Inequalities between a real number and an imaginary number

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Homework Help Overview

The discussion revolves around the comparison of a real number and an imaginary number, specifically the expression \(\sqrt{2}i \leq \sqrt{2}\). Participants are exploring the implications of this inequality and the definitions involved in comparing complex numbers.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to interpret the meaning of the inequality involving a complex number and questions its validity. Some participants suggest that without a clear definition of the inequality for complex numbers, the comparison may not hold meaning. Others propose considering absolute values as a possible approach.

Discussion Status

The discussion is active, with participants providing insights and questioning the assumptions behind the comparison of complex and real numbers. There is acknowledgment of the need for clarity regarding the definition of inequalities in this context, and the original poster indicates a potential error in a previous step of their problem.

Contextual Notes

Participants note that the set of complex numbers is not an ordered field, which raises questions about the appropriateness of using inequalities in this scenario. The original poster also mentions a mistake in an earlier part of their problem, suggesting that further clarification may be needed.

phosgene
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Homework Statement



I'm just having a problem with a step that's part of a larger problem. Specifically, if I have:

[itex]\sqrt{2}i\leq\sqrt{2}[/itex]

I'm not sure what this actually means. If I ignore the i, each side is the same distance from the origin if I imagine both points on a graph, implying that both sides are equal. But I don't know whether this is a correct interpretation.
 
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I don't think without a definition of ##\le## for complex numbers, that means anything, and there isn't a generally accepted one. Perhaps try to solve your problem a different way, or try to figure out what ##\le## means in the context of this problem.
 
As whovian said, the set of complex numbers is NOT an ordered field. Perhaps you intended absolute values? [itex]\left|\sqrt{2}i\right|= \sqrt{2}[/itex].
 
Sorry, yes, I forgot to include the absolute value signs! But it turns out I got a previous step in the problem wrong, anyway...so I'll just post a different topic on the whole thing. Thanks for the replies though :)
 

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