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