1/i= -i or i?

Thanks Max,

I will try to explain :)

 

 

\(i^0=1\\ i^1=i\\ i^2=-1\\ i^3=-i\\ i^4=1\\~\\ \text{for integer values of n the pattern can be continued}\\ i^{4n}=1\\ i^{4n+1}=i\\ i^{4n+2}=-1\\ i^{4n+3}=-i\\~\\ \text{for n=-1}\\ i^{(4*-1)+3}=i^{-1}=\frac{1}{i}=-i\)

 

Now I will try to explain your logic error when you found i^-1=i

 

When you square a negative number you get a positive number.  When you square root a positive number there are really 2 answers.  By convention we take the positive answer.

See if you can understand this with my example :)

 

\( \frac{1}{-4}=\sqrt{\frac{1}{(-4)^2}}=\frac{1}{4}\qquad \text{obviously not true} \\\frac{1}{i}=\sqrt{\frac{1}{(i)^2}}=\sqrt{\frac{1}{-1}}=\sqrt{-1}=i \qquad \text{Also not true}\)

\(\boxed{i^{4n-1} = -i}\)

\(1/i\\ =i^{4\cdot 0 - 1}\\ =-i\)

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