Double Counting

I ran into two instances of double counting in 24 hours. One is in mathematics, and the other pertains to a Toastmasters rating system.

The mathematical one involved this algorithm on numbers. Take the last digit and deduct it from the rest. For example, with the number 4257, take the 7 and subtract it from 425: 425 - 7 = 418. Repeat the process: 41 - 8 = 33; 3 - 3 = 0. The number is divisible by 11 if and only if you get zero at the end. Further, the digits you pull off along the way form the quotient: 387. Someone on Mathnerds wanted help in proving that this algorithm works. I suggested proving by mathematical induction. If the number is 10x + y (in our example, x = 425, y = 7, 10x+y = 4257), then the operation means subtracting y to obtain 10x, then subtracting y from the result (actually 10y since we chopped off a place): 10x - 10y = 10(x - y). By induction, x - y is divisible by 11 and the quotient is, say, the integer z. then 10x + y = 10(x-y) + 10y + y = 110z + 11y = 11(10z+y), so the number is divisible by 11 and the quotient has last digit y. The person who wanted help wanted to know why we subtract y twice. We do so because the first time represents taking the last digit off the number, and the second time represents subtracting the 10y. In fact, that is why 11 is involved: 11 consists of two 1s.

The Toastmasters instance involved discussion of the Distinguished Club Plan. A Toastmaster Club earns one point for each of the following: earning two Competent Toastmaster (CTM) awards, earning two more CTMs, earning an Advanced Toastmaster (ATM) award, earning another one, earning a leadership award, earning another one, getting four more members, getting four more members, getting at least 4/7 of the officers trained, and turning in required reports in on time. But - a Toastmaster club must be over 20 members or have increased by 5 members before it can even be considered for these awards. A Toastmaster in my club found our club had 3 new members in the year, but because of losses, we increased from 11 only to 13 members. We need 3 new members. If we get that, we earn the first membership requirement (four members) and earn two members towards the second four members. The other Toastmaster said that was double counting - first we count the members to qualify; then we count them towards membership points. I didn't think so. There are two separate requirements - the 20 or increase in 5 requirement and the membership points in the Distinguished Club Plan.

It is interesting that two of these double countings came in the same day. What causes these? Apparently there is the underlying assumption in our society that you need to do something only once - we don't sign a million dollar contract twice, we don't try a man twice for the same crime (that is called "double jeopardy"), and we don’t get married twice. So they generalize to other concepts.. For example, a retiree with a job is said to be a "double dipper". In each case, you must look at the conditions. If the first trial never occurred, then to try the person again is not double jeopardy. If the contract is signed, that is enough to bind the two parties to the requirements of the contract. In our number example, the two subtractions mean two different things - taking a digit off the number, and subtracting a number. The two membership requirements are separate - one is an absolute condition for getting Distinguished Club rating, and the other is one of the points that the club can earn.

Still it is confusing and one needs to check the conditions to see if a certain case really is one of double counting.