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 then is a contradiction, 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

Further reading

Please expand this article. Some suggested sources are given below. More information might be found in a section of the talk page. (February 2012)

• 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. http://www.research.ibm.com/journal/rd/276/ibmrd2706P.pdf.  Try also here.
• 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. http://www.research.ibm.com/journal/rd/276/ibmrd2706Q.pdf.  Try also here.