Math 2.0

Functions of a complex variable Proof

Proof that the modulus of the conjugate equals the modulus

Builds on Modulus of a complex numberComplex conjugateReal and imaginary parts

Proof

  1. Let z=a+biz = a + b i with aa and bb real.
  2. By the definition of the complex conjugate, zˉ=a+(−b)i\bar{z} = a + (-b) i. So Re⁡zˉ=a\operatorname{Re} \bar{z} = a and Im⁡zˉ=−b\operatorname{Im} \bar{z} = -b, by the definition of the real and imaginary parts.
  3. By the definition of the modulus, ∣zˉ∣=a2+(−b)2|\bar{z}| = \sqrt{a^2 + (-b)^2}.
  4. Since (−b)2=b2(-b)^2 = b^2, this is a2+b2\sqrt{a^2 + b^2}, which is ∣z∣|z| by the same definition. So ∣zˉ∣=∣z∣|\bar{z}| = |z|.

Idea

Conjugation changes the sign of the imaginary part only. The modulus depends on the square of that part, so the sign does not matter.

Reading path

To understand this page from the ground up, read these 39 pages in order. The foundations come first. The path takes the proofs Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the existence of square roots of nonnegative reals.

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Complex numbers, object, no evidence levelComplex numbersobjectConstruction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRational numbers, object, no evidence levelRational 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…definitionSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionThe reals form a field, statement, no evidence levelThe reals form a fieldstatementSupremum, definition, no evidence levelSupremumdefinitionComplex conjugate, definition, no evidence levelComplex conjugatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementBinary relation, definition, no evidence levelBinary relationdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionTotal order, definition, no evidence levelTotal orderdefinitionProof that the reals form an ordered field, with the Cauchy sequence construction, proof, no evidence levelProof that the reals …proofThe reals form an ordered field, statement, no evidence levelThe reals form an ord…statementProof of the existence of square roots of nonnegative reals, proof, no evidence levelProof of the existenc…proofExistence 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 …definitionProof that the modulus of the conjugate equals the modulus, proof, no evidence levelProof that the modulu…proof
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.
  • no evidence level
  1. Complex numbers object
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Integers object
  5. Ordered field definition
  6. Proposition concept
  7. Rational numbers object
  8. Real numbers object
  9. Set bounded above definition
  10. Real and imaginary parts definition
  11. Set definition
  12. Ring of integers statement
  13. Implication definition
  14. Predicate definition
  15. The reals form a field statement
  16. Supremum definition
  17. Complex conjugate definition
  18. Ordered pair definition
  19. The rationals form a field statement
  20. Universal quantifier definition
  21. Completeness axiom statement
  22. Cartesian product definition
  23. Subset definition
  24. The rationals form an ordered field statement
  25. Binary relation definition
  26. Order on the reals definition
  27. Antisymmetric relation definition
  28. Reflexive relation definition
  29. Transitive relation definition
  30. Preorder definition
  31. Partial order definition
  32. Total order definition
  33. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  34. The reals form an ordered field statement
  35. Proof of the existence of square roots of nonnegative reals proof
  36. Existence of square roots of nonnegative reals statement
  37. Square root definition
  38. Modulus of a complex number definition
  39. Proof that the modulus of the conjugate equals the modulus proof

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Complex_conjugate, Properties
About this page
TagNC1THAZVY9
URIm2://NC1THAZVY9
Kindproof
WritingW0 stub
MSC30A99
ProvesModulus of the conjugate
UsesModulus of a complex number, Complex conjugate, Real and imaginary parts
Changed4 November 2026

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