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## Sunday, January 10, 2016

### $i^i$

$i$ is that weird number whose square is $-1$. Without Googling, can you calculate $i^i$? Yes, $i$ to the $i$. Does it even make sense?

No, it didn't for me, until I found http://betterexplained.com.

There are a lot of great articles on this site, but I especially love two series: http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ and http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/.

This is how math should be taught in high school. Would love to hear your opinions if you disagree.

If you go down this road as I just did eventually you would come up with the most beautiful equation in all of mathematics:

$e^{i * \pi} + 1 = 0$

I don't even know how such an equation is possible. $i$ looks man-made, $\pi$ and $e$ are natural, but together with $0$ and $1$ they are the most important constants, and somehow after an addition, a multiplication, and an exponentiation, all of which are basic arithmetic operations, they all fit together. They all line up as if someone creates them exactly for this and only this equation.

The equation is called Euler's identity. Some fun facts about Euler, extracted from the above video (which is great, you should watch it!)

- In 1988, a math magazine ran a poll to vote for the top 10 most beautiful results in all of mathematics. Euler's identity is #1 and Euler's formula (V + F = E + 2) #2.

- Euler's work totals over 75 volumes and 25,000 pages. The Swiss Academy published the first volume in 1911. They still are not done. The grandchildren of the first editors are getting old.

- Euler produced on average a paper per week in the year of 1775. He was essentially blind since 1771.

- When Euler died there was a backlog of his works that were not yet published. So after he died, he published 228 papers. That's more than most of us will ever publish alive!

Euler is impossible.

By the way $i^i \approx 0.2$. Does it blow your mind that raising an imaginary number to an imaginary exponentiation returns a real number? Am I imagining? No, it's real.

#### 1 comment:

Thuong said...

The post reminds me something about the transcendental numbers, but I could not remember where I had read about this. Now, there is no doubt that i^i is transcendental number. See the link:

https://en.wikipedia.org/wiki/Gelfond%27s_constant.

I haven't read any proof for this, but my guess is that it is difficult. A useful link if you are curious about this kind of numbers: