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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
In this case, the number of primitive Pythagorean triples (a, b, c) with a < b is 2 k−1, where k is the number of distinct prime factors of c. [25] There exist infinitely many Pythagorean triples with square numbers for both the hypotenuse c and the sum of the legs a + b.
When u and v are real numbers, they can be viewed as vectors in , and the triangle inequality expresses a relationship between absolute values. In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem , and for general triangles, a consequence of the law of cosines , although it may be ...
According to the Pythagorean theorem, for a right triangle with side lengths and , the length of the hypotenuse can be calculated as +. This formula defines the Pythagorean addition operation, denoted here as : for any two real numbers and , the result of this operation is defined to be [3] = +.
Using the Pythagorean theorem to compute two-dimensional Euclidean distance. ... the same formula for one-dimensional points expressed as real numbers can be used, ...
Every non-negative real number is a square, so p(R) = 1. For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares, [1] so p = 2. By Lagrange's four-square theorem, every positive rational number is a sum of four squares, and not all are sums of three squares, so p(Q) = 4.
The name Pythagorean theorem arises from the geometric interpretation in Euclidean geometry. ... is a real number then the Cauchy–Schwarz inequality implies that ...
When u and v are real numbers, they can be viewed as vectors in , and the triangle inequality expresses a relationship between absolute values. Pythagorean theorem : It states that the area of the square whose side is the hypotenuse (the side opposite the right angle ) is equal to the sum of the areas of the squares on the other two sides.