ABACUS: MYSTERY OF THE BEAD
The Bead Unbaffled - An Abacus Manual
DIGIT SUM QUICK CHECK - Checking
answers
Checking the accuracy of answers in problems of addition, subtraction, multiplication and division often involves doing the problem again for a second or even a third time. This was my method for many years and I always found it tedious. In his book Speed Mathematics Simplified, Edward Stoddard shows us a quick and efficient way to determine if answers are correct. It involves calculating the Digit Sum and Casting out 9s. I found that with a little practice, this check can be performed in a few seconds; even when the problem involves large strings of numbers. This Digit Sum Quick Check is especially well suited to abacus work.
Basically the Quick Check works like this.
CALCULATING THE DIGIT SUM AND CASTING OUT 9s
This is best explained by example. Take the number 139. Using addition techniques as taught by Takashi Kojima, calculate the Digit Sum by adding all three digits together, 1 + 3 + 9 = 13. Then using Kojima's subtraction techniques, cast out the 9s (subtract 9 as many times as you can). In this case 9 can only subtracted once, 13 - 9 = 4. Since no more 9s can be subtracted the Digit Sum of 139 is 4.
In another example take 7849. Calculate the Digit Sum by adding all four digits together, 7 + 8 + 4 + 9 = 28. Cast out the 9s (subtract 9 as many times as you can). In this case, subtract 9 from 28 three times leaving 1. Since no more 9s can be subtracted the Digit Sum of 7849 is 1.
MORE EXAMPLES: Add the digits in each of these numbers and cast out 9s. The result is the Digit Sum.
The Digit Sum of 549 = 0 *
The Digit Sum of 928 = 1
The Digit Sum of 7468 = 7
The Digit Sum of 42702 = 6
The Digit Sum of 332173 = 1
* The technique requires casting out 9s, and 9s reduce to zero.
QUICK CHECKING THE ANSWER
ADDITION
Rule: Add together the Digit Sums of the numbers in the problem; the resulting Digit Sum must equal the Digit Sum of the of the answer.
Problem Digit Sum Check *Digit Sum of the answer = 6* |
In this example |
Problem Digit Sum Check *Digit Sum of the answer = 7* |
In this example |
SUBTRACTION
Rule: Subtract the Digit Sums of the numbers in the problem; the resulting Digit Sum must equal the Digit Sum of the of the answer.
Problem Digit Sum Check *Digit Sum of the answer = 3* |
In this example it's clear that 2 + 9 - 8 = 3 |
In the above example, an alternative would be to add the numbers starting at the bottom. Add 8 + 3 to equal 11, which would reduce to 2.
Problem Digit Sum Check *Digit Sum of the answer = 5* |
In this example, it's possible to 5 - 4 + 9 - 5 = 5 |
Once again an alternative would be to add the numbers starting at the bottom. Add 5 + 5 + 4 = 14, which reduces to 5.
MULTIPLICATION
Using multiplication techniques as taught by Takashi Kojima, each of these calculations is easily done on a soroban.
Rule: Multiply the Digit Sum of the multiplier x The Digit Sum of the multiplicand; the resulting Digit Sum must equal the Digit Sum of the of the product.
Problem Digit Sum Check *Digit Sum of the answer = 1* |
4 x 7 = 28, |
Problem Digit Sum Check
3,875 5 *Digit Sum of the answer = 3* |
5 x 6 = 30, |
DIVISION
First the terminology. In the problem 6 ÷ 3 = 2, 6 is the dividend, 3 is the divisor and 2 is the quotient.
Using division techniques as taught by Takashi Kojima, each of these calculations is easily done on a soroban.
Rule: Multiply the Digit Sum of the divisor x the Digit Sum of the quotient; the resulting Digit Sum must equal the Digit Sum of the dividend.
*Dividend 161 reduces to 8* |
Quotient 23 reduces to 5 |
The following example shows us how to deal with a division problem that involves a remainder. 877 ÷ 27 = 32 with a remainder of 13. In the case of a remainder add on more step to the division rule.
Rule: Multiply the Digit Sum of the divisor x the Digit Sum of the quotient then add the Digit Sum of the remainder; this Digit Sum must equal the Digit Sum of the dividend.
*Dividend 877 reduces to 4* |
Divisor 27 reduces to 0 (0 x 5) = 0 + R4 = 4 |
This is a great method and it works very well for those who take the time to learn it. But it's not infallible. Mr. Stoddard does caution us to be on the lookout for two instances where the Quick Check can run into problems.
1) Since 9 reduces to 0, this check will not catch an error in which one digit in an answer is written as 9 when it should have been a 0, or when a 0 should have been a 9.
2) This Quick Check will not find errors where two numbers have been reversed. For example, if the answer should have been 29 but was written incorrectly as 92, the error will not be found.
In addressing this issue, Mr. Stoddard rightly points out, "...years of experience have shown that the errors not caught by the digit sum are exceedingly rare. For most needs, it is perfectly adequate..."
He goes on to say that, "In return for these shortcomings, the digit-sum check offers a substantial bonus. The digit sum will not only tell you if your answer is wrong; it will tell you by how much it is wrong. If the digit sum of your answer is 4, and you find that it should be 7, then you know that one digit of your answer is too low by exactly 3. You do not know which digit it is, but the fact that one digit is precisely 3 less than it should be is helpful in locating the error quickly."
▪ Link to a Magic Trick that uses DigitSums
REFERENCES:
Stoddard,
Edward.
Thanks to
Edvaldo Siqueira,
|
November, 2005
Totton Heffelfinger
Toronto Ontario Canada
Email
totton[at]idirect[dot]com