Proof without words

In mathematics, a proof without words is a proof 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 more formal and mathematically rigourous proofs due to their self-evident nature.Harvnb|Dunham|1994|p=120] To be a proof, the case that the diagram demonstrates must be able to be generalised. [mathworld|title=Proof without Words|urlname=ProofwithoutWords Retrieved on 2008-6-20]

Examples

um of odd numbers

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, as shown on the right. Each colored strip represents the consecutive odd numbers, and since they form a square when combined, their sum is a perfect square.Harvnb|Dunham|1994|p=121]

Pythagorean theorem

The Pythagorean theorem can be proven without words as shown in the second diagram on right. The two different methods for determining the area of the large square give the relation : $a^2 + b^2 = c^2,$ between the sides. This proof is more subtle than the above, but still can be considered a proof without words. [Harvnb|Nelsen|1997|p=3]

Jensen's inequality

Jensen's inequality can also be proven graphically, as illustrated on the third diagram. The dashed curve along the "X" axis is the hypothetical distribution of "X", while the dashed curve along the "Y" axis is the corresponding distribution of "Y" values. Note that 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]

Usage

The "College Mathematics Journal" runs a regular feature entitled "Proof without words" containing, as the title suggests, proofs without words.Harvnb|Dunham|1994|p=121] The Art of Problem Solving and USAMTS websites run Java applets illustrating proofs without words. [citation|url=http://usamts.org/Gallery/G_Gallery.php|publisher=Art of Problem Solving|accessdate=2008-06-20|title=Gallery of Proofs (AoPS)] [citation|title=Gallery of Proofs (USAMTS)|url=|publisher=Art of Problem Solving|accessdate=2008-06-20]

Notes

References

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Proof without words

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