Math 2.0

Functions of a complex variable Definition

Argument of a complex number

Builds on Complex numbersModulus of a complex number

Definition

Let zz be a complex number with z≠0z \neq 0, and let ∣z∣|z| be its modulus. An argument of zz is a real number θ\theta with z=∣z∣(cos⁡θ+isin⁡θ).z = |z| (\cos \theta + i \sin \theta). The notation arg⁡z\arg z means the set of all arguments of zz, or one chosen argument.

Remarks

If θ\theta is an argument of zz, then the arguments of zz are exactly the numbers θ+2πk\theta + 2 \pi k with k∈ℤk \in \mathbb{Z}. The number 00 has no argument in this sense. Some conventions set arg⁡0=0\arg 0 = 0. For example, the arguments of ii are π/2+2πk\pi/2 + 2 \pi k, and those of −1-1 are π+2πk\pi + 2 \pi k.

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Complex numbers, object, no evidence levelComplex numbersobjectReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness 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…definition
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  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Real and imaginary parts definition
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  7. Existence of square roots of nonnegative reals statement
  8. Square root definition
  9. Modulus of a complex number definition
  10. Argument of a complex number definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Argument_(complex_analysis), Definition
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Tag0ZZRSX91QR
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Kinddefinition
WritingW1 stated
MSC30A99
DefinesArgument of a complex number
UsesComplex numbers, Modulus of a complex number

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