Russian abacus has been faithfully serving people for more than five centuries, helping to perform simple arithmetic operations faster. It is convenient and quick to add income and subtract expenses from them. Techniques that simplify multiplication were not given to everyone and were often replaced by the usual addition, and division was the lot of the “chosen ones” and was performed much faster on paper.
Accounts, in principle, work only with positive numbers, and if there is a need to take into account the excess of expenses over income (losses), then the calculations are carried out modulo the number. The corresponding sign is remembered or written down on paper, and at the necessary moment is inserted into the number. When multiplying and dividing, a thread (wire, rod, rod) with 4 bones - the discharge separator (hereinafter referred to as RR) is not taken into account, even if you have to work with fractions (they are converted to integers, and after the calculations are completed, the reverse procedure is performed).
Russian abacus - history
So what is it? Russian abacus is the simplest mechanical device forperforming calculations. These are addition, subtraction, division and multiplication. There are two theories of the appearance of the account in Russia:
- Borrowing them from the Chinese through intermediaries in the person of the Tatar-Mongols in the XIV century AD. Just a century before the appearance of our "ancestors" wooden accounts in China, they acquired the final form of a counting device. True, they had 8, not ten categories and 7 bones, separated by a partition in the ratio of 5 and 2. But just let a Russian person improve something - the result of the improvement will differ from the source like heaven and earth.
- According to another theory, simple abacus is a truly Russian invention. They are just based on the decimal number system (in China at that time the quinary was adopted), which arose in the Muscovite state, including from the 16th century spread to the monetary sphere. There are documented references to a "board count" (16th century).
As it was in reality, history is silent. But the “board” count until the middle of the 17th century (until it won) competed with the European counting system on lined boards such as abacus, where it was done using pebbles or special tokens.
How to count?
Sample is an old wooden abacus. They have 12 transverse rods (PP separates 8 upper from 3 lower) with ten white knuckles, except for two black ones in the middle on 11 of them (4 knuckles on PP). Thus, Russian abacus can fix any number up to 10 million. And ifexclude PP, then up to 10 billion.
So, how to count on the accounts? Postponing numbers is done by moving the bones from the right to the left position, and when typing 10 bones on the left, they are removed to their original position. In the next discharge, only one bone is transferred to the left position. RR separates integers (from above) from their tenths, hundredths and thousandths, respectively, and does not take part in the calculations (previously used to account for "half", which was equal to ½ "money" or ¼ kopeck).
Accounting accounts
They became widespread in the 19th-20th centuries, until they were supplanted by EKVMs (electronic keyboard computers). By the way, adding machines, which counted much faster, could not do this, but working on them required special and rather complex training to master the skills of working on them, unlike counting, which was many times easier and faster to learn to work on.
Actually, the art of working on accounting accounts is to know all the ways to achieve the exact result of actions by decomposing the general into private, easier operations. For example, multiplying by 25 is replaced by multiplying by 100 and dividing the result by 2 in succession twice. Or, both multiplying and dividing by any power of 2 is done in successive corresponding operations, the number of which is equal to that power.
How to count on the accounts? Another example. Multiplication by a two-digit number from the same digits "AA"(11, 22, and so on) is replaced by multiplying by "A", moving the result one digit up (multiplying by 10) and adding this sum to the previous one. The speed of calculations, as well as the use of special techniques by him, depend on the experience and training of the person working on the accounts, the method of his training.
Addition
Account addition is the easiest operation. The first number is typed, then knuckles are added to it, denoting the third, and so on. Only one condition must be observed. If there are not enough bones to move them to the left row, that is how many bones must be left in this row, after which one bone should be moved to the left on the upper rod. The execution takes place from top to bottom (professionals can and vice versa) and only equal digits are added (ones with ones, tens with tens, and so on).
