Casting out elevens.

Yesterday, we discussed casting out nines, a method for checking your work when doing addition. (It also works for multiplication and subtraction, as will the method we discuss today.) Casting out nines has a problem that it will not catch a transposition error. For example.

 321 casting out nines is 6
-235 casting out nines is 10 -> 1
  86 casting out nines is 14 -> 5 

check: 6 - 1 = 5, we are good

 321 casting out nines is 6
-253 casting out nines is 10 -> 1
  68 casting out nines is 14 -> 5 

check: 6 - 1 = 5, we are good... or are we?

Transposition can be caught be casting out elevens. Instead of adding up all the digits, we start with the digit in the ones place, then subtract the digit in the tens place, add the digit in the hundreds place, subtract the digit in the thousands place, and continue this alternation of addition and subtraction until we run out of digits.  Let me show examples with the numbers above, 321, 235, 86, 253 and 68.

321: start with the 1, subtract 2 (gives us -1), add 3.  Casting out elevens gives us 2.
235: start with 5, subtract 3 (gives us 2) add 2. Casting out elevens gives us 4.
86: take 6 subtract 8. Casting out elevens is -2. (If you don't like negative numbers, add 11 and get 9. It will still work.)
253: start with 3, subtract 5 (gives us -2) then add 2. Casting out elevens gives us 0. This means 253 is a multiple of eleven, though that is not needed for the work we are doing.
68: take 8, subtract 6, you get 2.  Let's do the problems above again.


 321 casting out elevens is 2
-235 casting out elevens is 4
  86 casting out elevens is -2

check: 2 - 4 = -2, we are good

 321 casting out elevens is 2
-253 casting out elevens is 0
  68 casting out elevens is 2

check: 2 - 0 = 2, a different result from above

In the era when this was used, you could check original invoices against the copied numbers in the ledger, or possibly the sum in the ledger would be compared with cash on hand. Transposition errors are fairly common and casting out nines won't catch them, but casting out elevens will. What this means is that when we divide 86 by 9, we get a remainder of 5, which is also the remainder when dividing 68 by 9. The remainder of 68 by 11 is 2, while the remainder of 86 by 11 is 9, which is 11 away from -2, another way to write the remainder.

Let's say 86 is the right answer and 68 is the wrong answer. If the only mistake you have made is a transposition, the difference between right and wrong will always be a multiple of 9, like 86-68 = 18. This tells you that if you have made a single mistake, you transposed two numbers in the ones place and the tens place, and the difference between them is 2.

Tomorrow: Two different "tricks" for multiples of 7.