An operation is a process that takes some input(s) and gives back an output(s). Many of these operations (especially arithmetic) are some type of relation (typically binary relations like functions). If we look at multiplication from this perspective, yes, multiplication is not repeated addition, however, we can define multiplication BY repeated addition. I say “by” because, yes, in some ways, Devlin’s slightly overzealous argument is correct, it is not defined AS repeated multiplication. In math, that tiny word change can alter the meaning completely. The other thing that everyone seems to over look is where are we using it. Multiplication being taught as repeated addition works when we are discussing natural numbers, and for a child it should work fine.

I do not support Devlin’s opinion that we should alter the education systems so multiplication isn’t taught as repeated addition, however I also feel some pedagogical bias to treat multiplication as only repeated addition. I think that the system we have in place is fine as long as we get the point across that multiplication ISN’T repeated addition. It is just EQUAL (equal for many of you who have forgotten, means they are the same but not necessarily the same thing) to it with natural numbers.

In summary (partially because I would much rather the majority of articles and comments have conclusions):

Multiplication is NOT repeated addition;

Multiplication is DEFINED its EQUALITY to the the repeated addition of numbers;

I feel the educational system in place currently works fine, albeit could use some clarity that multiplication is BASED on repeated addition;

This issue is just a confusion in semantics, either sides are sort of right and wrong

There are some who say pi was invented by Archamedes. In some of the research I looked at pi apparently was used by different cultures for area. Some of those numbers for pi calculations are different but the area is the correct. The Greeks, Hebrews, Egyptians, Babylonians, Indian and Persians all have varied factors for pi but very accurate ratios for area. Maybe this is by accident maybe not. All of these cultures used light and shadows to map time and measurements. Maybe having the answer based on earthen geometrics is a trail back to pi versus a formula first theory(chicken or egg theory).

Here’s my best guess:

I would say that pi is not so much a numeric value for every area but rather a numeric ratio for a given area. Can you have pi without an area first? Does not the area beg an equation rather than the equation an area.

]]>