David Wells. Translated from English: Aryeh Lerner. Mi-An Publishing (416, (09-7781239 p., NIS 94
:to the Joy of Numbers David Wells A Wise and Witty Introduction You are a Mathematician
by Aryeh Rokh
The joy of mathematics
The proof of the Pythagorean theorem, from the book
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The purpose of the book is to expose readers to mathematical problems and fascinating ways to solve them. The reader is invited to solve the problems on his own. If he does not succeed, he is presented with one or more solutions in a way designed to excite and delight him. The original name of the book in English is You Are Mathematician, to tell you: there is nothing to be afraid of mathematics, everyone is capable of solving problems in mathematics, not only professional mathematicians. For this purpose you have to work. You have to play with the problem. You have to use your imagination. Connections to other fields should be used, and similarities between seemingly unrelated fields should be found.
Therefore, this is a book recommended first of all for mathematics teachers. Through it they will be able to easily enrich the knowledge of their students as well as themselves. The book is also recommended for reading to anyone who likes mathematics and has basic knowledge. Even those who consider themselves to hate math may discover a pleasant surprise during reading, when it becomes clear to them that they are able to acquire considerable mathematical knowledge and even enjoy it.
To illustrate all this we will refer to the wonderful proof of the Pythagorean Theorem, presented in the book on page 240. AS Loomis presented 367 proofs of this theorem in his book "The Pythagorean Theorem", published in 1940. There is no other theorem in mathematics that has received so many proofs. The multitude of proofs indicates the special place of the theorem in the mathematical world. However, the proof presented in the book is the most beautiful in my humble opinion, as will be explained later.
The Pythagorean Theorem states that given a right triangle, the sum of the areas of the squares built on the perpendiculars is equal to the area of the square built on the remainder. There are more complicated proofs and there are relatively simple proofs. What they all have in common is that they are based on calculations, on overlapping triangles, on cumbersome auxiliary constructions, and more. None of this weighs on the wonderful proof found in the book. I allowed myself, for my own benefit and the benefit of the readers, to add clarifications that are missing in the book. And this is the proof:
The entire plane is "paved" with squares, in patterns of 4 large squares with a small square in between. Now connect the vertices of the squares diagonally. A grid of squares is obtained. It is extremely important to understand why these are squares. This is due to symmetry considerations. Turn the page 90 degrees in any direction. Has there been any change in form? The answer is negative. This form of flooring is symmetrical towards a 90 degree rotation. Hence, the diagonal squares created by connecting the vertices of the "tiles" diagonally must be squares, because otherwise we would feel the change when we turned the page. We would be able to determine each time in which state the page is, which is impossible with such symmetry (the truth is that this claim can be proven by overlapping triangles and calculating angles, but why spoil such a sterile proof with calculations and overlaps?)
Now we see that the entire plane is paved on one side by large squares (which is the perpendicular of the triangles formed), and by small squares (which is the other perpendicular of the same triangle). That is: the "area" of the plane is equal to the sum of the areas of such squares multiplied by the "number" of the rows. On the other hand, the "area" of the plane is equal to the area of the entire diagonal tiling of the squares, while the diagonal squares rest on the remainder of the same triangle. The "number" of the diagonal squares is "equal" to the "number" of the large squares, which is also the "number" of the small squares. This proved the Pythagorean theorem before our eyes, without us performing a single overlap or a single calculation. This is the ultimate proof of the Pythagorean theorem. Only because of this proof does the book deserve to be within the reach of any thirst for knowledge and challenge.
The book "The Man Who Loved Only Numbers" (by Paul Hoffman, Mater Publishing, 2001), tells the story of the prolific mathematician Paul Ardash. He was always looking for proofs from the "book", that is: heavenly proofs, from the "book" of God. May I suggest that this proof of the Pythagorean theorem is like this. It makes wonderful use of two fundamental elements in mathematics: symmetry, and reordering, the reorganization of elements. In addition, this proof convincingly and unambiguously explains why the Pythagorean theorem is true, why it must be true. This is a proof from the "book". The mathematician Hardy stated that "there is no place in the world for ugly mathematics". The proof of this theorem, like other theorems and other ideas in the book before us, proves it well.
However, the book is not without its shortcomings. It does not have a bibliographic list and there is no reference to mathematical literature. This makes it difficult for the reader to find the source of the abundance of mathematical information presented before him, and to expand his education. Sometimes the proof lacks detail, or essential details are omitted. A teacher who aims to impart knowledge from the book to his students is recommended to go through the proof and explain it well to himself before teaching it to his students. Also, the relationship between different parts of the book is not always clear, and there are cases where more careful editing was required.
Aryeh Lerner did an excellent job translating the book. On the other hand, the cover illustration lacks credit. The writer is also not presented at all to the public of his readers. All of this does not make me retreat in any way from the compliments with which this review opened. The study of mathematics and its love are at a great low point in Israel: there is a great and unjustified fear of it. There is a great lack of books like this, based on which there is a salute to the beauty inherent in this world. Every book, and in particular a fine book like this, deserves to be blessed and read, which causes a lot of pleasure and joy.
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