Ancient method of counting chips

Counting chips are an ancient Chinese program for decimal calculations using counting chips.

A counting chip is also known as a chip, a cue, a calculator, etc. It is initially a small bamboo stick or the like, archaeological findings, the ancient counting chips are the same length and thickness of the small stick, generally 13 ~ 14 cm long, but also wood, animal bone, ivory, metal and other materials, two hundred dozens of pieces for a bunch, in a cloth bag to carry around.

According to canonical records and archaeological discoveries, at least in the early years of the Warring States period chips have appeared. It uses the decimal system of counting invented by the Shang Dynasty in China, which can easily carry out the four operations as well as multiplication, square and other more complex operations, and can be zero, negative numbers and fractions to make representations and calculations. From the Warring States period until the Ming Dynasty was replaced by the bead counting, chip counting is the ancient Chinese method of daily calculations, counting chips are commonly used by ancient Chinese mathematicians in the study of mathematics, is the basis of a variety of important mathematical inventions in ancient China, created the ancient Chinese mathematical system centered on calculations, and the ancient Greek mathematical system centered on logical reasoning is different; calculation-centered mathematical systems are The calculation-centered mathematical system has been the mainstream of world mathematics for more than a thousand years

Chips were introduced from China to the Korean Peninsula and Japan in the 6th century AD. The method of multiplication and division was introduced to India, and became the Tupan algorithm. from the beginning of the 9th century to the 10th century, Arab mathematical works, such as the Principles of Indian Arithmetic, used Arabic numerals to represent the Tupan algorithm, but the concept of the decimal system, the representation of fractions, as well as the calculation method of the four operations of addition, subtraction, multiplication, and division were the same as those of the Chinese algorithm, and some of them also used the space " "Some scholars believe that the ancient Chinese system of calculations was exactly the same as that of Chinese chips. Some scholars believe that China's ancient calculations, through the Silk Road into India, Arabia, contributing to the Indian - Arab numerical system.

Numerical Representation

The chip counting number system is the only system in the world that uses only a combination of direction and position of a symbol to represent any decimal number or fraction. Unit numbers: the chip sticks are arranged vertically one stick represents 1, two sticks represent 2, and five sticks represent 5 as shown on the picture. However, the representation of the numbers from 6 to 9, instead of 6 to 9 sticks side by side, the same bit of pentadecimal, that is, a stick of chips to represent the digital 5, placed horizontally on top of the chips from 1 to 4. This already implies the prototype of the abacus.

Avoid confusion by using a straight horizontal arrangement

Numbers larger than 9 are represented in decimal, with a chip placed to the left of the single digit, representing ten times that number, and to the left of the ten-digit number, representing the hundredth digit, and so on. 231 is represented by placing a chip in the single digit, which represents 1, and in the ten-digit position by placing a chip in the three-digit number, representing 30, and in the hundredth position by placing a chip in the two-digit number, representing 200, and the total number, which is two hundred. 2, representing 200, the total number is two hundred and thirty-one (231).

The chip board is usually a tabletop or floor, usually without a grid. If chips 2, 3, 1 arranged side by side, it is possible to be misread as 51 or 24; in order to avoid misreading of the neighboring bits, the invention of alternating vertical code horizontal code every other bit, that is, the individual bit of vertical code, the ten bit with horizontal code, the hundred bit with vertical code, the thousand bit with horizontal code, so on and so forth, you can avoid the misinterpretation altogether.

Representation of zero[

Numbers followed by a diagonal stick to indicate a negative number

China has had "0" since the beginning of the calculation, i.e., the empty space indicates "0". Zero in the calculation is the position of zero and the result of zero, there is no specific symbol, which is different from the Arabic numerals have a symbol 0, Arabic numerals 0 is only the symbol of zero, not the result of the operation.

Positive and negative numbers

In the Song Dynasty, red chips were used to represent positive numbers, black chips were used to represent negative numbers, and black chips were used across the board, but the last digit of the number was marked as negative by adding a diagonal stick.

Decimals

Sun Zi arithmetic metrics have been decimalized concepts, such as feet, inches, minutes, centimeters, millimeters, silk, hu. Seven zhang, one foot, two inches, three minutes, four centimeters, five millimeters, and six silks is 71.23456 feet in modern representation, and the arithmetic chip arrangement - Image search Wikipedia

Addition

The arithmetic chip itself already contains addition, and the arithmetic is very convenient and fast. The biggest difference with Arabic numeral addition is that the counting chips themselves are additive, and only the sticks have to be moved mechanically. In contrast, Arabic numerals, such as 1 and 2, cannot be mechanically stacked to form the word 3.

Subtraction Multiplication Division Four operations with fractions Greatest common divisor Square root Open square root Higher equations Quadratic higher equations

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