Math 2.0

statement proposition

Modulus of the 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.

Explanations

L1 human

A number and its conjugate are mirror images in the real axis. A mirror keeps distances, so both numbers are at the same distance from 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
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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

Sources

  1. Web https://en.wikipedia.org/wiki/Complex_conjugate, Properties

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