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 ${\displaystyle a}$ and ${\displaystyle b}$, this length can be calculated as $${\displaystyle a\oplus b={\sqrt {a^{2}+b^{2}}},}$$ where ${\displaystyle \oplus }$ denotes the Pythagorean addition operation.^[1]

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 $${\displaystyle E=mc^{2}\oplus pc.}$$ 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;^[2] it is related to the quadratic mean or "root mean square".

Pythagorean addition (and its implementation as the hypot function) is often used together with the atan2 function to convert from Cartesian coordinates ${\displaystyle (x,y)}$ to polar coordinates ${\displaystyle (r,\theta )}$:^[3][4] $${\displaystyle {\begin{aligned}r&=x\oplus y=\operatorname {hypot} (x,y)\\\theta &=\operatorname {atan2} (y,x).\\\end{aligned}}}$$

Repeated Pythagorean addition can find the diameter of a rectangular cuboid. This is the longest distance between two points, the length of the body diagonal of the cuboid. For a cuboid with side lengths ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$, this length is ${\displaystyle a\oplus b\oplus c}$.

If measurements ${\displaystyle X,Y,Z,\dots }$ have independent errors ${\displaystyle \Delta _{X},\Delta _{Y},\Delta _{Z},\dots }$ respectively, the quadrature method gives the overall error, $${\displaystyle \varDelta _{o}={\sqrt {{\varDelta _{X}}^{2}+{\varDelta _{Y}}^{2}+{\varDelta _{Z}}^{2}+\cdots }}}$$ whereas the upper limit of the overall error is $${\displaystyle \varDelta _{u}=\varDelta _{X}+\varDelta _{Y}+\varDelta _{Z}+\cdots }$$ if the errors were not independent.^[5]

In signal processing, addition in quadrature is used to find the overall noise from independent sources of noise. For example, if an image sensor 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 $${\displaystyle \sigma =6\oplus 3\oplus 2={\sqrt {6^{2}+3^{2}+2^{2}}}=7}$$ digital numbers,^[6] showing the dominance of larger sources of noise.

The root mean square of a finite set of ${\displaystyle n}$ numbers is ${\displaystyle {\tfrac {1}{\sqrt {n}}}}$ times their Pythagorean sum. This is a generalized mean of the numbers.

The operation ${\displaystyle \oplus }$ is associative and commutative,^[7] and $${\displaystyle {\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}=x_{1}\oplus x_{2}\oplus \cdots \oplus x_{n}.}$$ This means that the real numbers under ${\displaystyle \oplus }$ form a commutative semigroup.

The real numbers under ${\displaystyle \oplus }$ are not a group, because ${\displaystyle \oplus }$ can never produce a negative number as its result, whereas each element of a group must be the result of applying the group operation to itself and the identity element. On the non-negative numbers, it is still not a group, because Pythagorean addition of one number by a second positive number can only increase the first number, so no positive number can have an inverse element. Instead, it forms a commutative monoid on the non-negative numbers, with zero as its identity.

Hypot is a mathematical function defined to calculate the length of the hypotenuse of a right-angle triangle. It was designed to avoid errors arising due to limited-precision calculations performed on computers. Calculating the length of the hypotenuse of a triangle is possible using the square root function on the sum of two squares, but hypot avoids problems that occur when squaring very large or very small numbers. If calculated using the natural formula, $${\displaystyle r={\sqrt {x^{2}+y^{2}}},}$$ the squares of very large or small values of ${\displaystyle x}$ and ${\displaystyle y}$ may exceed the range of machine precision when calculated on a computer, leading to an inaccurate result caused by arithmetic underflow and overflow. The hypot function was designed to calculate the result without causing this problem.^[8]

If either input to hypot is infinite, the result is infinite. Because this is true for all possible values of the other input, the IEEE 754 floating-point standard requires that this remains true even when the other input is not a number (NaN).^[9]

Since C++17, there has been an additional hypot function for 3D calculations:^[10] $${\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}.}$$

• Euclidean distance
• Alpha max plus beta min algorithm