Math 2.0

Field theory and polynomials Proposition

Inverse of a complex number

Builds on A complex number times its conjugateThe complex numbers form a field

Statement

Let zz be a complex number with z≠0z \neq 0. Then z−1=zˉ∣z∣2.z^{-1} = \frac{\bar{z}}{|z|^2}. With z=a+biz = a + b i and a,ba, b real, this is z−1=aa2+b2−ba2+b2 iz^{-1} = \dfrac{a}{a^2 + b^2} - \dfrac{b}{a^2 + b^2} \, i.

Remarks

Proof: zzˉ=∣z∣2z \bar{z} = |z|^2 by the formula for a complex number times its conjugate, and ∣z∣2>0|z|^2 > 0 because z≠0z \neq 0. Divide by ∣z∣2|z|^2. To compute a quotient w/zw / z, multiply the numerator and the denominator by zˉ\bar{z}.

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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…definitionThe reals form a field, statement, no evidence levelThe reals form a fieldstatementSupremum, definition, no evidence levelSupremumdefinitionComplex conjugate, definition, no evidence levelComplex conjugatedefinitionThe complex numbers form a field, statement, no evidence levelThe complex numbers f…statementCompleteness 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…statementInverse of a complex number, statement, no evidence levelInverse of a complex …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. The reals form a field statement
  6. Supremum definition
  7. Complex conjugate definition
  8. The complex numbers form a field statement
  9. Completeness axiom statement
  10. Existence of square roots of nonnegative reals statement
  11. Square root definition
  12. Modulus of a complex number definition
  13. A complex number times its conjugate statement
  14. Inverse of a complex number statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Complex_number, Complex conjugate, absolute value, argument and division
About this page
Tag0FQRAV5SKH
URIm2://0FQRAV5SKH
Kindstatement, proposition
WritingW1 stated
MSC12D99
StatusThis copy of the site has no build data. Run m2 status 0FQRAV5SKH.
UsesA complex number times its conjugate, The complex numbers form a field

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