The unreal world -'1'

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Evaluate i − 1 i^{ -1 }

-1 -i 1 i

Dinesh Nath Goswami by Dinesh Nath Goswami , 20, India

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2 solutions

i − 1 = 1 i = 1 i × i 3 i 3 = i 3 i 4 = i 3 [ ∵ i 4 = 1 ] = − i { i }^{ -1 }=\frac { 1 }{ i } =\frac { 1 }{ i } \times \frac { { i }^{ 3 } }{ { i }^{ 3 } } =\frac { { i }^{ 3 } }{ { i }^{ 4 } } ={ i }^{ 3 }\quad \quad \left[ \because \quad { i }^{ 4 }=1 \right] \\ =-i

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Why multiply numerator and denominator by i 3 i^3 when you can get the same answer by multiplying numerator and denominator by i i ?

i − 1 = 1 i = i i 2 = i − 1 = ( − i ) i^{-1}=\frac 1i=\frac i{i^2}=\frac i{-1}=(-i)

There are many ways to get the answer, some methods being complete overkill, for example, one can use Euler's Theorem here:

i − 1 = ( e π / 2 ) − 1 = e − π / 2 = cos ⁡ ( − π / 2 ) + i sin ⁡ ( − π / 2 ) = 0 + ( − 1 ) i = ( − i ) i^{-1}=\left(e^{\pi/2}\right)^{-1}=e^{-\pi/2}=\cos(-\pi/2)+i\sin(-\pi/2)=0+(-1)i=(-i)

Prasun Biswas - 6 years ago

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Yes, there can be many alternative solutions

Dinesh Nath Goswami - 6 years ago
Sai Raghava Puni
Jul 7, 2015

its 1/i=-i

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