Pythagorean addition

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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 ab, 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


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 ee = 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.

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