Math 2.0

statement proposition

A complex number times its conjugate

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.

Explanations

L1 human

When you multiply a complex number by its conjugate, the imaginary parts cancel. What remains is the square of the length of the number, for example (3+4i)(3−4i)=9+16=25(3 + 4 i)(3 - 4 i) = 9 + 16 = 25.

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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
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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

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Sources

  1. Web https://en.wikipedia.org/wiki/Complex_conjugate, Properties

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