{{Short description|Mathematical proof expressed visually}} [[File:Nicomachus_theorem_3D.svg|thumb|Proof without words of the Nicomachus theorem ({{harvtxt|Gulley|2010}}) that the sum of the first {{mvar|n}} cubes is the square of the {{mvar|n}}th triangular number]] 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.<ref name="dunham120">{{Harvnb|Dunham|1994|p=120}}</ref> When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable.<ref>{{mathworld|title=Proof without Words|urlname=ProofwithoutWords}} Retrieved on 2008-6-20</ref>
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=== thumb|upright|A proof without words for the sum of odd numbers theorem 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''<sup>2</sup>—can be demonstrated by a proof without words.<ref name="dunham121">{{Harvnb|Dunham|1994|p=121}}</ref>
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=== {{seemain|Pythagorean theorem#Rearrangement proofs}} thumb|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 <math>a^2 + b^2 = c^2</math> can be proven without words.<ref>{{Harvnb|Nelsen|1997|p=3}}</ref>
One method of doing so is to visualise a larger square of sides <math>a+b</math>, with four right-angled triangles of sides <math>a</math>, <math>b</math> and <math>c</math> in its corners, such that the space in the middle is a diagonal square with an area of <math>c^2</math>. The four triangles can be rearranged within the larger square to split its unused space into two squares of <math>a^2</math> and <math>b^2</math>.<ref>Benson, Donald. ''[https://books.google.com/books?id=8_vbuzxrpfIC&pg=PA172 The Moment of Proof : Mathematical Epiphanies]'', pp. 172–173 (Oxford University Press, 1999).</ref>
===Jensen's inequality=== thumb|upright|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''.<ref>{{citation|title=Jensen's Inequality|periodical=Bulletin of the American Mathematical Society|volume=43|issue=8|year=1937|publisher=American Mathematical Society|page=527|doi=10.1090/S0002-9904-1937-06588-8|doi-access=free|last1=McShane|first1=E. J.}}</ref>
==Usage== ''Mathematics Magazine'' and ''The College Mathematics Journal'' run a regular feature titled "Proof without words" containing, as the title suggests, proofs without words.<ref name="dunham121"/> The Art of Problem Solving and USAMTS websites run Java applets illustrating proofs without words.<ref>{{citation|url=http://artofproblemsolving.com/articles/proof-without-words|publisher=Art of Problem Solving|accessdate=2015-05-28|title=Gallery of Proofs}}</ref><ref>{{citation|title=Gallery of Proofs|url=http://usamts.org/Gallery/G_Gallery.php|publisher=USA Mathematical Talent Search|accessdate=2015-05-28}}</ref>
==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.<ref>{{cite book |last=Lang |first=Serge |title=Basic Mathematics |author-link=Serge Lang |date=1971 |publisher=Addison-Wesley Publishing Company |location=Reading, Massachusetts |page=94 |quote=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.}}</ref> 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.<ref>{{cite book |last1=Benson |first1=Steve |last2=Addington |first2=Susan |last3=Arshavsky |first3=Nina |last4=Cuoco |last5=Al |last6=Goldenberg |first6=E. Paul |last7=Karnowski |first7=Eric |title=Facilitator's Guide to Ways to Think About Mathematics |publisher=Corwin Press |edition=Illustrated |date=October 6, 2004 |url=https://books.google.com/books?id=PW80dP3YN2MC&dq=%22proof+without+words%22&pg=PA78 |page=78 |isbn=9781412905206 |quote=Proofs without words are not <em>really</em> proofs, strictly speaking, since details are typically lacking.}}</ref><ref>{{cite book |last=Spivak |first=Michael |author-link=Michael Spivak |date=2008 |title=Calculus |edition=4th |url=https://books.google.com/books?id=7JKVu_9InRUC&q=%22basing+the+argument+on+a+geometric+picture+is+not+a+proof%22&pg=PA136 |location=Houston, Texas |publisher=Publish or Perish, Inc. |page=138 |isbn=978-0-914098-91-1 |quote=Basing the argument on a geometric picture is not a proof, however...}}</ref> Rather, mathematicians use proofs without words as illustrations and teaching aids for ideas that have already been proven formally.<ref>{{cite book |last1=Benson |first1=Steve |last2=Addington |first2=Susan |last3=Arshavsky |first3=Nina |last4=Cuoco |last5=Al |last6=Goldenberg |first6=E. Paul |last7=Karnowski |first7=Eric |title=Facilitator's Guide to Ways to Think About Mathematics |publisher=Corwin Press |edition=Illustrated |date=October 6, 2004 |url=https://books.google.com/books?id=PW80dP3YN2MC&dq=%22proof+without+words%22&pg=PA78 |page=78 |isbn=9781412905206 |quote=However, since most proofs without words are visual in nature, they often provide a reminder or hint of what's missing.}}</ref><ref>{{cite magazine |last=Schulte |first=Tom |date=January 12, 2011 |title=Proofs without Words: Exercises in Visual Thinking (review) |url=https://www.maa.org/press/maa-reviews/proofs-without-words-exercises-in-visual-thinking |magazine=MAA Reviews |publisher=The Mathematical Association of America |access-date=October 26, 2022 |quote=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.}}</ref>
==See also== {{Commons category|Proof without words}}
* {{annotated link|Pizza theorem}} * {{annotated link|Philosophy of mathematics}} * {{annotated link|Proof theory}} * {{annotated link|Visual calculus}}
==Notes== {{reflist}}
==References== {{refbegin}} *{{citation|last=Dunham|first=William|authorlink=William Dunham (mathematician)|title=The Mathematical Universe|publisher=John Wiley and Sons|isbn=0-471-53656-3|year=1994|url-access=registration|url=https://archive.org/details/mathematicaluniv0000dunh}} *{{citation |last=Nelsen |first=Roger B. |title=Proofs without Words: Exercises in Visual Thinking |publisher = Mathematical Association of America |isbn = 978-0-88385-700-7 |year=1997 | pages=160}} *{{citation |last=Nelsen |first=Roger B. |title=Proofs without Words II: More Exercises in Visual Thinking |publisher=Mathematical Association of America |isbn=0-88385-721-9 |year=2000 |pages=[https://archive.org/details/proofswithoutwor0000nels/page/142 142] |url=https://archive.org/details/proofswithoutwor0000nels/page/142 }} *{{citation | last1 = Gulley | first1 = Ned | editor-last = Shure | editor-first = Loren | title = Nicomachus's Theorem | url = http://blogs.mathworks.com/loren/2010/03/04/nichomachuss-theorem/ | date = March 4, 2010 | publisher = Matlab Central}}. {{refend}}
{{Mathematical logic}}
Category:Proof without words Category:Articles containing proofs Category:Mathematical proofs Category:Visual thinking