For more than five centuries, Russian accounts have faithfully served people, helping them to complete simple arithmetic operations faster. Conveniently and quickly add income and subtract expenses from them. Techniques that simplified multiplication were not given to everyone and were often replaced by habitual addition, and division was the destiny of the “chosen ones” and was much faster performed on paper.
The 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 numbers. The corresponding character is remembered or recorded on paper, and at the right time is inserted into the number. In the process of multiplication and division, a thread (wire, rod, rod) with 4 bones - the separator of digits (hereinafter referred to as PP) is not taken into account, even if you have to work with fractions (they are converted to integers, and after the end of the calculations, the reverse procedure is performed) .
Russian scores - history
So what is it? Russian accounts are the simplest mechanical device for performing calculations. This is 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 the “ancestors” of wooden accounts in China, they acquired the final form of a counting device. True, they had 8, not ten digits and 7 stones, separated by a partition in the ratio of 5 and 2. But just give the Russian person something to improve - the result of the improvement will differ from the source like heaven and earth.
- According to another theory, simple scores are a true Russian invention. They are precisely based on the decimal number system (five was adopted in China at that time), which arose in the Moscow state, including from the sixteenth century it extended to the monetary sphere. There are documented references to the "board account" (XVI century).
As it really was, history is silent. But the “board” account until the middle of the 17th century (until it won) competed with the European system of counting on lined boards like abacus, where it happened with the help of pebbles or special tokens.
How to count?
The pattern is old wooden abacus. They have 12 transverse rods-wires (PP separates 8 upper from 3 lower) with ten white knuckles, except for two black in the middle on 11 of them (on the PP - 4 knuckles). Thus, Russian accounts can record any number up to 10 million. And if you exclude PP, then up to 10 billion.
So, how to count on the accounts? Numbers are postponed by moving the knuckles from the right to the left position, and when you set 10 pits on the left, they are removed to their original position. In the next discharge, only one bone is transferred to the left position. PP separates integers (above) from their tenths, hundredths and thousandths, respectively, and does not take part in the calculations (previously used to account for “half money”, which was equal to ½ “money” or ¼ penny).
Accounting Accounts
They became widespread in the 19th-20th centuries, until they were replaced by computers (electronic keyboard computers). By the way, this could not be done by arithmometers, which were considered much faster, but working on them required special and rather complicated training in mastering the skills of working on them, in contrast to counting, it was much easier and faster to teach them to work.
In fact, the art of working on accounting accounts consists in knowing all the ways to achieve an accurate result of actions by decomposing the general into private lighter transactions. For example, multiplying by 25 is replaced by multiplying by 100 and dividing the result by 2 consecutively. Or, both multiplying and dividing by any power of 2 is performed by the corresponding sequential action, the number of which is equal to this power.
How to count on the accounts? Another example. Multiplication by a two-digit number of the same digits “AA” (11, 22, and so on) is replaced by multiplication by “A” with the result transferred to the digit up (multiplication by 10) and adding this sum to the previous one. From the experience and training of the person working on the accounts, the method of his training depends on the speed of calculations, as well as his application of special techniques.
Addition
Adding accounts is the easiest operation. The first number is dialed, 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 - exactly so many bones must be left in this row, after which one knuckle should be moved to the left on the upper bar. Execution occurs from top to bottom (professionals can and vice versa) and only equal digits are added (units with units, tens with tens, and so on).
Subtraction
How is deduction on accounts performed? Remembering that accounts do not work with negative numbers, you should always keep in mind that subtraction is made from a larger number. But if you need to do the opposite, then the less is subtracted from the larger, and the character is remembered or recorded. Subtraction on Russian accounts is made from top to bottom, that is, from higher ranks to lower ranks. On the corresponding wire, the required number of seeds is discarded to the right, and if there are not enough, one bone is transferred to the right in the highest order, and on this wire everything is transferred to the left and the required number is removed from them.
Multiplication
Now about the multiplication on the accounts. Ancient accounts help to increase the speed of carrying out multiplication actions, which is much faster than the speed of carrying out the same actions on paper. In practice, multiplication is the multiple addition of the desired with itself in numerical terms. A few tips:
- It is better to take a larger number as a basis, then fewer operations will be performed. Multiplication starts from the lowest level and goes up.
- A number is added up by itself as many times as the number in this category “means” (we will talk about ways to reduce the number of these operations at the end of this section). When moving to the next category, the result is transferred one rod higher (multiplied by 10). And again the same procedure. If the discharge is “0”, 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 set as a result of all actions manually on paper.
Methods that simplify the process of multiplication:
- At 4 - double doubling.
- By 5 - transfer one rank higher and dividing the result by 2.
- By 6 - multiplication by 5 plus the initial number.
- At 7 - threefold doubling and minus the initial number.
- At 9, the transfer is one digit higher and minus the initial number.
Division
Just as multiplication is replaced by multiple additions, so division on accounts is a constant subtraction. It all starts with the upper discharge and goes down. The number of seeds equal to the divisor is transferred to the right (each time it succeeds on the topmost wire, one bone is transferred to the left) until there are less seeds on the left than the number by which the division (divisor) is performed.
Then the following discharge is connected to the process. And if there were stones in the previous wire, then the divisor is already subtracted from the double-digit number. If not, then, as in the previous time. If in the lower category the subtraction takes place without the remainder of the seeds on the left, then the division is done without a remainder. If the bones remain on the left, then in the case of optional receipt of a fractional number as a result, the remainder is ignored, and if it is necessary to obtain it, the subtraction continues to the desired accuracy on the rods below PP, indicating the fractional separator on paper. Similarly, division into two-digit, three-digit (etc.) numbers is performed, only the first subtraction is from the two, three, and so on higher digits, respectively.
How to simplify the division?
Methods that simplify the division process:
- At 2 - the process proceeds in the reverse order - from bottom to top. On each rod, half the seeds are discarded, and the "extra", with their odd number, is also discarded. In the lower category, 5 stones are transferred to the left for this.
- At 4 - twice the division by 2.
- By 5 - transfer the entire number by one rod down (dividing by 10) and multiplying it by 2.
- At 8 - three times the division by 2.
- At 9, the transfer is one digit higher and minus the initial number.
Improvement
Over a quarter of a millennium of popularity and practical necessity, the account has been repeatedly undertaken (often ending successfully), attempts to improve Russian scores. Let us dwell only on 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 operations for Russians, but quickly extracted cubic roots, raised numbers to a power, calculated compound percentages, and so on. This was achieved only by the methods of addition and subtraction with fixing intermediate results in a special field account. However, the commission was so struck by the speed of obtaining the desired result that it recommended this device for the production and introduction of a special course in military institutions. But the matter did not come to the real execution of the decision.
At present, in Russia, accounts are used only as a museum exhibit or as a heirloom. Very rarely, if they are present in someone in the house, they can be used by the younger generation to ride on the floor, or older to massage their legs or back. But in vain! In modern China, students of the lower grades are taught at Xuanpan, since it is believed that a child who has mastered this method of counting develops better and faster, who has not learned how to work on this ancient device.