Proof without words

Sum of odd numbers The statement that the sum of all positive odd numbers up to 2n − 1 is a perfect square—more specifically, the perfect square n2—can be demonstrated by a proof without words. In one corner of a grid, a single block represents 1, the first square. That can be wrapped on two sides by a strip of three blocks (the next odd number) to make a 2 × 2 block: 4, the second square. Adding a further five blocks makes a 3 × 3 block: 9, the third square. This process can be continued indefinitely. Pythagorean theorem The Pythagorean theorem that a^2 + b^2 = c^2 can be proven without words. One method of doing so is to visualise a larger square of sides a+b, with four right-angled triangles of sides a, b and c in its corners, such that the space in the middle is a diagonal square with an area of c^2. The four triangles can be rearranged within the larger square to split its unused space into two squares of a^2 and b^2. Jensen's inequality Jensen's inequality can also be proven graphically. A dashed curve along the X axis is the hypothetical distribution of X, while a dashed curve along the Y axis is the corresponding distribution of Y values. The convex mapping Y(X) increasingly "stretches" the distribution for increasing values of X. ==Usage==

Mathematics Magazine and The College Mathematics Journal run a regular feature titled "Proof without words" containing, as the title suggests, proofs without words. ==Compared to formal proofs==

For a proof to be accepted by the mathematical community, it must logically show how the statement it aims to prove follows totally and inevitably from a set of assumptions. A proof without words might imply such an argument, but it does not make one directly, so it cannot take the place of a formal proof where one is required. Rather, mathematicians use proofs without words as illustrations and teaching aids for ideas that have already been proven formally. ==See also==