers reasoned, 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. In fact, what is a number? Nowhere else do we see anything like this that could be called a number; rather, a number is a way of the human mind to study the world. Or maybe this is a study of ourselves from within? Philosophers have been arguing about this for centuries in a row, so we don’t undertake to give an exhaustive answer. One way or another, arithmetic managed to take its positions so firmly that in the modern world no one can be considered socially adapted without knowledge of its fundamentals.
How did a natural number appear
Of course, the main object that arithmetic operates with is a natural number, such as 1, 2, 3, 4, ..., 152 ... etc. Arithmetic of natural numbers is the result of counting ordinary objects, such as cows in a meadow. Still, the definition of “many” or “few” once ceased to suit people, and it was necessary to invent more advanced counting techniques.
But a real breakthrough came when human thought reached the point that two kilograms, two bricks, and two details could be designated with the same number “two”. 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. So the arithmetic of numbers was born, which further developed and expanded, occupying ever greater positions in the life of society.
Such in-depth concepts of numbers as zero and negative numbers, fractions, designations of numbers by numbers and other methods have a rich and interesting history of development.
Arithmetic and practical Egyptians
Two of the oldest human companions in the study of the 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. So, the papyrus of Rind of Egyptian origin (named so because it belonged to the owner of the same name), dating from the 20th century. BC, in addition to other valuable data, contains the decomposition 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 meaning of such a complex decomposition? The fact is that the Egyptian approach did not tolerate abstract thoughts about numbers, on the contrary, the calculations were carried out only for practical purposes. That is, the Egyptian will be engaged in such a thing as calculations, solely in order to build a tomb, for example. It was necessary to calculate the length of the rib of the structure, and this forced the person to sit behind the papyrus. As you can see, the Egyptian progress in the calculations was caused, rather by massive construction, rather than a love of science.
For this reason, the calculations found on papyrus cannot be called reflections on the topic of fractions. Most likely, this is a practical preparation, which helped to solve problems with fractions in the future. The ancient Egyptians, who did not know the multiplication tables, performed rather long calculations, decomposed into many subtasks. Perhaps this is one of those subtasks. It is easy to notice that calculations with such blanks 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. Practice interested them no less, 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 with reality. Of course, you can multiply 10 cows and 100 liters of milk, but you can’t go far.
The deep-thinking Greeks left a significant mark in history, and their works reached us:
- Euclid and the Beginnings.
- Pythagoras.
- Archimedes.
- Eratosthenes.
- Zeno
- Anaxagoras.
And, of course, the Greeks, who turned everything into philosophy, and especially the successors of the Pythagoras work, were so passionate about numbers that they considered them to be the mystery of the harmony of the world. The numbers were so studied and researched that some of them and their pairs were assigned special properties. For instance:
- Perfect numbers are those that are equal to the sum of all its divisors, except for the number itself (6 = 1 + 2 + 3).
- Friendly numbers are those numbers, one of which is equal to the sum of all the 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 to be with it for the sake of profit, achieved great success by researching, playing and adding numbers. It should be noted that not all of their research found wide application, some of them remained only "for beauty".
Eastern thinkers of the Middle Ages
In the same way, in the Middle Ages, arithmetic owed its development to eastern contemporaries. The Indians gave us the numbers that we are actively using, such a thing as “zero”, and the positional version of the calculus, familiar to modern perception. From Al-Kasha, who worked in Samarkand in the 15th century, we inherited decimal fractions, without which it is difficult to imagine modern arithmetic.
In many ways, the acquaintance of Europe with the achievements of the East was made possible thanks to the work of the Italian scientist Leonardo Fibonacci, who wrote the work “The Book of the Abacus,” introducing oriental innovations. It has become 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 was rooted in Europe, began to spread to Russian lands. The first Russian arithmetic was published in 1703 - it was a book on the arithmetic of Leonty Magnitsky. For a long time, it remained the only teaching manual in mathematics. It contains the starting points of algebra and geometry. The numbers used in the examples in Russia’s first arithmetic textbook are Arabic. Although Arabic numerals were found earlier, in engravings dating back to the 17th century.
The book itself is decorated with images of Archimedes and Pythagoras, and on the first page is an image of arithmetic in the form of a woman. She sits on the throne, under it is written in Hebrew a word denoting the name of God, and on the steps that lead to the throne are the words "division", "multiplication", "addition", etc. You can only imagine what significance such truths, which are now considered commonplace.
A 600-page textbook describes both the basics, such as the addition and multiplication tables, and the applications to the navigation sciences.
It is not surprising that the author chose the 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, unenviable art ...". 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.
Complicated 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 compound. Examples of primes: 2, 3, 5, 7, 11, and all others that have no other divisors except the number 1 and itself.
As for the number 1, it is on a special account - there is an agreement that it must be considered neither simple nor compound. Simple at first glance, a simple number conceals many unsolved secrets within itself.
