The Pythagorean Theorem And Its Converse: Fill & Download for Free

Erm… how hard have you looked?If you pick up any physics, engineering, or computer science textbook, you will easily find examples. For that matter, right here on Quora, you can find Mathematical Applications, which was a Space that I started with the express purpose of giving detailed descriptions of how mathematics can be applied to various fields.Here’s another fun exercise: look up books or websites on carpentry. For example, you might look at the Complete Book of Framing: An Illustrated Guide for Residential Construction, or MyCarpentry.com. Whatever resource you choose, try to look up “geometry” or “Pythagorean theorem”—I can all but guarantee you that you will find some section talking about it.Here’s a rough idea of why this should be true. Suppose that you are building something and you want to connect two straight pieces so that they are perpendicular to one another. If the pieces are comparatively small, you use a carpenter’s square or maybe a level.However, if the pieces are quite large (such as if you are building a roof or similar), then you might have a hard time getting your pieces lined up closely enough via such methods, making them untenable. In this case, the Pythagorean theorem is your best friend. (Technically, it’s the converse of the Pythagorean theorem—more on this in a second.) Recall what the Pythagorean theorem says: if you have a right triangle with sides with lengths [math]a,b,c[/math] where [math]c[/math] is the length of the hypotenuse, then [math]a^2 + b^2 = c^2[/math].A closely related theorem is that if you have a triangle with side lengths [math]a,b,c[/math] and it happens to be that [math]a^2 + b^2 = c^2[/math], then this triangle is a right triangle—specifically, the angle opposite to the side with length [math]c[/math] is a right angle. This is actually the theorem that everyone uses, although I don’t think I have ever seen a carpentry text ever actually acknowledge that they make use of the converse of the Pythagorean theorem and not the Pythagorean theorem itself. Consider this a minor gripe of a grouchy mathematician.In any case, once you know this theorem, all you need to do to ensure that you construct a perfect right angle is to measure the side lengths of the triangles in your construction and make sure that they satisfy the constraint of [math]a^2 + b^2 = c^2[/math]—or, as would be more common to write in a carpentry setting, that the square of the diagonal is the sum of the squares of the rise and the run.This is an absolutely ancient application of geometry. It was being used by the ancient Egyptians over 4,000 years ago—this can be surmised from the fact that if you look at the pyramids, they are clearly built with the simple 3–4–5 right triangle in mind. And, as you can see, it is still in regular use today.

Do you have the time to devote to a serious study of plane geometry? In spite of it often being called "elementary", it's not very elementary. Something that we all know, like the Pythagorean theorem, is not easy to prove rigorously. Yes, we've all seen various cut and paste proofs of it, but how rigorous are they, really? For example, they all rely on the existence of squares, but how do you prove that squares exist?Euclid knew the answer to that. Euclid's Elements, Book I, Proposition 46, the proposition preceding his proof of the Pythagorean theorem:Look at all the things that went into proving it. I.Post.4 is the fourth postulate, that all right angles are equal. Euclid based his geometry on axioms (i.e. postulates and common notions). The axioms were explicitly stated assumptions. For the most part, they can be easily stated, but one of them, the parallel postulate, I.Post.5, has a fairly complicated statement. It's actually used in this proposition, not directly, but indirectly though proposition I.34. The Pythagorean theorem is false if the parallel postulate doesn't hold.Unless you've studied mathematics based on an axiomatic theory (like Euclid's, or number theory based on the Dedekind-Peano axioms, or set theory based on Zermelo-Fraenkel axioms) you haven't seen real mathematics. All that arithmetic, algebra, trigonometry, and calculus you've studied for years and years is not real mathematics until every statement is proved, and that requires axioms, definitions, and theorems with proof.One way you can learn some real mathematics is by studying Euclid's Elements. The first book will show you a lot. It ends with the Pythagorean theorem and its converse. (The converse states that for a triangle if [math]c^2=a^2+b^2[/math], then the triangle is a right triangle with hypotenuse [math]c[/math].) You'll also see flaws in Euclid's presentation. There are several places where he makes conclusions without justification. Recognizing flaws is important, too. Euclid's presentation wasn't improved until about 1900 when mathematicians fixed those flaws. See Hilbert's Foundations of Geometry for more on that.Yes, Euclid's Elements is still worth studying.Your second question, what books can you recommend for Euclidean geometry, depends on what you mean by Euclidean geometry. If you mean Euclidean geometry the way Euclid did it, then his Elements is the best text.The problem with most modern textbooks is that they use coordinates or otherwise build on assumed properties of real numbers. Since they usually don't prove anything about he real numbers, that means the exposition is not self-contained. Look for textbooks that don't use coordinates or real numbers, but build from stated axioms of geometry.

Yes, that's the goal of the first book of his Elements. It's Proposition 47. Its converse is Proposition 48. Most of the propositions in Book I are used in Euclid's proof of the Pythagorean theorem.