Math 2.0

statement theorem

Existence of nth roots of complex numbers

Statement

Let zz be a complex number and let nn be a positive integer. Then there is a complex number ww with wn=zw^n = z.

Remarks

For z=reiθz = r e^{i \theta} with r>0r > 0, one solution is w=r1/neiθ/nw = r^{1/n} e^{i \theta / n}, by the polar form. For z≠0z \neq 0, there are exactly nn solutions. They are ww times the nnth roots of unity. This fact is a special case of the fundamental theorem of algebra.

Explanations

L1 human

Every complex number has a square root, a cube root, and so on. For example, −1-1 has the square roots ii and −i-i, and ii has the square roots ±22(1+i)\pm \frac{\sqrt{2}}{2}(1 + i). To take an nnth root, take the nnth root of the length and divide the angle by nn.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionEuler's formula, statement, no evidence levelEuler's formulastatementProduct of exponentials, statement, no evidence levelProduct of exponentia…statementReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionRoot of unity, definition, no evidence levelRoot of unitydefinitionSupremum, definition, no evidence levelSupremumdefinitionde Moivre's formula, statement, no evidence levelde Moivre's formulastatementCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionArgument of a complex number, definition, no evidence levelArgument of a complex…definitionPrincipal value of the argument, statement, no evidence levelPrincipal value of th…statementPolar form, statement, no evidence levelPolar formstatementThe nth roots of unity, statement, no evidence levelThe nth roots of unitystatementNumber of nth roots of unity, statement, no evidence levelNumber of nth roots o…statementExistence of nth roots of complex numbers, statement, no evidence levelExistence of nth root…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Euler's formula statement
  5. Product of exponentials statement
  6. Real and imaginary parts definition
  7. Root of unity definition
  8. Supremum definition
  9. de Moivre's formula statement
  10. Completeness axiom statement
  11. Existence of square roots of nonnegative reals statement
  12. Square root definition
  13. Modulus of a complex number definition
  14. Argument of a complex number definition
  15. Principal value of the argument statement
  16. Polar form statement
  17. The nth roots of unity statement
  18. Number of nth roots of unity statement
  19. Existence of nth roots of complex numbers statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Nth_root, Complex roots
  2. Web https://en.wikipedia.org/wiki/Complex_number, Powers and roots

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