Subtraction
How is abacus subtracted? Remembering that abacus does not work with negative numbers, one must always keep in mind that the subtraction is made from a larger number. And if you need to do the opposite, then still the smaller is subtracted from the larger, and the sign is remembered or written down. Subtraction in Russian accounts is done from top to bottom, that is, from the highest to the lowest digits. On the corresponding wire, the required number of bones is discarded to the right, and if there are not enough of them, then one bone is transferred to the right in the highest digit, and on this wire everything is transferred to the left and the required number is removed from them to the right.
Multiplication
Now about multiplication on the abacus. ancient abacuscontribute to an increase in the speed of carrying out multiplication actions, which significantly exceeds the speed of carrying out the same actions on paper. In practice, multiplication is the repeated addition of the desired with itself in numerical terms. Some tips:
- It is better to take a larger number as a basis, then fewer operations will be performed. Multiplication starts from the lowest digit and goes up.
- A number is added to itself as many times as the number in this digit “means” (we will discuss ways to reduce the number of these operations at the end of this section). When moving to the next digit, the result is transferred one rod higher (multiplied by 10). And again the same procedure. If there is a “0” in the discharge, then the transfer to the senior rod occurs, but the addition does not, and it is necessary to proceed to the further multiplication procedure.
- Fractional numbers are multiplied as integers, and the corresponding separator is put as a result of all manual actions on paper.
Methods that simplify the process of multiplication:
- On 4 - double doubling.
- By 5 - move one digit higher and divide the result by 2.
- For 6 - multiply by 5 plus the initial number.
- On 7 - triple doubling and minus the initial number.
- On 9 - transfer one digit higher and minus the initial number.
Division
As multiplication is replaced by repeated addition, so division on accounts is a constant subtraction. It all starts at the top and goes down. Move number to the rightpits equal to the divisor (each time it succeeds on the topmost wire, one tile is moved to the left) until there are pits on the left less than the number by which the division is made (divisor).
Then the next bit is connected to the process. And if bones remain in the previous wire, then the divisor is already subtracted from a two-digit number. If not, then as before. If in the lowest category the subtraction passes without the remainder of the stones on the left, then the division is made without a remainder. If there are bones left on the left, then in case of optional receipt of a fractional number as a result, the remainder is ignored, and if it is mandatory to receive it, then the subtraction continues to the required accuracy on the bars below the PP, indicating the fractional separator on paper. Similarly, division into two-digit, three-digit (etc.) numbers is performed, only first the subtraction comes from respectively two, three and so on higher digits.
How to simplify division?
Methods that make dividing easier:
- On 2 - the process proceeds in the reverse order - from bottom to top. On each rod, half of the bones are discarded, and the “extra”, with an odd number of them, is also discarded. In the lower category, 5 bones are transferred to the left for this.
- By 4 - double division by 2.
- By 5 - transferring the whole number down one rod (dividing by 10) and multiplying it by 2.
- By 8 - three times division by 2.
- On 9 - transfer one digit higher and minus the initial number.
Improvement
During a quarter of a millennium of popularity and practical necessity, abacus has been repeatedly attempted (often successfully) to improve Russian abacus. Let's focus on just one of them. In 1828, Major General F. M. Svobodsky presented to the appropriate authority a counting device that not only carried out the usual Russian counting actions, but rather quickly extracted cube roots, raised numbers to a power, calculated compound interest, and so on. This was achieved only by the methods of addition and subtraction with the fixation of intermediate results on a special account field. However, the commission was so impressed by the speed of obtaining the desired result that it recommended this device for production and the introduction of a special course in military institutions. But the matter did not reach the real implementation of the decision.
Currently in Russia, abacus is used only as a museum exhibit or family heirloom. Very rarely, if someone has them in the house, they can be used by the younger generation for rolling on the floor, or by the older ones for foot or back massage. But in vain! In modern China, "Suanpan" is taught to elementary school students, as it is believed that a child who has mastered this method of counting develops better and faster, who has not learned to work on this ancient device.
