Concept
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 and , this length can be calculated as where 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 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. Pythagorean addition (and its implementation as the hypot function) is often used together with the atan2 function to convert from Cartesian coordinates to polar coordinates : If measurements have independent errors respectively, the quadrature method gives the overall error, whereas the upper limit of the overall error is if the errors were not independent. This is equivalent of finding the magnitude of the resultant of adding orthogonal vectors, each with magnitude equal to the uncertainty, using the Pythagorean theorem. In signal processing, addition in quadrature is used to find the overall noise from independent sources of noise. For example, if an gives six digital numbers of shot noise, three of dark current noise and two of Johnson–Nyquist noise under a specific condition, the overall noise is digital numbers, showing the dominance of larger sources of noise. The root mean square of a finite set of numbers is just their Pythagorean sum, normalized to form a generalized mean by dividing by . The operation is associative and commutative, and This means that the real numbers under form a commutative semigroup.
