Is there anything you've recently learned that you should've

I should've spent more time with my father.

>>8769954
I should have learned Euler's formula for sine and cosines

I learned that science is no less than arbitrary than any other human practice.

All trig identities are algebraic consequences of their exponential definitions

>>8769954
Yee I learned yo momma a bitch

>>8769954
grober bases

it will probably be part of the standard high school curriculum in 50 years

>>8770291
>>8770219

This. I don't know why they don't teach this back when you start trig or put more emphasis on it during calc 1 and 2.

Seriously, you can derive pretty much every fucking identity in two minutes flat or less whenever you happen to need it.

>>8769954
I recently learned that every even number greater than 2 can be expressed as the sum of two prime numbers

>>8769954
>Is there anything you've recently learned that you should've known from a young age?
Consequences for past and present; and for who we will be and will do.
Harsh that it seems, no one is innocent, only ignorant and uncivilized, to some degree. Ignorance of civility is not an excuse.

>>8769954
>Is there anything you've recently learned that you should've known from a young age?

Learning a language is good for the brain, fun, and useful.

>>8770337
Can you please explain to a brainlet what it is. Is it something like Taylor series where you have polynomial basis?

>>8770361
Archaic system of education. They should actually illustrate how and why identities exist so students can get a sense of how creativity works in math.

>>8769954
that a donkey ISNT a baby horse.
I know i'm fucking retarded

>>8769954
I learned that math is derived from applying logical steps starting from an axiom.

Suddenly it all makes sense.

>>8770361
>>8770291
What does this mean?

Basically I should have read Lockhart's Lament back in middle school instead of giving up on maths from ages 15 to 25.

>>8770361
>>8772376
true, i was always so happy and surprised whenever i found out why some formulas were the way they were and everyting became very, very simple once you created those mental connections between those formulas and other formulas and their explanations and origins

human brains learn by comparing currently existing information in the brain with newly memorized information, you use your building blocks to explain new information

"study x = something because this is how it is faggot" is a lot harder and more tedious than "x = something because of this and that, can you see the pattern?"

there are also other similar connections you can make, like, describing a shape as the intersection of some graphs and calculating the area of those subgraphs using integrals, instead of trying to memorize the formulas for areas

>>8772352
he obviously cannot if he spelled it wrong so ridiculously

>>8772490
you can find things out from their definitions using only logic. I'll show you an example with log rules.

1) let x = log(A) and y = log(B)
2) 10^x = A and 10^y = B
3) AB = 10^x10^y =10^(x+y)
4) log(AB) = x+y = log(A) + log(B)

>>8770165
That's depressing

>>8772490
Johann Bernoulli started out with some function that happened to be the derivative of arctan(x), and expanded it with partial fractions. He then told Euler about it, but he already knew it, and neither of them did anything with it. Then some guy actually evaluated the integral to get arctan(x) in terms of natural log, which lead to his formula of [math]ix=\ln(\cos(x)+i\sin(x))[/math] which is basically Euler's formula. 30 years later Euler actually needed that and derived it himself some other way, and it was just named after him for some reason after that as well (just like Euler's number itself). Eulers formula is simply [math]e^{ix}=\cos(x)+i\sin(x)[/math]. Knowing that cos(-x)=cos(x), we can rearrange this formula easily to get [math]\cos(x)=\frac {e^{ix}+e^{-ix}} {2}, \sin(x)=\frac {e^{ix}-e^{-ix}} {2i} [/math].

We can derive a couple trig identities using this. Just as an example, when trying to find the half angle sine identity, putting in 2x into that formula should tell you that you should use an exponent rule. After taking a look at it, you should already be recognizing that the original formulas are (a+b)/c and (a-b)/d, and you have a^2-b^2 in the expanded function. When you expand them out, you get sin(x)*cos(x) = sin(2x)/2, which gives you the sin(2x) half angle formula. We know that cos(2x) will equal sqrt(1-sin^2(2x)), so after enough algebraic hell, you'll get cos(2x)=cos^2(x)-sin^2(x). Almost every other trig identity can be found through this algebraic manipulation as well.

>>8769954
>What is 3.

There goes your entire "thing you learned" motherfucker

>>8771805
AFAIK this conjecture still isn't formally proven.

>>8774550
Can you not read? OP said that every positive number greater than 1 can be expressed as the product of 1 or more primes. 3 is the product of 3 which is 1 prime number.

>>8774552
>thatsthejoke.jpg

>>8774524
delet this

>>8774558
>No upper limit on the more
Pfff, why even have that as a theorem if you don't restrict yourself.

>>8774572
Reading through his actual derivation (http://www.17centurymaths.com/contents/euler/introductiontoanalysisvolone/ch8vol1.pdf ), I was surprised he didn't claim pi and name it "Euler's ratio".

Also, if you get confused at the whole [math]\displaystyle \frac {z\sqrt{-1}} {i} [/math] part, in an earlier chapter (http://www.17centurymaths.com/contents/euler/introductiontoanalysisvolone/ch7vol1.pdf ) he defined i as an "infinitely large number"

In all honesty though, I don't understand why it took him 40 years after already seeing half the proof to discover it, and I definitely don't understand why it's named after Euler specifically.

>>8774644
>40 years
30*

bump, i like this thread

>>8773523
This is why my math teachers were shit. They just tell you how to use the formula but not how the formula actually works.