Finger speed algorithm

Chapter 3 Addition and Subtraction (1)
Addition and subtraction refers to the basis of multiplication and division. A large number of calculations in actual work are generally addition and subtraction.Therefore, it is of great significance to learn the addition and subtraction of counting well to master the skill of counting.

There is a law of inverse operation between addition and subtraction.In daily computing work, addition is often used to check whether the subtraction that has been done is correct, and subtraction is used to check whether the addition that has been done is correct.At the same time, in counting, the operation of addition often contains the operation of subtraction; in the operation of subtraction, the operation of addition often contains.Therefore, when learning to count, addition and subtraction should be explained and practiced together.

For the convenience of narration and study in the future, we will define, explain and explain the relevant terms to be used in counting by referring to the terminology of abacus calculation.

The "dial bead" in abacus calculation is called "pinching finger" in finger calculation.For example, "dial out 5" is called "pinch out 5"; "dial up 3" is called "pinch up 3".The "beads" in abacus calculations are called "calculation points" in the calculation, and the "calculation points" are the contacts between the fingertips of the counting finger and the middle finger at a certain position. "Beads" means that the counting point of these numbers is called "lower point"; the counting bead representing 1 is called "upper bead" and represents the counting point of 2 (when the counting finger is up to represent 3, the tip of each counting finger does not touch the middle finger, and there is no Counting points, here is just a rule) called "up point".Beads with beams to represent numbers are called “inner beads”; beads without beams and without numbers are called “outer beads”.The calculation point that touches the middle finger to represent the number is called "inside point", and the calculation point that has not yet expressed the number is called "outside point".For example, when we express the number 4 on the counting finger, the dots on the first line, the second line, and the third line on the middle finger, that is, the dots used to represent the numbers 5, 5, and 5 are the "inside points". The counting point used to represent the numbers 3 and 1 is the "outside point"; The calculation point representing the numbers 2 and 3 is exactly the "external point".

(Section [-]) Basic addition and subtraction of arithmetic

Basic arithmetic addition and subtraction refers to the addition and subtraction between single digits (numbers), including direct addition, direct subtraction, full 5 force, broken 5 subtraction, carry addition, abdication subtraction, break 5 carry addition, abdication A total of eight types are deducted from 5.

[-]. Direct addition and direct subtraction
1. Direct addition

The direct addition category refers to the calculation problem of directly adding a few pinches to the counting finger on the basis of the original calculation point.Example 1, 1+2=3.

Features: The outer point is enough to add, directly pinch the number to be added from the outer point.There are 35 situations in the direct addition category:
0+1，0+2，0+3，0+4，0+5，0+6，0+70+80+9。

1+1，1+2，1+3，1+5，1+6，1+7，1+8。

2+1，2+2，2+5，2+6，2+7。

3+1，3+5，3+6。

4+5.

5+1，5+2，5+3，5+4。

6+1，6+2，6+3。

7+1，7+2，8+1。

Direct addition calculations can also be performed by formulas, and there are nine sentences in the formulas:

One on one, two on two, three on three, four on four, five on five, six on six, seven on seven, eight on eight, nine on nine.

Direct addition formulas are abbreviated as "几上几", which can be used to directly add two numbers at this finger position. The first word of each formula indicates the number to be added, and the following words indicate the action of pinching fingers and the number to be added. Count the numbers represented by the dots.

2. Direct reduction

The direct subtraction category refers to calculation problems that directly pinch off the upper point, the lower point, or the upper and lower points.

Features: The inner point is enough to subtract, directly pinch the number to be subtracted from the inner point.There are 35 situations in the direct reduction category:
1—1，2—1，3—1，4—1，6—1，7—18—19—1。

2—2，3—2，4—2，7—2，8—2，9—2。

3—3，4—3，8—3，9—3。

4-49-4.

5—5，6—5，7—5，8—5。

6—6，7—6，8—6，9—6。

7—7，8—7，9—7。

8-8, 9-8.

9—9.

Direct subtraction calculations are calculated with formulas, and the formulas have nine sentences:
One goes to one, two goes to two, three goes to three, four goes to four, five goes to five, six goes to six, seven goes to seven, eight goes to eight, nine goes to nine.

The direct reduction formula is simply written as "a few to a few".Two numbers can be directly subtracted when used in this finger position. The first word of each formula indicates the number to be subtracted, and the subsequent words indicate the action of pinching the fingers and the number represented by the pinched point.

