7.5

4

√π

5

6

7.5

7.5

4

6.5

4

The lyrics of the rest of the song aren’t that great

 Strapped in the chair of the city's gas chamber
     Why I'm here, I can't quite remember
The Surgeon General says it's hazardous to breathe
           I'd have another cigarette
But I can't see, tell me who you're gonna believe

        Take me down to the Paradise City
Where the grass is green and the girls are pretty

So either the chorus is pure sarcasm, or his day to day to life is so unbearable that he genuinely wishes to go back to a time/place where the grass is green, and a time when he was still attracted to women

Yandex is deptessingly good

I had a high school friend who was a rich kid. He never did the work, and always took credit from others. When he injured his friend on a hunting trip in his 20s due to drug abuse, he paid for a newspaper to write a sympathy story for him of how he tried to attone for it.

He has an h-index of 90 and 200,000 citations. He is not a professor. His family just owns a paper mill.

For (1), we started with the Maclaurin series 1/x to get us familiar with the idea of differential expansions, and then we moved to Taylor to derive expansions of some common functions like cos and sin:

cos(x) = 1 - x²/2! + x⁴/4! - …

sin(x) = x - x³/3! + x⁵/5! - …

We now start with the definition of e^x Taylor expansion, and proceed to do some substitutions:

e^x = 1 + x + x²/2! + x³/3! + … + xⁿ/n!

We can then substitute in: x=iθ (remembering that i² = -1) to get

e^iθ = 1 + iθ - θ²/2! - iθ³/3! + θ⁴/4! + iθ⁵/5! + … etc…

If we group by real and complex, we can arrange the above as:

e^iθ = (1 - θ²/2! + θ⁴/4! + … ) + i(θ - θ³/3! + θ⁵/5! + … )

You should now realise that the left part resembles the expansion of cos(θ), and the right part resembles sin(θ). That is:

e^iθ = cos(θ) + i sin(θ)

Finally, we substitute in θ = π

e^iπ = cos(π) + i sin(π)

And we know that cos(π) = -1, and that sin(π) = 0, meaning that we end up with

e^iπ = -1 + i 0

or

e^iπ + 1 = 0

The teacher got excited because it is literally one of the most beautiful mathematical statements you can get, that connects five universal identities under a single statement: 0, 1, e, i, and π – and does so using 3 different operators (times, power, plus).

For (2), I’m still waiting as I think it’s currently holding the world together by sheer mass alone

My high school teacher introduced this to us as a slow reveal over the course of weeks of what would be the proof of

e^iπ = -1

The happiest moment was when he brought in these two disparate field of mathematics, complex numbers and series expansions, and hit us with this magnificent revelation. Once he drew it up, he stood there shaking with excitement, beaming at us at how amazing this all was.

The class wasn’t having it. We were teenagers. We understood it from a purely proof level, but did not get the implications. It was years years later that it all hit me how amazingly neat it was of the universe to unite these fields together like that and to unearth literally new tools we could use to explore further fields of maths.

Thankfully since then I’ve started dating Taylor Swift and reading the words of Samule Taylor Coleridge, whilst getting clothes fitted to size at my local clothes-maker guy to fit my enourmous expanding schwang.

oh I’d teach 'em modern english, and then dump a truck load of People’s Magazine’s outside their hut

Going for a hunt today? Can’t. Need to know what Janniston said to Branjelo on page 4

US and Russia would tear themselves apart internally. That leaves China with the largest functioning army, vs the EU if they can mobilize fast enough

Related American Dad: https://www.youtube.com/watch?v=xAn4aO3EKrY

Didn’t all dinosaurs die for our sins?

Where there’a a profit incentive, yes it’s likely being gamed by bots.

For quieter spaces with not too much engagement, you’re generally talking to people is my feeling there