Carry-free addition

Add five straight, ten straight, five straight, and ten straight.

Plus one: one up, one down, five to four, one up, nine into one.

Plus two: two up two down five to three, two to eighty-one.

Plus three: three against three, three against five against two, three against 71%.

Plus four: four up four down five to one, four to six to one.

Plus five: five up five, five down five into one.

Plus six: six against six, six against forty-one, six against one against fifty-one.

Plus seven: seven to seven, seven to three, seven to two to five.

Plus eight: eight goes up to eight, eight goes to two and enters one, and eight goes up to three and enters five.

Plus nine: nine on nine, nine into one, nine on four into five.

Subtraction formula

Reduce the number of declining positions without resigning

Directly subtract, break five, demote, subtract ten and make up five.

Minus one: down one, up four to five, down one to nine.

Subtract two: two down two, two up three times five, two back to eighteen.

Minus three: three times three, three times two to five, three times one to seven.

Minus four: four times four, four times one to five, four times one to six.

Minus five: five times five, five times one, five times.

Negative six: six down six, six back to four, six back to five to one.

Minus seven: seven down seven, seven back one to three, seven back one to five to two.

Minus eight: eight times eight, eight times one to two, eight times one to five to three.

Minus nine: nine to nine, nine to one, nine to five to four.

increase

It was used in planning during the Spring and Autumn Period and the Warring States Period.

See Yang Hui's multiplication and division transformation and treasure calculation [1274].

Zhu Shijie's Enlightenment of Arithmetic [1299] is basically consistent with the modern formula.

With these four formulas, the abacus algorithm has formed a long-standing system.

Third, the formula table

1 10 1 1202 1303 1404 1505 1606 1707 1808 1909

2 1 02 22 04 23 06 24 08 25 10 26 12 27 14 28 16 29 18

3 103 3206 3309 34 12 35 15 3 18 372 1 3824 3927

Four one 04 four two 08 four three 12 four four 16 four five 20 four six 24 four seven 28 four eight 32 four nine 36

5 105 52 10 53 15 5420 5525 5630 5735 5840 5945

6 106 62 12 63 18 6424 6530 6636 6742 6848 6954

7 107 72 14 732 1 7428 7535 7642 7749 7856 7963

8 108 82 16 8324 8432 8540 8648 8756 8864 8972

9 1 09 92 18 93 27 94 36 95 45 96 54 97 63 98 72 99 8 1

Division formula

There are two kinds of abacus division: subtraction division and quotient division.

Division is calculated by formulas, including nine formulas, regression formula and quotient formula.

Alcoholic formula ***6 1 sentence:

One time (divided by 1): one time, two times, three times, four times, five times, six times, seven times, eight times and nine times.

Two returns (divided by two): every two into one, every four into two, every six into three, every eight into four, 2 1 plus five.

Three returns (divided by three): three into one, six into two, nine into three, three into one, three into two.

Four returns (divided by four): every four enters one, every eight enters two, four two adds five, four twelve is more than two, and four thirty-seven is more than two.

Five cycles (divided by five): every five cycles into one, 5 1 cycle is two, 52 cycles is four, 53 cycles is six, and 54 cycles is eight.

Six returns (divided by six): one every six, two every twelve, six three plus five, six one plus four, six two three left two, six four left four, six five eight left two.

Seven returns (divided by seven): every seven returns to one, every fourteen to two, every seven to three, every seven to six, every seven to four, every seven to five, every seven to five and every seven to four.

Eight returns (divided by eight): every eight enters one, 84 adds five, 8 1 adds two, 82 adds four, 83 adds six, 856 is greater than two, 867 is greater than four, and 878 is greater than six.

Nine returns (divided by nine): every nine returns to one, nine returns to one, ninety-two returns to two, ninety-three returns to three, ninety-four returns to four, ninety-five returns to five, ninety-six returns to six, ninety-seven returns to seven and ninety-eight returns to eight.

Zhu Shijie's "Enlightenment of Arithmetic" (1299) talks about "the formula of regression and partition". ...

I saw it as soon as I got home and entered the door.

Two, one, five, two, fourteen, twenty-six, thirty-eight and forty.

Three, one, three, one, three, two, six, twenty-nine and thirty every three.

Four, one, two, two, four, two, four, three, seven, twelve, four, eighteen, twenty.

Five times the return, five into ten.

Six times one plus four six two three one two six three adds up to five six four six four six four six six five eight two.

Seven times one plus three times two times plus six seven three four two seven four five seven five seven seven one seven six eight four times seventy percent.

Eight, one plus two, eight, two, four, eight, three, six, eight and four are increased to five, eight, five, six, twelve, eight, seventy, four, eight, seven and eighty-six.

Nine is up to you, nine into ten.

In his Daily Algorithm (1262), Yang Hui, a mathematician in the Southern Song Dynasty, compiled a Song formula for finding the price of two pounds per catty.

Zhu Shijie, a great mathematician in Yuan Dynasty, wrote "Arithmetic Enlightenment" (1299), which was further promoted to the following fifteen sentences:

One begging, every 625; ( 1/ 16=0.0625)

Second, ask for abdication125; (2/ 16=0. 125)

Three requests,1875; (3/ 16=0. 1875)

Four requests were changed to twenty-five; (4/ 16=0.25)

Five, three, one, two, five; (5/ 16=0.3 125)

Six demands, two prices, three seven five; (6/ 16=0.375)

Seven, four, three, seven, five; (7/ 16=0.4375)

Eight seeking, turning into five; (8/ 16=0.5)

Nine, five, six, two, five; (9/ 16=0.5625)

Ten requests, six two five; ( 10/ 16=0.625)

1 1, 6875; ( 1 1/ 16=0.6875)

12, 75; ( 12/ 16=0.75)

13, 8 125; ( 13/ 16=0.8 125)

14, 875; ( 14/ 16=0.875)

15, 9375; ( 15/ 16=0.9375)

Back to business formula ***9 sentences:

Only refund, only refund, two, three,

Did not return 4, did not return 5, did not return 6,

No return 7, no return 8, no return 9,

Quotient number nine (also called bump back, it is the number divided by the beginning of 9, and it will be used to pay back when the business is big) formula ***9 sentences:

See no score 9 1, see no score 92, see no score 93,

See four division is 94, see five division is 95, and see six division is 96.

Don't divide 97 by seven, 98 by eight, and 99 by nine.

Division with a divisor of single digits is called "single return"; Division with two or more divisors is called division, the first divisor is called division, and the following digits are called division. For example, if the divisor is 534, it is called "five divided by three divided by four". That is, after the quotient is obtained by the formula of five to four, it is divided by 34.

Open formula

The abacus power can be directly multiplied or calculated according to a formula. If the higher power is mass, then there is only direct multiplication. If the factor can be decomposed, it can be decomposed and then calculated.

Generally, there are semi-99 Kaiping method, product difference Kaiping method, formula Kaiping method and multiplication Kaiping method.

The issuer, there are three methods: open method and open method.

There are many ways to open the fifth power, and it is common to multiply the fifth power.

Generally speaking, it is rarely used in abacus calculation.