How much does the abacus know?

This article was published in the 9th issue of King of Mathematics 20 19.

In our study and life, we often encounter complex math problems. At this time, we can get the answer quickly with the help of electronic calculator. But in ancient times without calculators, what should our ancestors do? Don't worry, they also have an artifact-abacus.

The abacus looks simple, with strings of wooden beads [1] arranged side by side in a rectangular wooden frame, much like a box of candied haws. The middle beam divides each bead into two parts: the upper two and the lower five. The lower bead stands for 1 and the upper bead stands for 5.

When different numbers of beads are attached to the beams facing each other across the bank, different numbers are formed. For example, 9 comes from this:

A string of abacus corresponds to a number. From right to left, it is a number, ten numbers and a hundred numbers. ...

Of course, if you want to use decimals, the rightmost string can also be reserved for decimals, percentiles and thousandths. ...

We might as well simplify it and only discuss integers, so 20 19 looks like this:

Smart you may have noticed that 0~9 only needs 1 on top and 4 on the bottom. Why is there an extra bead up and down? It turns out that the ancients used hexadecimal [2] when calculating the weight. We often say "half a catty is eight taels", eight taels equals half a catty, and sixteen taels equals one catty. The upper two and the lower five can just represent 0~ 15.

Not only that, in the process of doing some decimal calculations, these two "extra" beads will also be used temporarily, such as borrowing and carrying.

Then why do you think of separating two beads to represent 5 instead of directly stringing 15 on each string? Because our hands have five fingers. In daily life, we handle numbers below 5 with one hand and numbers above 5 and below 10 with two hands. As soon as your little friend reaches out, you can instantly know how much his new toy costs. Our brains are used to counting in groups of five, so it is very clever to let the beads on it represent five.

Although the abacus is simple in structure, it is very clever to use. The clever ancients came up with many catchy formulas. The idiom "three times five divided by two" is actually an addition formula in abacus calculation:

For example, calculate:

In addition to the most basic addition, subtraction, multiplication and division, abacus can do more complicated mathematical calculations. In the hands of the older generation of scientists, it even made great contributions to the development of China's atomic bomb!

Abacus calculation was also listed in the UNESCO Intangible Cultural Heritage List in 20 13, and it is known as the fifth invention in China!

In fact, abacus was not born as lean as it is now.

A long time ago, in ancient times, people thought of using pebbles to represent the number of prey, 1 pebble represents 1 prey. The population of the tribe and the fruits picked can be represented by such piles of pebbles.

With the evolution of human civilization, people are no longer satisfied with counting stones, but put stones on boards, draw horizontal and vertical lines on the boards, use the position of stones on the lines to represent numbers, and sometimes use stones of different colors to represent different numbers. When calculating, I moved the stone around. If I didn't know, I thought it was chess. [4]

This "abacus" has one drawback. If you accidentally touch it, stones can easily run around, so people dig grooves in the board, round the stones and put them in the grooves. Stones can only move in their own ditches, and no matter how restless they are, they can't jump into the ditch next door.

However, this abacus is still not portable enough, so you should carefully put away the beads every time you use it. The clever ancients thought of stringing them together and fixing them on the board one by one. This is the final appearance of the abacus.

Actually, abacus is not unique to China. Other countries and regions have similar inventions, but none of them lived as long as China. Looking at these abacus calculations full of "exotic customs", it seems to witness the evolutionary history of abacus calculations.

Two thousand five hundred years ago, the Greeks carved lines with special meanings on marble slabs, put pebbles on slate, and constantly moved their positions during calculation. This is the earliest appearance of abacus.

Two thousand years ago, the Romans dug a groove in a copper plate, put small copper beads in the groove, and pushed the copper beads up and down as if playing marbles when calculating.

On the distribution of beads, the Romans and China thought of going together. They put 65,438+0 beads representing 5 in each ditch on the upper side and 4 beads representing 65,438+0 in each ditch on the lower side, and each bead can represent 0~9.

The Inca is an ancient and mysterious empire, and their abacus is very strange, like a reduced terrace. Historians have not figured out how to use it until today.

The Russian abacus looks very similar to China's abacus, except that the Russians did not think of separating one or two beads to represent five, but "foolishly" strung each string of 10 without a beam in the middle. When using it, it is actually placed vertically, and the ball is dialed from right to left. The beaded stick is deliberately bent, with a slight bulge in the middle to ensure that the beads stay on the left and right sides.

The abacus was introduced into Japan from China in the 4th century/kloc. Japanese abacus is essentially China's abacus, but they seldom use those two extra beads, so they often delete them directly.