proof without words

Image:Proofwithoutwords.svg

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.{{Harvnb|Dunham|1994|p=121}}

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.

{{seemain|Pythagorean theorem#Rearrangement proofs}}

File:Diagram of Pythagoras Theorem.png

The Pythagorean theorem that $a^2\; +\; b^2\; =\; c^2$ can be proven without words.{{Harvnb|Nelsen|1997|p=3}}

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 $b^2$.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).

File:Jensen graph.svg

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.{{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.}}

Mathematics Magazine and The College Mathematics Journal run a regular feature titled "Proof without words" containing, as the title suggests, proofs without words. The Art of Problem Solving and USAMTS websites run Java applets illustrating proofs without words.{{citation|url=http://artofproblemsolving.com/articles/proof-without-words|publisher=Art of Problem Solving|accessdate=2015-05-28|title=Gallery of Proofs}}{{citation|title=Gallery of Proofs|url=http://usamts.org/Gallery/G_Gallery.php|publisher=USA Mathematical Talent Search|accessdate=2015-05-28}}

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.{{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.}} 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.{{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 really proofs, strictly speaking, since details are typically lacking.}}{{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...}} Rather, mathematicians use proofs without words as illustrations and teaching aids for ideas that have already been proven formally.{{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.}}{{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.}}

{{Commons category|Proof without words}}

• {{annotated link|Pizza theorem}}
• {{annotated link|Philosophy of mathematics}}
• {{annotated link|Proof theory}}
• {{annotated link|Visual calculus}}

{{reflist}}

{{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:Articles containing proofs

Category:Mathematical proofs

Category:Visual thinking