Yeah, I got that, I was almost there to, now I am saying that is flat out wrong. Why do you say that 1 x 1 = 1 is conventional? That doesn't look like convention, unless you are saying that 1 + 1 = 2 is also convention. I don't think anyone decided that the unit square was a unit of area. It just so happened that 1 x 1 = 1 looks like a unit square and the name stuck. And the definite integral of S over the interval (0,S) ends up to be S^2 which for a unit square is also 1.
So far I am seeing silliness, but I certainly can be persuaded. Show me the formula for area of an equilateral triangle with sides of length 1 that computes to an area of 1. And then show me the calculus solution of the same area. And then explain where the heck some constant came from out of thin air. I say that if you introduce such a constant then you are inventing a unit where there was none. The problem with this thread is the assumption that the unit square is a unit of area, it is not, and never was.
On Jun 29, 2012, at 11:51 AM, Joe Niederberger wrote:
> But, more to your point and apparent consternation, Kirby, Wayne, myself, realize that this taking of the unit square as the unit of area *is* conventional, and *can* be taken to be other things, like setting a equilateral triangle of side 1 to be the unit area.