Math 2.0

Functions of a complex variable Proposition

Formula for the modulus

Builds on Modulus of a complex number

Statement

Let xx and yy be real numbers. Then the modulus of the complex number x+yix + y i is ∣x+yi∣=x2+y2.|x + y i| = \sqrt{x^2 + y^2}.

Remarks

This follows from the definition of the modulus, because Re⁡(x+yi)=x\operatorname{Re}(x + y i) = x and Im⁡(x+yi)=y\operatorname{Im}(x + y i) = y for real xx and yy. The hypothesis that xx and yy are real is necessary. For example, ∣i+i⋅i∣=∣−1+i∣=2|i + i \cdot i| = |-1 + i| = \sqrt{2}, but i2+i2=−2\sqrt{i^2 + i^2} = \sqrt{-2} is not even real.

Reading path

To understand this page from the ground up, read these 10 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 levelSupremumdefinitionCompleteness 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 …definitionFormula for the modulus, statement, no evidence levelFormula for the modul…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.
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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. Completeness axiom statement
  7. Existence of square roots of nonnegative reals statement
  8. Square root definition
  9. Modulus of a complex number definition
  10. Formula for the modulus statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Absolute_value, Complex numbers
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Tag034FGCEYX4
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Kindstatement, proposition
WritingW0 stub
MSC30A99
StatusThis copy of the site has no build data. Run m2 status 034FGCEYX4.
UsesModulus of a complex number

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