Proofs Without Words

Summary

The following demonstrate proofs of various identities and theorems using pictures, inspired from this gallery. Summations The sum of the first odd natural numbers is . The sum of the first positive integers is . The sum of the first positive integers is .[1] The alternating sum of the first odd natural numbers is . (Source) Nichomauss' Theorem: the sum of the first cubes can be written as the square of the sum of the first integers, a statement that can be written as . Here, we use the same re-arrangement as the first proof on this page (the sum of first odd integers is a square). Here's another re-arrangement to see this: This also suggests the following alternative proof: An animated version of this proof can be found in this gallery. The th pentagonal number is the sum of and three times the th triangular number. If denotes the th pentagonal number, then . The identity , where is the th Fibonacci number. Back to Top Geometric Series The infinite geometric series . The infinite geometric series . The infinite geometric series . Another proof of the identity . The infinite geometric series . The arithmetic-geometric series , also known as Gabriel's staircase.[2] Back to Top Geometry The Pythagorean Theorem (first of many proofs): the left diagram shows that , and the right diagram shows a second proof by re-arranging the first diagram (the area of the shaded part is equal to , but it is also the re-arranged version of the oblique square, which has area ).[3] Another proof of the Pythagorean Theorem (animated version). Another proof of the Pythagorean Theorem; the left-hand diagram suggests the identity , and the right-hand diagram offers another re-arrangement proof. A dissection proof of the Pythagorean Theorem.[6] (Cut-the-knot) COMING: The last proof of the Pythagorean Theorem we shall present on this page, this one by dissection. The area of a triangle is given by , where is the inradius and is the semiperimeter.[10] (: we do not need to re-arrange the triangl...