Really?-2

Algebra Level 3

The imaginary part of ln ( 1 ) \ln(-1) is A i Ai . Find the smallest positive value of A A .


The answer is 3.141.

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2 solutions

Swagat Panda
Aug 4, 2016

( 1 ) = e ( 2 k + 1 ) i π , k Z + ln ( 1 ) = ln e ( 2 k + 1 ) i π \left( -1 \right) ={ e }^{ \left( 2k+1 \right) i\pi }, \quad k\in {Z}^{+}\Rightarrow \ln { \left( -1 \right) } =\ln { e^{ \left( 2k+1 \right) i\pi } } The first real value is at k = 0 \text{The first real value is at } k=0 ln ( 1 ) = i π A = π \Rightarrow \ln { \left( -1 \right) }=i\pi \Rightarrow \boxed{A=\pi}

z = ln ( 1 ) e z = 1 = 1 + 0 i = cos π + i sin π e z = e π i z = ln ( 1 ) = π i A = π 3.141 \begin{aligned} z & = \ln(-1) \\ e^z & = -1 \\ & = - 1 + 0i \\ & = \cos \pi + i \sin \pi \\ \implies e^z & = e^{\pi i} \\ z & = \ln(-1) = \pi i \\ \implies A & = \pi \approx \boxed{3.141} \end{aligned}

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