Proof without words

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.

Examples

Sum of odd numbers

A proof without words for the sum of odd numbers theorem

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.^[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.

Pythagorean theorem

Rearrangement proof of the Pythagorean theorem. The uncovered area of gray space remains constant before and after the rearrangement of the triangles: on the left it is shown to equal c², and on the right a²+b².

The Pythagorean theorem that $a^{2}+b^{2}=c^{2}$ can be proven without words.^[4]

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 $b^{2}$ .^[5]

Jensen's inequality

A graphical proof of 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.^[6]

Usage

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]

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.^[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]

Notes

^ 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.

References

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.

[dunham120-1] Dunham 1994, p. 120

[2] Weisstein, Eric W. "Proof without Words". MathWorld. Retrieved on 2008-6-20

[dunham121-3] Dunham 1994, p. 121

[4] Nelsen 1997, p. 3

[5] Benson, Donald. The Moment of Proof : Mathematical Epiphanies, pp. 172–173 (Oxford University Press, 1999).

[6] 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

[7] Gallery of Proofs, Art of Problem Solving, retrieved 2015-05-28

[8] Gallery of Proofs, USA Mathematical Talent Search, retrieved 2015-05-28

[9] 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.

[10] 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.

[11] 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...

[12] 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.

[13] 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.

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