Is this too Obvious? (additive inverse)

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Homework Help Overview

The discussion revolves around the properties of complex numbers, specifically focusing on the concept of the additive inverse as presented in a Linear Algebra context. The original poster expresses uncertainty about the necessity of verifying this property, having previously accepted it as a given.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the definition of the additive inverse by expressing a complex number in terms of its real and imaginary components. There is a questioning of the verification process and its perceived simplicity.

Discussion Status

Some participants have offered insights into the representation of the additive inverse, suggesting alternative forms for clarity. There is a recognition of the straightforward nature of the proof, yet a sense of questioning remains regarding its significance.

Contextual Notes

One participant notes that visualizing the additive inverse as a vector on the Argand diagram may not be directly relevant to the problem at hand, indicating a potential divergence in focus among participants.

Saladsamurai
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Homework Statement



In my new Linear Algebra book, it is discussing the basic properties of complex numbers

It says that "you should verify, using the familiar properties of Real numbers, that addition and multiplication on C satisfy the following properties..."

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...
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additive inverse: for every z[itex]\epsilon[/itex]C, there exists a unique number w[itex]\epsilon[/itex]C such that z+w=0

I have never been asked to show this before and have always just taken it as a given.
 
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I think you can assume real numbers have additive inverses. Write z=a+bi and say what w must is.
 
Right. So let z = (a + bi) then w = -(a + bi)

That just seems silly to me. Did I really verify anything?

Anyway, thanks! :smile:
 
Saladsamurai said:
Right. So let z = (a + bi) then w = -(a + bi)

That just seems silly to me. Did I really verify anything?

Anyway, thanks! :smile:

It's bit better to write that as w=(-a)+(-b)i rather than -(a+bi). That way you've shown w is a real number plus a real number times i. I'll certainly admit it's not a difficult proof.
 
I like to think of the additive inverse of a complex number visually. If a complex number is represented as a vector on the Argand diagram, then its additive inverse is the "opposite" of that vector.
 
Yes, but that is irrelevant to this particular problem.
 

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