Math 2.0

Functions of a complex variable Proposition

Modulus of the conjugate

Builds on Modulus of a complex numberComplex conjugate

Statement

For each complex number zz, ∣zˉ∣=∣z∣|\bar{z}| = |z|. Here zˉ\bar{z} is the complex conjugate of zz and ∣⋅∣|\cdot| is the modulus.

Remarks

Proof: for z=a+biz = a + b i, ∣zˉ∣=a2+(−b)2=a2+b2=∣z∣|\bar{z}| = \sqrt{a^2 + (-b)^2} = \sqrt{a^2 + b^2} = |z|. In the complex plane, this says that a reflection in the real axis does not change the distance to 00.

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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 levelSupremumdefinitionComplex conjugate, definition, no evidence levelComplex conjugatedefinitionCompleteness 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 …definitionModulus of the conjugate, statement, no evidence levelModulus of the conjug…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Complex numbers object
  2. Real numbers object
  3. Set bounded above definition
  4. Real and imaginary parts definition
  5. Supremum definition
  6. Complex conjugate definition
  7. Completeness axiom statement
  8. Existence of square roots of nonnegative reals statement
  9. Square root definition
  10. Modulus of a complex number definition
  11. Modulus of the conjugate statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Complex_conjugate, Properties
About this page
Tag00KRCVE7CK
URIm2://00KRCVE7CK
Kindstatement, proposition
WritingW0 stub
MSC30A99
StatusThis copy of the site has no build data. Run m2 status 00KRCVE7CK.
UsesModulus of a complex number, Complex conjugate

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