$i^i$

$i$ is that weird number whose square is $-1$. Without Googling, can you calculate $i^i$? Yes, $i$ to cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 $i$. 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 cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 most beautiful equation in all of macá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365matics:

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

I don't even know how such an equation is possible. $i$ looks man-made, $\pi$ and $e$ are natural, but togecá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365r with $0$ and $1$ cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365y are cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 most important constants, and somehow after an addition, a multiplication, and an exponentiation, all of which are basic arithmetic operations, cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365y all fit togecá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365r. They all line up as if someone creates cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365m exactly for this and only this equation.



The equation is called Euler's identity. Some fun facts about Euler, extracted from cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 above video (which is great, you should watch it!)

- In 1988, a math magazine ran a poll to vote for cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 top 10 most beautiful results in all of macá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365matics. 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 cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 first volume in 1911. They still are not done. The grandchildren of cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 first editors are getting old.

- Euler produced on average a paper per week in cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 year of 1775. He was essentially blind since 1771.

- When Euler died cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365re 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 cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 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.

Comments

Thuong said…
The post reminds me something about cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 transcendental numbers, but I could not remember where I had read about this. Now, cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365re is no doubt that i^i is transcendental number. See cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 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:

http://euclid.colorado.edu/~tubbs/courses/courses.html