Date: Sat 11/28/98 3:59 PM

From: Alex Bogomolny

Thank you for the kind words.

Say 3*5 = 15 regardless of the method used to actually evaluate the product. This is a clue: if a problem has a unique solution then any method has to lead to that solution. For problems that have more than 1 solution, various methods may lead to different solutions. x_(n+1) = (x_n + 2/x_n)/2 converges to either sqrt(2) or -sqrt(2) depending on the starting point x_0.

I once attended a parent workshop run by the New Jersey Mathematical Coalition. Parents were supposed to estimate the amount of water leaked in a day by a dripping faucet. Each group of parents came up with a different answer. The presenter noted encouragingly that not always there is one single correct answer. Which was a complete nonsense because different faucets are supposed to leak at different rates. If we were asked about one particular faucet and came up with different answers, there would be a reason to look for an error in somebody's calculations, formulas, or measurements.

Finally, methods for solving math problems may have different sensitivity to errors of calculation. When implemented on a computer they may lead to different results. For example,

two formulas may be used for solving the equation ax^2 + 2bx + c = 0:

- (-b ± sqrt(b^2 - 4ac)/(2a)
- 2c/(-b ± sqrt(b^2 - 4ac)

Numerically, they may produce different results.

All the best,

Alexander Bogomolny