Pythagorean addition

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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 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

E = mc^2 \oplus pc.


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 ee = 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.

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