When students check multi-digit addition and subtraction, they mostly follow the algorithm or reciprocal relationship. This actually changed the original problem to some extent and did it again. In order to reduce the mistakes in calculation, it is naturally worth doing it twice. However, it takes too much time. Is there a simpler way to check? Yes This method is called "abandoning nine methods".

In order to understand this method, we must first know how to count nine. Add up the digits of a number until the total is a single digit. We call this number "divided by nine" of the original number. For example:

278: 2+7+8 =17 →1+17 = 8 (divided by nine)

361:3+6+1=10 →1=1= (decimal)

5674: 5+6+7+4 = 22 → 2+2 = 4 (divided by 9)

The divisor can also be obtained by crossing out 9 of the numbers whose sum is equal to 9, and adding up the remaining numbers to get a number less than 9, which is the divisor of the original number.

The method of abandoning nine is to use the number nine.

1. addition problem

Whether the result of adding two multi-digits is correct or not, you can use the discard nine method. The specific method is: first find the divisor of each addend, and then add them up. If the divisor of this sum is equal to the divisor of the original calculated sum, then the original calculation is correct, otherwise the original calculation is wrong.

Example 1 Judge whether the calculation results of the following two questions are correct:

( 1)872+654 1=74 13; (2)3705+6428= 10 123。

Generally speaking, the original calculation result of this problem is correct because the last two divisors are equal.

So the calculation of this problem is wrong. The correct answer is 10 133.

In order to facilitate observation, the above two questions can also be written in the following form:

Among them, the left is the divisor of the first addend, the right is the divisor of the second addend, the above is the divisor of the original addition sum, and the following is the divisor of the left and right sums.

2. Subtraction problem

As we all know, subtraction and addition are reciprocal operations:

Subtraction+difference = minuend.

So the calculation subtraction can still be done by addition.

Example 2 judges whether the results of the following two questions are correct.

( 1)8675-5489=3 186; (2) 10439-9996=443。

Generally speaking, the original calculation result of this problem is correct because the last two divisors are the same.

Similarly, on the whole, the original calculation result of this problem is correct.

Of course, the above practice can also be written in a simple form:

However, at this time, the deduction is on the left, the deduction is on the right, the deduction is above, and the deduction is below.

What is the basis for this abandonment of the nine laws? It takes advantage of the fact that a number can be divisible by 9. Careful students must have seen that a number divided by nine is the remainder of this number divided by nine. If the original calculation is correct, then the remainder on both sides of the addition equal sign is the same; If the remainder on both sides of the equal sign is different, it means that there must be an error in the calculation.

It should be noted that this method is not omnipotent:

1. It is impossible to find out if you write more and write less as 0 in the answer;

2. You can't find out whether the numbers in the answer are written backwards;

3. The wrong number you wrote is just in line with the discard nine method, and it is impossible to find out (although this possibility is very small).

However, as an auxiliary method, it should be said that it is still useful to abandon the nine methods in most cases.

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Abandon the Nine Calculation Method (2)

The method of discarding nine can not only check the addition and subtraction of multiple digits, but also check the multiplication and division.

1. multiplication problem

Whether the result of the multiplication of two multi-digit numbers is correct or not can still use the discard nine method. The specific method is: first find the divisor of two multipliers, and then multiply them. If the divisor of this product is equal to the divisor of the product originally calculated, then the original calculation is correct. Otherwise, the original calculation is wrong.

Example 1 Judge whether the result of the following operation is correct:

( 1)2467×429= 1058343;

(2)8459×376=3 180584。

Because the last two divisors are equal, the original calculation result is correct.

Similarly, the original calculation result of this problem is correct. & lt/PGN008 1。 TXT/PGN & gt；

In order to facilitate observation, the above two questions can be written in the following form:

Among them, the divisor of the first multiplier is on the left, the divisor of the second multiplier is on the right, the divisor of the original product is above, and the divisor of the left and right products is below.

2. Zoning problem

We know divisor × quotient = dividend. So the division can still be done by the multiplication. In addition, the division with remainder can also use the discard nine method, because

Divider × quotient+remainder = dividend.

Example 2 judges whether the following operation result is correct.

( 1)229026÷93 1=246;

(2) 16262 1÷467=348…… 105。

So, on the whole, the original calculation result of this problem is correct. & lt/pgn 0082 . TXT/PGN & gt；

So, similarly, generally speaking, the calculation result of this problem is correct.

Of course, the above practice can also be written in a simple form:

However, these two bifurcations have different meanings.

(1) formula has the divisor on the left, the divisor of quotient on the right, the divisor of the original dividend on the top and the product of left and right numbers on the bottom.

(2) The left side of the formula is the divisor of divisor and quotient divisor, the right side is the divisor of remainder divisor, the upper side is the divisor of divisor, and the lower side is the sum of left and right numbers.

It should be said that there is no complete simple expression for division with remainder.

Of course, abandoning the nine methods is not a panacea for multiplication and division. I won't go into details here.

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