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| In [[mathematics]], '''Pythagorean addition''' is the following [[binary operation]] on the [[real number]]s:
| | I'm Mellissa and I live with my husband and our 2 children in S-Hertogenbosch, in the NB south part. My hobbies are Creative writing, Roller Derby and Games Club - Dungeons and Dragons, Monopoly, Etc..<br><br>Also visit my web-site - [http://www.datingsidorna.com/ kontaktannonser] |
| :<math>a \oplus b = \sqrt{a^2+b^2}.</math>
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| The name recalls the [[Pythagorean theorem]], which states that the length of the [[hypotenuse ]]of a [[right triangle]] is {{nowrap|''a'' ⊕ ''b'',}} where ''a'' and ''b'' are the lengths of the other sides.
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| This operation provides a simple notation and terminology when the summands are complicated; for example, the [[energy-momentum relation]] in [[physics]] becomes
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| :<math>E = mc^2 \oplus pc.</math>
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| ==Properties==
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| The operation ⊕ is associative and commutative, and
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| :<math>\sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} = x_1 \oplus x_2 \oplus \cdots \oplus x_n</math>.
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| This is enough to form the real numbers into a [[commutative]] [[semigroup]]. However, ⊕ is not a [[Group (mathematics)|group]] operation for the following reasons.
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| 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 <math>\sqrt{2}e=e</math>, but if ''e'' is nonzero that implies <math>\sqrt{2}=1</math>, 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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| ==See also==
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| *[[Euclidean distance]]
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| *[[Hypot]] function
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| ==Further reading==
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| *{{cite journal |author=Moler, Cleve and Donald Morrison |title=Replacing Square Roots by Pythagorean Sums |journal=IBM Journal of Research and Development |volume=27 |issue=6 |pages=577–581 |year=1983 |url=http://www.research.ibm.com/journal/rd/276/ibmrd2706P.pdf |doi=10.1147/rd.276.0577 | id = {{citeseerx|10.1.1.90.5651}} }}.
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| *{{cite journal |first=Augustin A. |last=Dubrulle |title=A Class of Numerical Methods for the Computation of Pythagorean Sums |journal=IBM Journal of Research and Development |volume=27 |issue=6 |pages=582–589 |year=1983 |url=http://www.research.ibm.com/journal/rd/276/ibmrd2706Q.pdf |doi=10.1147/rd.276.0582 | id = {{citeseerx|10.1.1.94.3443}} }}.
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| [[Category:Binary operations]]
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I'm Mellissa and I live with my husband and our 2 children in S-Hertogenbosch, in the NB south part. My hobbies are Creative writing, Roller Derby and Games Club - Dungeons and Dragons, Monopoly, Etc..
Also visit my web-site - kontaktannonser