Math 2.0

Functions of a complex variable Proposition

Modulus is multiplicative

Builds on Modulus of a complex numberA complex number times its conjugate

Statement

For all complex numbers zz and ww, ∣zw∣=∣z∣ ∣w∣,|z w| = |z| \, |w|, where ∣⋅∣|\cdot| is the modulus.

Remarks

Proof: ∣zw∣2=zwzw‾=(zzˉ)(wwˉ)=∣z∣2∣w∣2|z w|^2 = z w \overline{z w} = (z \bar{z})(w \bar{w}) = |z|^2 |w|^2, and both sides are nonnegative. For z=a+biz = a + b i and w=c+diw = c + d i, this is the identity (a2+b2)(c2+d2)=(ac−bd)2+(ad+bc)2(a^2 + b^2)(c^2 + d^2) = (a c - b d)^2 + (a d + b c)^2. On the real numbers, it gives that the absolute value is multiplicative.

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To understand this page from the ground up, read these 12 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…statementModulus is multiplicative, statement, no evidence levelModulus is multiplica…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. 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
  12. Modulus is multiplicative statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Absolute_value, Complex numbers
About this page
Tag0HYNZE940T
URIm2://0HYNZE940T
Kindstatement, proposition
WritingW0 stub
MSC30A99
StatusThis copy of the site has no build data. Run m2 status 0HYNZE940T.
UsesModulus of a complex number, A complex number times its conjugate

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