# Pythagorean addition

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

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

## Properties

The operation ⊕ is associative and commutative, and

- .

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 , but if *e* is nonzero that implies , 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

## 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..