Санның 9-ға бөліну қасиетін пайдаланып амалдарды тексеру әдісі

Arithmetic procedure of verifying operations using modulo characteristics of digit 9
Casting out nines is any of three arithmetical procedures:
Adding the decimal digits of a positive whole number, while optionally ignoring any 9s or digits which sum to a multiple of 9. The result of this procedure is a number which is smaller than the original whenever the original has more than one digit, leaves the same remainder as the original after division by nine, and may be obtained from the original by subtracting a multiple of 9 from it. The name of the procedure derives from this latter property. Repeated application of this procedure to the results obtained from previous applications until a single digit number is obtained. This single digit number is called the "digital root" of the original. If a number is divisible by 9, its digital root is 9. Otherwise, its digital root is the remainder it leaves after being divided by 9. A sanity test in which the above mentioned procedures are used to check for errors in arithmetical calculations. The test is carried out by applying the same sequence of arithmetical operations to the digital roots of the operands as are applied to the operands themselves. If no mistakes are made in the calculations, the digital roots of the two resultants will be the same. If they are different, therefore, one or more mistakes must have been made in the calculations.

If the procedure described in the preceding paragraph is repeatedly applied to the result of each previous application, the eventual result will be a single digit number from which all 9s, with the possible exception of one, have been "cast out". The resulting single digit number is called the digital root of the original. The exception occurs when the original number has a digital root of 9, whose digit sum is itself, and therefore will not be cast out by taking further digit sums. The number 12565, for instance, has digit sum 1+2+5+6+5 = 19, which, in turn, has digit sum 1+9=10, which, in its turn has digit sum 1+0=1, a single digit number. The digital root of 12565 is therefore 1, and its computation has the effect of casting out (12565 1)/9 = 1396 lots of 9 from 12565.

To check the result of an arithmetical calculation by casting out nines, each number in the calculation is replaced by its digital root and the same calculations applied to these digital roots. The digital root of the result of this calculation is then compared with that of the result of the original calculation. If no mistake has been made in the calculations, these two digital roots must be the same. Examples in which casting out nines has been used to check addition, subtraction, multiplication, and division are given below.

In each addend, cross out all 9s and pairs of digits that total 9, then add together what remains. These new values are called excesses. Add up leftover digits for each addend until one digit is reached. Now process the sum and also the excesses to get a final excess. 2 and 4 add up to 6. 8+1=9 and 4+5=9; there are no digits left. 2, 4, and 6 make 12; 1 and 2 make 3. 2 and 0 are 2. 6, 0, 3 and 2 make 11; 1 and 1 add up to 2. The excess from the sum should equal the final excess from the addends.

First, cross out all 9s and digits that total 9 in both minuend and subtrahend (italicized). Add up leftover digits for each value until one digit is reached. Now follow the same procedure with the difference, coming to a single digit. Because subtracting 2 from zero gives a negative number, borrow a 9 from the minuend. The difference between the minuend and the subtrahend excesses should equal the difference excess.

First, cross out all 9s and digits that total 9 in each factor (italicized). Add up leftover digits for each multiplicand until one digit is reached. Multiply the two excesses, and then add until one digit is reached. Do the same with the product, crossing out 9s and getting one digit. *The excess from the product should equal the final excess from the factors. *8 times 8 is 64; 6 and 4 are 10; 1 and 0 are 1.

Cross out all 9s and digits that total 9 in the divisor, quotient, and remainder. Add up all uncrossed digits from each value until one digit is reached for each value. The dividend excess should equal the final excess from the other values. In other words, perform the same procedure as in a multiplication, only backwards. 8x4=32 which is 5, 5+3 = 8. And 8=8.