5. Full 5 plus, broken [-] minus
1. Make up the number
The sum of two numbers is 5, and the two numbers are called make up numbers.

The original number, 1, 2, 3, 4.

Make up the number, 4, 3, 2, 1.

2. Full 5 plus classes
Full 5 plus category refers to pinch a upper point, and pinch out one or several lower points when a word is used.Features: If the lower point is not enough, add the upper point, pinch up 5, and pinch off the addend to make up the number.The two numbers to be added are both less than 5, and the sum is not less than 5, but less than 10.

There are 5 situations for full 10 plus categories:
4+1，4+2，4+3，4+4，3+2，3+3，3+4，2+3，2+4，1+4。

There are four sentences in total for 5 plus calculation questions:
One go five go four two go five go three.The first number refers to the addition to be added, and "lower five" refers to pinching the upper and upper points (in abacus calculation, it means 5, and the next upper bead is dialed, which is called "lower five". In counting, it means 5, It is necessary to pinch the upper and upper points, which should be "upper five", but customarily, we still use the same formula as the abacus calculation, called "lower five"), "go" refers to pinch down to pinch up 5, and pinch off 4 Make up the number 1 (four to five to one)

Features: When the lower point of the subtrahend is not enough to subtract the subtrahend, that is to pinch the sum of the subtrahend, and then pinch out 5.

There are 5 cases of broken 10 reduction categories:
5—1，5—2，5—3，5—4。

6—2，6—3，6—4。

7-3, 7-4.

8—4.

When this kind of calculation problem is calculated by formula, the formula has four sentences:

One up four go five two go three go five three go two go five four go one go five formula is simply recorded as "a few go up a few go five".The first number in the formula refers to the subtrahend to be subtracted,
点，末了的数指掐去的算点所代表的数。例1、2+4=2+（5—1）=6，四下五去一。

3. Break 5 and reduce the category
Breaking 5 subtraction refers to the calculation problem of pinching the upper and lower points and pinching off the upper point.

“上几”指掐上的下点个数，“去五”指掐去上点。例27—4=（7+1）—5=3，四上一去五。

Pinch off 5, pinch 4 to make up the number 1 (four on one to five)

[-]. Direct carry plus, direct abdicate subtraction

1's complement
The sum of two numbers is 10, and these two numbers are complements of each other.

Original numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Complement, 9, 8, 7, 6, 5, 4, 3, 2, 1.

2. Direct carry plus class
Direct carry and addition refers to the arithmetic problem of removing the upper point or lower point, or removing the upper and lower points at the same time on the ones counting finger, and then carrying forward 1 counting finger (digit).

Features: Adding a number means removing the complement of the number, and then counting forward by 1.There are 35 cases of direct carry addition:
9+1.

8+2, 9+2.

7+3，8+3，9+3。

6+4，7+4，8+4，9+4。

5+5，6+5，7+5，8+5。

4+6, 9+6.

3+7，4+7，8+7，9+7。

2+8，3+8，4+8，7+8，8+8，9+8。

1+9，2+9，3+9，4+9，6+9，7+9，8+9，9+9。

Direct carry and addition calculations are performed with formulas, and there are nine sentences in the formulas:

一去九进一二去八进一三去七进一四去六进一五去五进一六去四进一七去三进一八去二进一九去一进一此类口诀可简记为“几去几进一”。第一个数指要加的加数，“去几”指掐去内点的个数，“进一”指在前一算指（位）上掐入一个下点。例1、3+8=（3—2）+10=11，八去二进一。

Pinch off 8's complement 2-way then one 1 counts into one (eight removes two into one)

例2、9+5=（9—5）+10=14，五去五进一。

Pinch off the complement of 5, count 5 forward and add one (eight to five into one)

3. Direct abdication and deduction

Direct abdication and subtraction refers to the calculation problem that when the original finger is not enough to subtract, the previous finger is retreated by one, and then the original finger is added directly.

Features: The ones digit is not enough to subtract, the tens digit is backed up by one (pinch off one lower point), and the ones digit is added with the complement of the subtrahend.

There are 35 cases of direct abdication and reduction:
10—1.

10-2, 11-2.

10—3，11—3，%12—3。

10—4，11—4，12—4，13—4。

10—5，11—5，12—5，13—5，14—5。

10-6, 15-6.

