Find the truth gets simpler and complex

Algebra Level 4

Which of the following are always true?

a) $i^x$ is imaginary if $x$ is imaginary.
b) $x^i$ is imaginary if $x$ is imaginary.
c) $x^x$ is imaginary if $x$ is imaginary.

This is a part of set Find the Truth .

c

only None All of these choices

a

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c

b

only

b

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c

a

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a

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b

2 solutions

Ivan Koswara
Mar 19, 2015

We claim that $i^i$ is real (and thus not imaginary).

By Euler's formula, $i = \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} = e^{\frac{\pi}{2} i}$ . Thus, $i^i = \left( e^{\frac{\pi}{2} i} \right)^i = e^{\frac{\pi}{2} i \cdot i} = e^{-\frac{\pi}{2}}$ . A positive real number raised to a real number is also a real number, so $i^i = e^{-\frac{\pi}{2}}$ is a real number.

This proves that all three statements are false, by taking $x=i$ .

This is the perfect solution I was waiting for!

Archit Boobna - 6 years, 2 months ago

Parth Bhardwaj
Mar 17, 2015

If we take x=i in all cases then we get all the terms as i^i which is a real number approximately =.207, thus the answer is real.

Oh ho... Wrong option clicked

Mehul Arora - 6 years, 2 months ago

Find the truth gets simpler and complex

This is a part of set Find the Truth .

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