Math 2.0

Functions of a complex variable Proof

Proof of the formula for the modulus

Builds on Modulus of a complex numberReal and imaginary parts

Proof

  1. Let xx and yy be real numbers, and let z=x+yiz = x + y i.
  2. Since xx and yy are real, the definition of the real and imaginary parts gives Re⁡z=x\operatorname{Re} z = x and Im⁡z=y\operatorname{Im} z = y.
  3. By the definition of the modulus, ∣z∣=(Re⁡z)2+(Im⁡z)2|z| = \sqrt{(\operatorname{Re} z)^2 + (\operatorname{Im} z)^2}.
  4. Substituting step 2 into step 3 gives ∣x+yi∣=x2+y2|x + y i| = \sqrt{x^2 + y^2}.

Idea

The modulus is defined through the real and imaginary parts. When xx and yy are real, these parts of x+yix + y i are simply xx and yy.

Reading path

To understand this page from the ground up, read these 38 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 levelSupremumdefinitionOrdered 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 of the formula for the modulus, proof, no evidence levelProof of the formula …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. Ordered pair definition
  18. The rationals form a field statement
  19. Universal quantifier definition
  20. Completeness axiom statement
  21. Cartesian product definition
  22. Subset definition
  23. The rationals form an ordered field statement
  24. Binary relation definition
  25. Order on the reals definition
  26. Antisymmetric relation definition
  27. Reflexive relation definition
  28. Transitive relation definition
  29. Preorder definition
  30. Partial order definition
  31. Total order definition
  32. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  33. The reals form an ordered field statement
  34. Proof of the existence of square roots of nonnegative reals proof
  35. Existence of square roots of nonnegative reals statement
  36. Square root definition
  37. Modulus of a complex number definition
  38. Proof of the formula for the modulus proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Absolute_value, Complex numbers
About this page
TagR7EVNDA9S9
URIm2://R7EVNDA9S9
Kindproof
WritingW0 stub
MSC30A99
ProvesFormula for the modulus
UsesModulus of a complex number, Real and imaginary parts
Changed4 November 2026

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