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In mathematics, a **proof without words** (or **visual proof**) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self-evident nature.^{[1]} When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable.^{[2]}

A proof without words is not the same as a mathematical proof, because it omits the details of the logical argument it illustrates. However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof.

The statement that the sum of all positive odd numbers up to 2*n* − 1 is a perfect square—more specifically, the perfect square *n*^{2}—can be demonstrated by a proof without words.^{[3]}

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.

The Pythagorean theorem that can be proven without words.^{[4]}

One method of doing so is to visualise a larger square of sides , with four right-angled triangles of sides , and in its corners, such that the space in the middle is a diagonal square with an area of . The four triangles can be rearranged within the larger square to split its unused space into two squares of and .^{[5]}

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*.^{[6]}

*Mathematics Magazine* and *The College Mathematics Journal* run a regular feature titled "Proof without words" containing, as the title suggests, proofs without words.^{[3]} The Art of Problem Solving and USAMTS websites run Java applets illustrating proofs without words.^{[7]}^{[8]}

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.^{[9]} 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.^{[10]}^{[11]} Rather, mathematicians use proofs without words as illustrations and teaching aids for ideas that have already been proven formally.^{[12]}^{[13]}

Wikimedia Commons has media related to Proof without words.

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**^**Dunham 1994, p. 120**^**Weisstein, Eric W. "Proof without Words".*MathWorld*. Retrieved on 2008-6-20- ^
^{a}^{b}Dunham 1994, p. 121 **^**Nelsen 1997, p. 3**^**Benson, Donald.*The Moment of Proof : Mathematical Epiphanies*, pp. 172–173 (Oxford University Press, 1999).**^**McShane, E. J. (1937), "Jensen's Inequality",*Bulletin of the American Mathematical Society*, vol. 43, no. 8, American Mathematical Society, p. 527, doi:10.1090/S0002-9904-1937-06588-8**^***Gallery of Proofs*, Art of Problem Solving, retrieved 2015-05-28**^***Gallery of Proofs*, USA Mathematical Talent Search, retrieved 2015-05-28**^**Lang, Serge (1971).*Basic Mathematics*. Reading, Massachusetts: Addison-Wesley Publishing Company. p. 94.We always try to keep clearly in mind what we assume and what we prove. By a 'proof' we mean a sequence of statements each of which is either assumed, or follows from the preceding statements by a rule of deduction, which is itself assumed.

**^**Benson, Steve; Addington, Susan; Arshavsky, Nina; Cuoco; Al; Goldenberg, E. Paul; Karnowski, Eric (October 6, 2004).*Facilitator's Guide to Ways to Think About Mathematics*(Illustrated ed.). Corwin Press. p. 78. ISBN 9781412905206.Proofs without words are not

*really*proofs, strictly speaking, since details are typically lacking.**^**Spivak, Michael (2008).*Calculus*(4th ed.). Houston, Texas: Publish or Perish, Inc. p. 138. ISBN 978-0-914098-91-1.Basing the argument on a geometric picture is not a proof, however...

**^**Benson, Steve; Addington, Susan; Arshavsky, Nina; Cuoco; Al; Goldenberg, E. Paul; Karnowski, Eric (October 6, 2004).*Facilitator's Guide to Ways to Think About Mathematics*(Illustrated ed.). Corwin Press. p. 78. ISBN 9781412905206.However, since most proofs without words are visual in nature, they often provide a reminder or hint of what's missing.

**^**Schulte, Tom (January 12, 2011). "Proofs without Words: Exercises in Visual Thinking (review)".*MAA Reviews*. The Mathematical Association of America. Retrieved October 26, 2022.This slim collection of varied visual 'proofs' (a term, it can be argued, loosely applied here) is entertaining and enlightening. I personally find such representations engaging and stimulating aids to that 'aha!' moment when symbolic argument seems not to clarify.

- Dunham, William (1994),
*The Mathematical Universe*, John Wiley and Sons, ISBN 0-471-53656-3 - Nelsen, Roger B. (1997),
*Proofs without Words: Exercises in Visual Thinking*, Mathematical Association of America, p. 160, ISBN 978-0-88385-700-7 - Nelsen, Roger B. (2000),
*Proofs without Words II: More Exercises in Visual Thinking*, Mathematical Association of America, pp. 142, ISBN 0-88385-721-9 - Gulley, Ned (March 4, 2010), Shure, Loren (ed.),
*Nicomachus's Theorem*, Matlab Central.