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

Further reading

• Moler, Cleve and Donald Morrison (1983). "Replacing Square Roots by Pythagorean Sums" (PDF). IBM Journal of Research and Development. 27 (6): 577–581. CiteSeerX 10.1.1.90.5651. doi:10.1147/rd.276.0577..
• Dubrulle, Augustin A. (1983). "A Class of Numerical Methods for the Computation of Pythagorean Sums" (PDF). IBM Journal of Research and Development. 27 (6): 582–589. CiteSeerX 10.1.1.94.3443. doi:10.1147/rd.276.0582..