By Mac Lane, Birkhoff (WEIL, HOCQUEMILLER)
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Extra info for Algebre, solutions developpees des exercices, 1ere partie, ensembles, groupes, anneaux, corps [Algebra]
The point of a measurement problem is not what the measurement is; it’s how to figure out what it is. The answer to the question about the diagonal of _ a square is not √2; it’s the mosaic design. ) 50 M E A S U R E M E N T The solution to a math problem is not a number; it’s an argument, a proof. We’re trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying.
So the area of a rectangle is just the product of its sides. It doesn’t matter whether the sides come out even or not. What about the area of a triangle? My favorite way to think about it is to imagine a rectangular box built around the triangle. It turns out that the area of the triangle is always half that of the rectangle. Do you see why? Why does a triangle take up exactly half of its box? What happens to the area of the triangle as we slide the tip horizontally? What if it goes past the sides of the box?
The point is that these two big squares are identical; they both have sides equal to the two sides of the rectangle added together. In particular, this means that the two mosaics have the same total area. Now, if we remove the four triangles from each, the remaining areas must also match, so the two smaller squares really do take up exactly as much space as the larger one. c a b Let’s call the sides of the rectangle a and b and the diagonal c. 48 M E A S U R E M E N T Then the square of side a together with the square of side b has the same total area as the square of side c.
Algebre, solutions developpees des exercices, 1ere partie, ensembles, groupes, anneaux, corps [Algebra] by Mac Lane, Birkhoff (WEIL, HOCQUEMILLER)