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Difference between revisions of "Pythagorean theorem"

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(Revised text to exhibit the more common name "the Pythagorean theorem.")
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The Pythagorean theorem is a special case of the [[Cosine theorem|cosine theorem]]; its infinite-dimensional analogue (in [[Hilbert space|Hilbert space]]) is the [[Parseval equality|Parseval equality]] (i.e. the completeness theorem for orthonormal systems).
 
The Pythagorean theorem is a special case of the [[Cosine theorem|cosine theorem]]; its infinite-dimensional analogue (in [[Hilbert space|Hilbert space]]) is the [[Parseval equality|Parseval equality]] (i.e. the completeness theorem for orthonormal systems).
  
The problem of solving Phytagoras' equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759401.png" /> in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759402.png" /> leads to the [[Pythagorean numbers|Pythagorean numbers]]. The problem of solving its generalization, the Diophantine equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759404.png" />, is called Fermat's last (or great) theorem, cf. [[Fermat great theorem|Fermat great theorem]].
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The problem of solving Phytagoras' equation $a^2+b^2=c^2$ in integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759402.png" /> leads to the [[Pythagorean numbers|Pythagorean numbers]]. The problem of solving its generalization, the Diophantine equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075940/p0759404.png" />, is called Fermat's last (or great) theorem, cf. [[Fermat great theorem|Fermat great theorem]].
  
 
A right-angled triangle with sides having integer lengths is called a Phytagorean triangle.
 
A right-angled triangle with sides having integer lengths is called a Phytagorean triangle.

Revision as of 17:26, 17 August 2011

Also known as Pythagoras' theorem, the Pythagorean theorem is a theorem in geometry that gives a relationship between the sides of a right-angled triangle. The Pythagorean theorem was evidently known before Pythagoras (6th century B.C.), but the proof in general form is ascribed to him. Originally the theorem established a relationship between the areas of the squares constructed on the sides of a right-angled triangle: The square on the hypotenuse is equal to the sum of the squares on the other sides. Sometimes, the Pythagorean theorem is formulated briefly as follows: The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the catheti. The theorem converse to the Pythagorean theorem is also true: If the square of a side of a triangle is equal to the sum of the squares of the other two sides, then that triangle is right-angled.

Comments

The Pythagorean theorem is a special case of the cosine theorem; its infinite-dimensional analogue (in Hilbert space) is the Parseval equality (i.e. the completeness theorem for orthonormal systems).

The problem of solving Phytagoras' equation $a^2+b^2=c^2$ in integers leads to the Pythagorean numbers. The problem of solving its generalization, the Diophantine equation , , is called Fermat's last (or great) theorem, cf. Fermat great theorem.

A right-angled triangle with sides having integer lengths is called a Phytagorean triangle.

References

[a1] W.H. Greub, "Linear algebra" , Springer (1967)
How to Cite This Entry:
Pythagorean theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pythagorean_theorem&oldid=19489
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article