- 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 on2008 -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 : 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 andUSAMTS websites runJava applet s 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
*citation|last=Dunham|first=William |authorlink =William Dunham (mathematician)|title=The Mathematical Universe|title=John Wiley and Sons|isbn=0-471-53656-3|year=1974
*citation|last=Nelsen|first=Roger B.|publisher=Mathematical Association of America|isbn=978-0883857007|year=1997
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