Math 2.0

Functions of a complex variable Proposition

A complex number times its conjugate

Builds on Complex conjugateModulus of a complex number

Statement

Let zz be a complex number, let zˉ\bar{z} be its conjugate and let ∣z∣|z| be its modulus. Then zzˉ=∣z∣2.z \bar{z} = |z|^2. In particular, zzˉz \bar{z} is a nonnegative real number.

Remarks

With z=a+biz = a + b i: (a+bi)(a−bi)=a2−(bi)2=a2+b2(a + b i)(a - b i) = a^2 - (b i)^2 = a^2 + b^2. This identity gives the formula for the inverse of a complex number. It also gives a quick proof that the modulus is multiplicative.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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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 …definitionA complex number times its conjugate, statement, no evidence levelA complex number time…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.
  • no evidence level
  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. A complex number times its conjugate statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Complex_conjugate, Properties
About this page
Tag0ZB7J071WQ
URIm2://0ZB7J071WQ
Kindstatement, proposition
WritingW0 stub
MSC30A99
StatusThis copy of the site has no build data. Run m2 status 0ZB7J071WQ.
UsesComplex conjugate, Modulus of a complex number

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