File:ComplexUnitRoots.svg
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Captions
Captions
Complex unit roots
Summary[edit]
| Description |
Dansk: Løsningerne til ligningen zⁿ = 1 for n = 1, 2, 3, 4, 5 og 6 ligger alle på de viste enhedscirkler. De kaldes de komplekse enhedsrødder af grad n. Tallet 1 (markeret med en grøn prik) er en triviel løsning i alle tilfældene. De komplekse tal ε₂ = –1, ε₃ = –1/2 + √3/2·i, ε₄ = i, ε₅ = (√5 – 1)/4 + √(10 + 2√ 5)/4·i og ε₆ = √3/2+ 1/2·i er markeret med røde prikker. De og deres potenser (markeret med sorte prikker) er de komplekse enhedsrødder af grader mellem 2 og 6.
English: The solutions to the equation zⁿ = 1 for n = 1, 2, 3, 4, 5 and 6 all lie on the unit circles shown. They are designated complex unit roots of degree n. The number 1 (marked with a green dot) is a trivial solution in all cases. The complex numbers ε₂ = –1, ε₃ = –1/2 + √3/2·i, ε₄ = i, ε₅ = (√5 – 1)/4 + √(10 + 2√ 5)/4·i and ε₆ = √3/2+ 1/2·i are identified by red dots. They and their powers (marked with black dots) are the complex unit roots of degrees between 2 and 6.
Deutsch: Die Lösungen der Gleichung zⁿ = 1 für n = 1, 2, 3, 4, 5 und 6 liegen alle auf den gezeigten Einheitskreisen. Sie werden komplexe Einheitswurzeln des Grades n genannt. Die Zahl 1 (mit einem grünen Punkt markiert) ist in allen Fällen eine triviale Lösung. Die kompleksen Zahlen ε₂ = –1, ε₃ = –1/2 + √3/2·i, ε₄ = i, ε₅ = (√5 – 1)/4 + √(10 + 2√ 5)/4·i und ε₆ = √3/2+ 1/2·i sind mit roten Punkten markiert. Sie und ihre Potenzen (mit schwarzen Punkten markiert) sind die komplexen Einheitswurzeln von Graden zwischen 2 und 6. |
| Date | |
| Source | Own work |
| Author | AstroOgier |
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 14:19, 18 December 2019 | | 331 × 638 (89 KB) | AstroOgier (talk | contribs) | User created page with UploadWizard |
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