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By Black T. H.

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Briefly, however, the symbol 2π represents a complete turn, a full circle, a spin to face the same direction as before. Hence 2π/4 represents a square turn or right angle. You may be used to the notation 360◦ in place of 2π, but for the reasons explained in Appendix A and in footnote 15 of Ch. 3, this book tends to avoid the former notation. 10. 40) which was to be demonstrated. Extending the same technique to the case of an n-sided polygon, we have that n φk = 2π, k=1 φk + ψ k = 2π . 2 Solving the latter equations for φk and substituting into the former yields n k=1 or in other words 2π − ψk 2 = 2π, n k=1 ψk = (n − 2) 2π .

The area of each of the four triangles in the figure is evidently ab/2. The area of the tilted inner square is c 2 . The area of the large outer square is (a + b) 2 . But the large outer square is comprised of the tilted inner square plus the four triangles, hence the area 34 CHAPTER 2. 3: A right triangle. 4: The Pythagorean theorem. 11. FUNCTIONS 35 of the large outer square equals the area of the tilted inner square plus the areas of the four triangles. 42). 43) where h is an altitude perpendicular to both a and b, thus also to c; and where r is the corresponding three-dimensional diagonal: the diagonal of the right triangle whose legs are c and h.

In the event that the reverse order of multiplication is needed, we shall use the notation b k=a f (k) = [f (a)][f (a + 1)][f (a + 2)] · · · [f (b − 2)][f (b − 1)][f (b)]. Note that for the sake of definitional consistency, N N f (k) = 0, f (k) = 0 + k=N +1 k=N +1 N N f (k) = (1) k=N +1 f (k) = 1. k=N +1 This means among other things that 0! = 1. 5) On first encounter, such and notation seems a bit overwrought. ” However, experience shows the latter notation to be extremely useful in expressing more sophisticated mathematical ideas.

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