Pythagorean addition

In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its two sides. According to the Pythagorean theorem, for a triangle with sides $a$ and $b$ , this length can be calculated as

a\oplus b={\sqrt {a^{2}+b^{2}}},

where $\oplus$ denotes the Pythagorean addition operation.

This operation can be used in the conversion of Cartesian coordinates to polar coordinates. It also provides a simple notation and terminology for some formulas when its summands are complicated; for example, the energy-momentum relation in physics becomes

E=mc^{2}\oplus pc.

It is implemented in many programming libraries as the hypot function, in a way designed to avoid errors arising due to limited-precision calculations performed on computers. In its applications to signal processing and propagation of measurement uncertainty, the same operation is also called addition in quadrature; it is related to the quadratic mean or "root mean square".

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