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Proof without Words


A proof that is only based on visual elements, without any comments.

ProofWithoutWordsPentagonal

An arithmetic identity can be demonstrated by a picture showing a self-evident equality between numerical quantities. The above figure shows that the difference between the nth pentagonal number and n is equal to three times the (n-1)th triangular number. Of course, the situation depicted is a particular case of the formula (here it corresponds to n=5), but it is presented in a way that can be immediately generalized.

Another form of proof without words frequently used in elementary geometry is the dissection proof.


See also

Arithmetic-Logarithmic-Geometric Mean Inequality, Dissection Proof, Gabriel's Staircase, Odd Number Theorem, Proof, Prosthaphaeresis Formulas, Trigonometric Addition Formulas

This entry contributed by Margherita Barile

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References

Ayoub, A. B. "Proof Without Words: Arithmetic Mean-Geometric Mean Inequality." Math. Computer Educ. 31, 191, 1997.Bogomolny, A. "Proofs Without Words." http://cut-the-knot.org/ctk/pww.shtml.Chilaka, J. O. "Proof Without Words." Math. Computer Educ. 30, 312, 1996.Chilaka, J. O. "Proof Without Words: A Combinatorial Identity for (3; n)." Math. Computer Educ. 35, 43, 2001.Harvard University Instructional Computing Group. "Proof without Words: Sum of Squared Integers." http://icg.harvard.edu/~gov3009/spring02/sumsq.pdf.Nelsen, R. B. Proofs Without Words: Exercises in Visual Thinking. Washington, DC: Math. Assoc. Amer., 1997.Nelsen, R. B. "Proof Without Words: Sums of Integers as Sums of Cubes." Math. Mag. 71, 65, 1998.Nelsen, R. B. Proofs Without Words II: More Exercises in Visual Thinking. Washington, DC: Math. Assoc. Amer., 2001.Morey, J. "Proof Without Words." http://www.math.ubc.ca/~morey/talk/proofwowords.html.Sher, D. "Proof Without Words: 3^n-1=sum_(k=0)^(n-1)2×3^k. Math. Computer Educ. 31, 190, 1997.Sher, D. "Proof Without Words: 1+sum_(chi=0)^(k)(n-1)n^chi=n^(k+1)." Math. Computer Educ. 32, 51, 1998.Wilson, J. http://jwilson.coe.uga.edu/emt725/AMGM/AMGM.1.html.Wise, D. S. "Proof Without Words: A Generalization from Pythagoras." Math. Mag. 71, 64, 1998.

Referenced on Wolfram|Alpha

Proof without Words

Cite this as:

Barile, Margherita. "Proof without Words." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ProofwithoutWords.html

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