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The natural logarithm of -1

Whilst going over old material, I remember a friend had told me something that was I found quite interesting a while ago about logarithms, and it's troubled me enough for the past few minutes for me to come to fellow mathematics enthusiasts for some insight regarding this conundrum.

If you were to enable imaginary numbers on a calculator, or even do this out by hand, it's found that:

\(ln { (-1) } =\pi i\)

I have a good understanding of logarithms to know what this technically means, but I'm curious as to why it is.

How come \({ e }^{ \pi i } = { (-1) }\) ? How does the ratio of the circumference of a circle to it's diameter and a complex number come into play in exponential and logarithmic functions?

Note by Andrew Tawfeek
2 years, 2 months ago

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Hmm , complex numbers is not my forte too .

I'm sure you must be knowing that \(e^{ix} = \cos x + i \sin x\) . So replace \(x\) by \(\pi\) ,to get your answer . Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M This is actually my first time hearing of that identity! Is that where the Eulurs number came from? Andrew Tawfeek · 2 years, 2 months ago

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@Andrew Tawfeek Do you know MacLaurin's expansion? You may use it to get that identity. Pranjal Jain · 2 years, 2 months ago

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@Pranjal Jain This is my first time hearing of it, I'll be sure to look into it, it looks really interesting! Andrew Tawfeek · 2 years, 2 months ago

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Complex logarithms are actually periodic with period \(2\pi\) Joel Tan · 2 years, 2 months ago

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Khanacademy teaches this part well. On how the Euler identity comes about. Look up a few of the videos. Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan What all topics did you learn from Khan Academy ? Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M idk... This was a long time ago, and one of the first thing i learnt from there. Afterwards I did a lot of organic chemistry. But that's it. Haven't logged onto it for many months. Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan I didn't know that they had Org Chem videos , I just learned Math from there . Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M Chem is good, but not good enough for JEE, or even board. Math class at our coaching institute is very good, so I didn't have to look to other sources. I finally had to learn all of chemistry on my own though.Hey, I you got my email id right? If not: raghav1900@gmail.com Raghav Vaidyanathan · 2 years, 2 months ago

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@Deepanshu Gupta ,@Raghav Vaidyanathan ,@Shashwat Shukla ,@Mvs Saketh

What are your views on a possible geometric interpretation ? Azhaghu Roopesh M · 2 years, 2 months ago

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