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Week 13: More Proofs without Words Mathematics is abstract thought, mathematics is pure logic, mathematics is creative art. All these statements are wrong, but they are all a little right, and they are all nearer the mark than “mathematics is numbers” or “mathematics is geometric shapes.” Paul Halmos Figurate Numbers (Continued) We continue now with a few more “proofs without words” related to figurate numbers. Exercise 1: The diagram to the left demonstrates a certain formula, which we have never seen before, relating two consecutive triangular numbers to another triangular number. What is the formula? Explain exactly how the diagram demonstrates the formula. Exercise 2: Give a formal (completely non-visual) proof of the formula from Exercise 1. Exercise 3: Give a “proof without words” that the sum of two consecutive tetrahedral numbers is always a square pyramidal number. Infinite Sums Proofs without words can be a powerful tool for grasping the truth of many mathematical statements and formulas, especially those involving limits, such as infinite series. The proof without words shown to the right may be the most famous of all of them. Exercise 4: Consider the proof without words shown to the right. Give a detailed explanation of exactly how the diagram demonstrates the fact that it claims to demonstrate. 1 1 1 1 + 2+ 3+⋯ = 4 4 4 3 Week 13 Page 2 Exercise 5: As with previous proofs without words, explain the one given below. Exercise 6: The fact “proved” to the left can be demonstrated just as well by drawing a line segment (with length 1) and dividing it over and over again in much the same way. Show this version of the proof without words. Exercise 7: Draw a new diagram like the one in the previous exercise, but this time taking two-thirds of the line segment, and then two-thirds of what remains, and so on and so on. What infinite sum formula does this new diagram prove? 𝟏 𝟏 𝟏 𝟏 𝟏 + + + + +⋯ =𝟏 𝟐 𝟒 𝟖 𝟏𝟔 𝟑𝟐 Exercise 8: Create a new diagram, similar to the previous one, to “prove” that 0.9999 … = 1. Exercise 9: What does the diagram to the right demonstrate about the sum 30 + 31 + 32 + … + 3𝑛 ? Explain how the diagram shows this. Exercise 10: Give a proof by induction for the fact from Exercise 9. Week 13 Page 3 Geometry and Algebra Perhaps the most famous theorem in all of mathematics is the Pythagorean Theorem. Dozens of very different proofs of the theorem have been devised, and many of them can be given in the form of “proofs without words.” The diagram to the right shows one of the most straightforward and accessible proofs (without words) of the Pythagorean Theorem 𝒂2 + 𝒃2 = 𝒄2 . It is probably no surprise that a geometric diagram can serve as a proof without words for a geometric theorem. But there is also an ancient tradition of using simple geometric diagrams to illustrate facts that are more purely in the realm of algebra. For example, consider the following “proof without words” of a basic fact from algebra: To complete the square 𝒙(𝒙 + 𝒃), or 𝒙2 + 𝒃𝒙, we can add (𝒃/𝟐)2 . Some of the first cultures to discover the idea of “completing the square,” in order to solve a quadratic equation, thought about the process in geometric terms, much like what is shown above. The expression that we write in modern notation as 𝑥 2 + 𝑏𝑥 can be visualized as the area of a rectangle with width 𝑥 and length 𝑥 + 𝑏. The sequence of three illustrations above show how we can rearrange this area, and add a bit more, in order to form a square with area (𝑥 + 𝑏/2)2 . This proof without words, along with the one to be considered next, is available in animated form on myCMU. Week 13 Page 4 Here is a proof without words about a somewhat less familiar algebraic fact – or rather, a fact about all positive real numbers: The sum of a positive number and its reciprocal is always at least 2. Exercise 11: Using your basic knowledge of the areas of triangle and squares, and a little algebra, explain in full detail exactly how the “proof without words” given earlier demonstrates the Pythagorean Theorem. Exercise 12: Explain in the fullest possible detail exactly how the diagram given directly above demonstrates that the sum of a positive number and its reciprocal is always at least 2. Exercise 13: Give a formal (completely non-visual) proof that the sum of a positive number and its reciprocal is always at least 2.