Pythagorean addition

In mathematics, Pythagorean addition is the following binary operation on the real numbers:

\( a\oplus b={\sqrt {a^{2}+b^{2}}}. \)

The name recalls the Pythagorean theorem, which states that the length of the hypotenuse of a right triangle is a ⊕ b, where a and b are the lengths of the other sides.

This operation provides a simple notation and terminology when the summands are complicated; for example, the energy-momentum relation in physics becomes

( E=mc^{2}\oplus pc. \)

Properties

The operation ⊕ is associative and commutative, and

( {\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}=x_{1}\oplus x_{2}\oplus \cdots \oplus x_{n}. \)

This is enough to form the real numbers into a commutative semigroup. However, ⊕ is not a group operation for the following reasons.

The only element which could potentially act as an identity element is 0, since an identity e must satisfy e⊕e = e. This yields the equation( {\sqrt {2}}e=e \), but if e is nonzero that implies ( {\sqrt {2}}=1 \), so e could only be zero. Unfortunately 0 does not work as an identity element after all, since 0⊕(−1) = 1. This does indicate, however, that if the operation ⊕ is restricted to nonnegative real numbers, then 0 does act as an identity. Consequently, the operation ⊕ acting on the nonnegative real numbers forms a commutative monoid.
See also

Euclidean distance
Hypot function
Alpha max plus beta min algorithm
Metafont has Pythagorean addition and subtraction as built-in operations, under the names ++ and +-+ respectively.