Namespaces
Variants
Actions

Difference between revisions of "Pythagorean theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (moved Pythagoras theorem to Pythagoras' theorem: Titled incorrectly)
m (typo)
 
(5 intermediate revisions by 2 users not shown)
Line 1: Line 1:
A theorem in geometry that gives a relationship between the sides of a right-angled triangle. Pythagoras' 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, Pythagoras' 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 Pythagoras' 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.
+
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====
 
====Comments====
Phytagoras' 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]]; 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 <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]].
+
The problem of solving Pythagoras' equation $a^2+b^2=c^2$ in integers $a,b,c$ leads to the [[Pythagorean numbers]]. The problem of solving its generalization, the Diophantine equation $a^n+b^n=c^n$, $n\ge3$, 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.
+
A right-angled triangle with sides having integer lengths is called a Pythagorean triangle.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.H. Greub,  "Linear algebra" , Springer  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.H. Greub,  "Linear algebra" , Springer  (1967)</TD></TR></table>

Latest revision as of 19:38, 17 December 2016

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 Pythagoras' equation $a^2+b^2=c^2$ in integers $a,b,c$ leads to the Pythagorean numbers. The problem of solving its generalization, the Diophantine equation $a^n+b^n=c^n$, $n\ge3$, is called Fermat's last (or great) theorem, cf. Fermat great theorem.

A right-angled triangle with sides having integer lengths is called a Pythagorean 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=19342
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article