Euclid’s theorem says that there are an infinite number of primes, and Eratosthenes came up with a special arithmetic “sieve” that eliminates difficult numbers, leaving only simple ones.
Its essence is to emphasize the first uncrossed number, and subsequently cross out those that are multiples of it. Repeat this procedure many times - and get a table of primes.
The main theorem of arithmetic
Among the observations about primes, the main theorem of arithmetic must be mentioned in a special way.
The main theorem of arithmetic states that any integer greater than 1 is either prime or can be decomposed into the product of primes up to the order of the factors, and uniquely.
The basic theorem of arithmetic is proved quite cumbersome, and understanding it no longer looks like the simplest foundations.
At first glance, prime numbers are an elementary concept, but this is not so. Physics also once considered an atom to be elementary until it found the whole universe inside it. A beautiful story by the mathematician Don Zagir, “The First Fifty Million Primes,” is dedicated to primes.
From the “three apples” to deductive laws
What truly can be called the reinforced foundation of all science - these are the laws of arithmetic. Even in childhood, 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 then goes into 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 provides. The study of numbers is a "baby arithmetic" that 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 perhaps we do not know their exact wording.
The law of addition and multiplication
Any two natural numbers a and b can be expressed as the sum a + b, which will also be a natural number. Regarding addition, the following laws apply:
- Commutative , which says that from a rearrangement of the terms in places the sum does not change, or a + b = b + a.
- Associative , which says that the sum does not depend on the way the terms are grouped together, or a + (b + c) = (a + b) + c.
The rules of arithmetic, such as addition, are some of the elementary ones, 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 addition:
- a * b = b * a;
- a * (b * c) = (a * b) * c.
Interestingly, there is a law that combines addition and multiplication, also called distribution, or distribution law:
a (b + c) = ab + ac
This law actually teaches us to work with brackets, revealing them, thereby we can work with more complex formulas. These are precisely the laws that will lead us through the bizarre and difficult world of algebra.
Arithmetic Law
Human logic uses the law of order every day, checking clocks and counting bills. And, nevertheless, and it needs to be issued in the form of specific formulations.
If we have two positive integers a and b, then the following options are possible:
- a is b, or a = b;
- a is less than b, or a <b;
- 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 <b and b <c, then a <c.
There are also laws connecting order with the actions of multiplication and addition: if a <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 systems
We can say that numbers are a mathematical language, on whose convenience a lot depends. There are many calculus systems, which, like alphabets of different languages, differ from each other.
Consider the number system from the point of view of the influence of a position on the quantitative value of a digit at that position. So, for example, the Roman system is non-positional, where each number is encoded by a specific 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, a digit does not change its quantitative determination depending on what position it is in: first, second, etc. To get other numbers, you need to add the base ones. For instance:
The more familiar system for us using the Arabic numerals is positional. In such a system, the digit category determines the number of digits, for example, three-digit numbers: 333, 567, etc. The weight of any digit depends on the position at which one or another digit is located, for example, the digit 8 in the second position has a value of 80. This is typical for the decimal system, there are other positional systems, such as binary.
Binary arithmetic
We are familiar with the decimal system of calculus, consisting of single-digit numbers and multi-digit ones. The digit 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. A completely different logic and approach to the section, which is called "binary arithmetic." This is not surprising, because binary arithmetic was created not for human logic, but for computer. If the arithmetic of numbers came from the counting of objects, which was further abstracted from the properties of the object to the "naked" arithmetic, then this will not work with a computer. In order to be able to share his knowledge with computers, a person had to invent such a calculus model.
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 and decimal arithmetic is that the significance 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:
- The first digit on the left is 1 * 8 = 8, remembering that the fourth digit, which means it needs to be multiplied by 2, we get position 8.
- The second digit is 1 * 4 = 4 (position 4).
- The third digit is 0 * 2 = 0 (position 2).
- The fourth digit is 0 * 1 = 0 (position 1).
- So, our number is 1100 = 8 + 4 + 0 + 0 = 12.
That is, when switching to a new digit on the left, its significance in the binary system is multiplied by 2, and in the decimal - by 10. Such a system has one minus: this is too large an increase in the digits that are needed to write numbers. Examples of representing decimal numbers as doubles can be found in the following table.
Binary decimal numbers are shown below.
Octal and hexadecimal calculus systems are also used.
This mysterious arithmetic
What is arithmetic, double two, or unknown secrets of numbers? As you can see, arithmetic may seem simple at first glance, but its non-obvious lightness is deceptive. It can be studied for children along with Aunt Sova from the cartoon "Arithmetic-Baby", or you can immerse yourself in deep scientific research of almost a philosophical order. In history, she went from counting objects to worshiping the beauty of numbers. One thing is precisely known: with the establishment of the basic tenets of arithmetic, all science can rely on its strong shoulder.