Imaginary number -i raised to negative power

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Homework Help Overview

The discussion revolves around evaluating the expression involving the imaginary unit, specifically the term 1 - (-i)^-4. Participants are exploring how to handle the negative sign associated with the imaginary unit and its implications in the calculation.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are questioning how to treat the negative sign in the expression involving -i, with some suggesting to distribute the power and others considering the properties of i and -i. There is also a discussion about the relationship between i and -i as reciprocals.

Discussion Status

Several participants have offered insights into the evaluation process, with some providing steps that lead to the conclusion that the expression simplifies to 0. However, there is no explicit consensus on the approach to take regarding the negative sign in the context of the imaginary unit.

Contextual Notes

Participants are grappling with the treatment of negative powers and the implications of the negative sign in the context of imaginary numbers. There is an underlying assumption that the rules of exponents apply similarly to complex numbers as they do to real numbers.

CE Trainee
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Homework Statement



I came across this expression in homework and for the life of me I can't figure out how this evaluates to 0: 1 - ( -i )^-4 = 1 - 1 = 0

I know that i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. I'm just not sure how to treat the negative on the i. Do I just treat i as if it were just a regular number, i.e. (-1)^4 = 1?
Or can I just say ( -i )*( -i )*( -i )*( -i ) = 1? Can anyone shed some light on this. I know it has to be painfully simple but for some reason I just can't see it.

Thanks
 
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CE Trainee said:

Homework Statement



I came across this expression in homework and for the life of me I can't figure out how this evaluates to 0: 1 - ( -i )^-4 = 1 - 1 = 0

I know that i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. I'm just not sure how to treat the negative on the i. Do I just treat i as if it were just a regular number, i.e. (-1)^4 = 1?
Or can I just say ( -i )*( -i )*( -i )*( -i ) = 1? Can anyone shed some light on this. I know it has to be painfully simple but for some reason I just can't see it.

Thanks
Distribute the power:
(-i)^4 = (-1)^4i^4
 
i(-i)= -i^2= -(-1)= 1 so i and -i are reciprocals. In particular, (-i)^{-1}= i and so (-i)^{-4}= i^4= 1.
 
it is very easy problem. See,

<br /> <br /> 1 - i^{-4}

= 1 - \frac {1}{i^4}

= 1 - \frac {1}{1}

= 1 - 1

= 0
 

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