BOOKKEEPER'S CHECK: A CASTING-OUT-NINES TRICK

I show elsewhere a special property of numbers written in the DECIMAL NOTATION.

  1. Any re-arrangement of the digits of a number is a PERMUTATION OF THE FORM.
  2. THE DIFFERENCE BETWEEN ANY TWO PERMUTATIONS OF A DECIMALLY WIRTTEN NUMBER IS ALWAYS A MULTIPLE OF NINE.
Example: Take the number 123. This has 5 other PERMUTAIIONS: 213, 231, 321, 312. 132.

Here are the differences:

  1. 213 - 123 = 90 = 9 x 10.
  2. 231 - 123 = 108 = 9 x 12.
  3. 321 - 123 = 198 = 9 x 22.
  4. 312 - 123 = 189 = 9 x 21
  5. 132 - 123 = 9.
I'll state that as a CONDITIONAL, so we can understand the LOGIC of it. IF YOU PERMUTE DIGITS IN A DECIMALLY FORMULATED NUMBER, THEN THE DIFFERENCE "BEFORE" AND "AFTER" IS A MULTIPLE OF NINE.

This statement has both a CONVERSE and a CONTRAPOSITIVE.

CONVERSE: IF THE DIFFERENCE BETWEEN TWO NUMBERS IS A MULTIPLE OF NINE, THEN EACH IS A PERMUTATION OF THE OTHER. MAYBE WRONG!!! as I'll show.

CONTRAPOSITIVE: IF THE DIFFERENCE BETWEEN TWO NUMBER IS NOT A MULTIPLE OF NINE, THEN THEY ARE NOT PERMUTATIONS OF EACH OTHER.

ANY DECLATION CAN BE DISPROVED BY A SINGLE COUNTEREXAMPLE! Take that number, 213, and the number, 141, which is clearly a permutation of 123. But note: 141 - 123 = 18 = 9 x 2.

Elsewhere, I define THE DIGITAL ROOT OF A NUMBER as INVOLVING THE SUM OF THE DIGITS OF THE NUMBER. TWO "DECIMAL" NUMBERS WITH THE SAME DIGITAL ROOT ALWAYS DIFFER BY NINE.

Take the number, 123: 1 + 2 + 3 = 6. And the number, 141: 1 + 4 + 1 = 6. They have the SAME DIGITAL ROOT, so DIFFER BY A MULTIPLE OF NINE, EVEN THOUGH THEY ARE NOT MUTUALLY PERMUTIVE.

To see why the CONVERSE OF A STATEMENT CAN BE INCORRECT, BUT ITS CONTRAPOSITIVE ALWAYS CORRECT, I DIAGRAM THE GENERAL CONDITIONAL STATEMENT: IF A, THEN B. This says "SET A IS INCLUDED IN SET B":

		-------------------------
                |   ----------------  y  |
                |   |         x     |    | z
                |   |      A        |    |
                |   |_______________|    |
                |________________________|
If an element (say, x) is in set A, then it's in set B (The CONDITIONAL FORM). If an element (say, y) is in set B, it's on set A. WRONG! (The CONVERSE FORM.) If an element (say z) is not in set B, then it's not in set A. CORRECT! (The CONTRAPOSITIVE FORM.)
BOOKKEEPER'S CHECK

Bookkeeper's set up a CREDIT COLUMN OF INCOME ENTRIES; and DEBIT COLUMN of INCOME GOING OUT. The sum of the TWO COLUMNS SHOULD AGREE. If they do not, AN ERROR HAS OCCURRED.

Now, a very common error is a PERMUTATION OF AN ENTRY, and it's not too difficult to detect. But should you look for it?

THE BOOKKEEPER TAKES THE DIFFERENCE BETWEEN THE TWO COLUMNS. IF THE DIFFERENCE IS NOT A MULTIPLE OF NINE, IT'S FOOLISH TO LOOK FOR A PERMUTATION. IF THE DIFFERENCE IS A MULTIPLE OF NINE, IT'S EITHER DUE TO A PERMUTATION OR AN INCORRECT ENTRY WITH THE SAME DIGITAL ROOT AS THE CORRECT VLUE. (Dig?)

While going to Graduate School at NYU in NYC, I worked as HEAD OF THE CANCELLATION DEPARTMENT at an INSURANCE BROKERAGE one block from Wall Street. Over and over, I had to explain to the BOOKKEEPERS THAT ONLY THE CONTRAPOSITIVE CAN BE TRUSTED, BUT MAKING UP EXAMPLES SUCH AS THE ABOVE.

Every child in Elementary School should be taught this.