Complex Number Problem #1

I’ve come across many questions from students regarding complex numbers, and here is one thats quite fun to deal with!

$\sqrt{-1}^{\sqrt{-1}}$

Now this is really interesting, but can be solved with a bit of manipulation. We all know this is akin to $i^i$

We know that $\sqrt{-1}=e^{i\frac{\pi}{2}}$

$\sqrt{-1}^{\sqrt{-1}}$

$= e^{i\frac{\pi}{2}\sqrt{-1}}$

$= e^{i\frac{\pi}{2}i}$

$= e^{i^2\frac{\pi}{2}}$

$= e^{-\frac{\pi}{2}}$

Now our end result is a real number!

Like mentioned with regards to Euler’s Identity, it is really very amazing how complex numbers actually work!

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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Complex Number Problem #2 – The Culture
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February 3rd, 2016 09:47 PM

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