10—7，11—7，15—7，16—7。

10—8，11—8，12—8，15—8，16—8，17—8。

10—9，11—9，12—9，13—9，15—9，16—9。

17-918-9.

Direct abdication and reduction calculations are performed with formulas, and there are nine sentences in the formulas:

One return one return nine two return one return eight three return one return seven four return one return six five return one return five six return one return four seven return one return three eight return one return two nine return such formulas can be shortened as " How much is returned and how much is returned."The first number refers to the subtrahend to be subtracted; "return one" means that the current index is not enough to subtract, and borrow 1 from the previous index as 10; After that, add the subtracted number to the number on the index (complement of the subtrahend).

例3、11一8=（11—10）+2=3，八退一还二个位不够减，十位掐去1，个位掐入8的补数。

2. (Eight back one for two)

例4、13—5=（13—10）+5=8，五退一还五。

The above examples are operations on the front and back fingers.If the previous finger or two fingers are all zero, it is necessary to operate on three or four fingers before and after. At this time, the formula can be deduced according to mathematical principles.For example: 104-8=96, use the formula of "eight returns, one return nine two"; another example: 1007-9=998, use the formula "nine returns, one return nine nine one".

Four, broken 5 carry addition, abdication full 5 subtraction
1. Breaking the 5-carry addition class
Breaking the 5-carry addition category, refers to the math problem of pinching the ones digit into the lower point, pinching out the upper point, and carrying 1 forward.

Features: Add a number, that is, pinch the complement of this number to make up the number, and then pinch 5 and carry 1 to the previous finger.

There are 5 cases of broken 10-carry addition:
5+6，6+6，7+6，8+6，5+7，6+7，7+7，5+8，6+8，5+9。

The arithmetic of breaking 5-carry addition is performed with formulas, and there are four sentences in the formulas:

Six to one goes to five to one, seven to two to five to one, eight to two to five to one, nine to four to five to one.

This kind of formula can be shortened as "several up, few down, five into one".The first number refers to the number to be added, and the "upper number" is the number that should be added after "removing five into one" (the make-up of the complement of the number to be added).

当被加数不小于5而小于10，其加数大于5而小于10，两数和大于10而小于15时，适用于破5进位加类口诀。

例1、6+7=（6+2）—5+10=13，七上去五进一。

2. Abdication full 5 reductions
The deduction category of abdicating a full 5 means that the ones digit is not enough to subtract, the tens digit is reduced by 1, and the ones digit pinches the upper point and removes the lower point.

Features: The ones place is not enough to subtract, the tens place is backed off by 1, and the ones place is added to the complement of the subtrahend.

There are a total of 5 cases of abdication at least 10 reductions:
11—6，12—6，13—6，14—6。

12—7，13—7，14—7。

13-8, 14-8.

14—9.

In the above 10 cases, the subtrahend is greater than 5 but less than 10.

Abdication and full 5 reduction calculations are calculated with formulas, and there are four sentences in the formulas:

7K refund one to five to go one seven to one to five to go.
Eight returns one to five to three nine returns one to five to four
This kind of formula is actually a combination of two formulas, such as "eight back one back five go three", which is the combination of "eight back one back two" and "two down five back three", abbreviated as "A few refunds, one return, five returns."The first number is the subtraction; "Back one to five" means to pinch off a lower point on the previous finger position, and pinch the upper point on the current finger position; "Quo Ji" means to pinch off the current finger position The number of next points.

When borrowing 1 from the previous finger and subtracting the difference after subtracting the subtrahend from 10, and adding it back to the current finger, the lower point is not enough, so it is suitable for the abdication and full 5 subtraction formula.

例2、14—8=（14—10）+5—3=6，八退一还五去三。

Table of basic addition and subtraction formulas

Addend or subtrahend, add without carry, subtract without borrow, add with carry, subtract with borrow.

Direct addition and subtraction, full 5 addition, breaking 5 subtraction, direct carry addition, direct abdication subtraction, breaking 5 carry addition, abdication full 5 subtraction.

Direct addition, direct subtraction, full 5 plus, breaking 5 subtraction, direct carry plus, direct abdication subtraction, broken 5 carry plus, abdication full 5 subtraction.

