Understanding the Relationship between Absolute Value and Complex Numbers

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Homework Help Overview

The discussion centers around the relationship between the absolute value (or modulus) of complex numbers and their algebraic properties, specifically examining the expression |z|^2 and its comparison to z*z and z^2.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the definitions and properties of the modulus of a complex number, questioning the relationship between |z|^2, z*z, and z^2. Some participants attempt to clarify the distinction between modulus and absolute value, while others express confusion about the implications of these relationships.

Discussion Status

The discussion is ongoing, with various interpretations being explored. Some participants have provided clarifications regarding the definitions involved, but there is no explicit consensus on the relationship between the expressions discussed.

Contextual Notes

Participants are navigating the definitions of modulus and absolute value, and there appears to be some confusion regarding the implications of these definitions in the context of complex numbers. The original poster's question remains somewhat unclear, as participants are interpreting it in different ways.

jaejoon89
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Why is |z|^2 = z*z?

z = a + ib
z*z = (a - ib)(a + ib) = a^2 + b^2
z^2 = (a + ib)^2 = a^2 + 2iab - b^2

So it must have something to do with the absolute value, but I don't understand what or why.
 
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|z| does not define the "Absolute Value" of a complex number. The notation |z| refers to the modulus of z, which is by definition

|z| = sqrt(a^2 + b^2)

Geometrically it gives the distance of the complex number from the origin on the Argand Plane.

And quite obviously |z|^2 is NOT EQUAL TO z^2
 
|z|^2 is conjugate(z)*z=a^2+b^2. It's not equal to z^2.
 
although the modulus of z when z is of the form a + 0i (i.e. it is only in the reals), then wouldn't that be essentially like an absolute value?
 
Sure. Modulus of z is |z| is sqrt(a^2+b^2). It's still not the same as z^2. What's the question again?
 

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