1. One to one, one down to five to four, one up to four to five, one to nine to one, one back to one to nine.

2. Two to two, two to two, two to five to three, two to three to five, two to eight to one, two to one to eight.

3. Erbu two, ￡, three down five go two, three up two go five, three go seven into one, two back one back seven.

4. Four to four, four to four, four to five to one, four to one to five, four to six to one, four to one to six.

5. Five to five, five to five, five to five to one, five to one to five.

6. Go six, go four, go one, return six, one back four, six up one, go 53, and return five.

7. Seven to seven, seven to seven, seven to three to one, seven to one to three, seven to two to seven, seven to one to five to two.

8. Eight to eight, eight to eight, eight to two to one, eight to one to two, eight to divination - to five, eight to one to five to three.

9. Nine on nine, nine to nine, nine to one, nine to one to one, nine to four to five, nine to one to five to four.

practice one
1. Write the sum of the following numbers:

2. Write the complement of the following numbers: 1, 23, 4, 5, 67, 8, 9.

Calculate the following questions by means of arithmetic:

3.2+1，3+1，5+1，8+1，2+2，6+2，5+3，6+3，5+4，3+5，4+5，2+6，2+7，3+6。

4，3—1，6—1，8—1，3—2，7—2，9—2，8—3，9—4，7—5，9一5，8—6，7—7，9—7，9—8。

5.4+1，3+2，4+2，2+3，3+3，4+3，1+4，2+4，7—5，3+4，4+4。

6.5—1，5—2，6—2，5—3，6—3，7—3，5一4，6—4，7—4，8—4。

7.8+2，7+3，7+4，9+4，6+5，8+5，9+6，4+7，8+7，3+8，9+8，4+9，7+9，8+9。

8，10—2，11—3，13—4，12—5，14一5，10—6，15—6，11—7，16—7，12—8，17—8，13—9，15—9，18—9。

9.5+6，6+6，7+6，8+6，5+7，6+7，7+7，5+8，6+8，5+9。

10.11—6，12—6，13—6，14一6，12—7，13—7，14一7，13—8，14一8，14—9。

117+2，9一6，3+4，7—3，9+1，8+3，16—8，15—8，6+7，6+8，5+9，13—6，14—7，14—9。

(Second Section) Refers to multi-digit addition and subtraction

After learning the basic addition and subtraction of arithmetic in the previous section, now let's learn multi-digit addition and subtraction of arithmetic.

The addition and subtraction of two or more multi-digit numbers is called multi-digit addition and subtraction.Multi-digit arithmetic addition and subtraction must be calculated by pinching the counting finger according to the algorithm. The algorithm is that the ones digit is fixed, the digits are aligned, and the same digit is added and subtracted from the high digit to the low digit.

Example 1, 467+326=793.

指算过程：先在右手无名指（百位）、拇指（十位）、食指（个位）上依次布入4、6、7，然后在无名指、拇指、食指上依次加上3、2、6，算指上，得到结果793（图2—15）。其两种方法运算过程如下：

Wujue plus: ring finger, pinch in 5, pinch out the number 3 of 2; thumb, directly pinch in 2; index finger, pinch in the complement of 6 to make up the number 4 of 1, then pinch out 5 and count forward (thumb) Enter 1.Addition to the formula: ring finger, three down five go two; thumb, two up two; index finger, six up one go five into one.

Example 2, 653—467=186.

指算过程：先在右手无名指（百位）、拇指（十位）、食指（个位）上依次布入6、5、3，然后在无名指、拇指、食指上依次减去4、6、7，得到结果186。

The two methods operate as follows:
No formula subtraction: ring finger, pinch 4 to make up the number 1, pinch out 5; thumb, the previous counting finger (ring finger) back one, pinch in the complement of 6 4; index finger, the previous counting finger (thumb) back 1, pinch Enter 5, pinch out 7's complement 3 to make up 2.

Mantra minus: ring finger, four up, one down to five; thumb, six back up one back to four; index finger, seven back up one back five down two.

例3、3624—2389+8497=1235+8497=9732。

指算过程：先在左无名指（千位）、右无名指（百位）、右拇指（十位）、右食指（个位）上依次布人3、6、2、4；然后，在左无名指、右无名指、右拇指、右食指上依次减去2、3、8、9，得到结果1235，再在左无名指、右无名指、右拇指、右食指上依次加上8、4、9、7，得到结果9732。

The operation process of the two methods is as follows:
Addition and subtraction without tricks:

(End of this chapter)