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Anonymous Sun 27 Mar 02:01:07 2022 No.14339310 View Reply Original Report

Quoted By: >>14339313 >>14339326 >>14339351 >>14339513 >>14339704 >>14340627 >>14340650 >>14340771 >>14341461 >>14341485 >>14341975 >>14343247 >>14343251

what does it mean?

Anonymous

Anonymous Sun 27 Mar 2022 02:02:47 No.14339313 Report

Quoted By:

>>14339310
Something something complex plane unit circle equals i^2

Anonymous

Anonymous Sun 27 Mar 2022 02:11:39 No.14339326 Report

Quoted By: >>14341618

>>14339310
It's a special case of $e^{ix} = \cos x + i \sin x$ for $x = \pi$

Anonymous

Anonymous Sun 27 Mar 2022 02:26:56 No.14339351 Report

Quoted By: >>14340549 >>14341618

>>14339310
e^ix can be used to represent a sine wave.

Anonymous

Anonymous Sun 27 Mar 2022 03:36:10 No.14339504 Report

Quoted By: >>14339534 >>14340566 >>14341544

I used to think and think and think about this, trying to wrap my brain around "raising something to the ith power"

There's a shortcut to all of it though, and it's simply understanding that all of this crap comes from derivatives. e^x is its own derivative, and sin(x) is its own 4th derivative, and that's how it makes sense.

Anonymous

Anonymous Sun 27 Mar 2022 03:39:17 No.14339513 Report

Quoted By:

>>14339310
something to the power of somethingsomething +1 anything = nothing

i dont understand how you can believe such things

Anonymous

Anonymous Sun 27 Mar 2022 03:50:08 No.14339534 Report

Quoted By:

>>14339504
(cont)

Also this formula didn't blow my mind when I first saw it. I thought it was just some mathematical convention and happens to be defined that way.
But when some dude explained to me and randomly started mentioning the Taylor series of e^x and sin(x), THAT'S when my mind was completely blown to pieces, because I saw the connection between e^x and sin(x)

it's the derivatives dude
both of them are kind of rotations dude

Anonymous

Anonymous Sun 27 Mar 2022 05:43:14 No.14339704 Report

Quoted By: >>14340549

>>14339310
-1 + 1 = 0

Anonymous

Anonymous Sun 27 Mar 2022 13:57:49 No.14340549 Report

Quoted By: >>14340638 >>14340657

>>14339351
Or a cosine wave.

>>14339704
>-1 + 1 = 0
No brainer that everyone can solve. But assume i = SQRT(-1)

Anonymous

Anonymous Sun 27 Mar 2022 14:04:10 No.14340566 Report

Quoted By: >>14341544

>>14339504
The way I had it described to me is because powers don't mathematically mean "take base number and multiply it by itself the exponent amount of times". Sure in practice that's what you're doing with integers, but the way we defined it to work with all real numbers (and complex numbers) is a different process entirely.

Anonymous

Anonymous Sun 27 Mar 2022 14:28:51 No.14340627 Report

Quoted By:

>>14339310
Two 180 degree rotations gets you back to the orientation you started with.

Anonymous

Anonymous Sun 27 Mar 2022 14:30:32 No.14340638 Report

Quoted By: >>14341658

>>14340549
A cosine wave is a sine wave, you dope.

Anonymous

Anonymous Sun 27 Mar 2022 14:33:48 No.14340650 Report

Quoted By: >>14341372

>>14339310
$e^{i\pi} = \phi - \phi^2$

Anonymous

Anonymous Sun 27 Mar 2022 14:35:50 No.14340657 Report

Quoted By: >>14341377

>>14340549
>assume i = SQRT(-1)
Wtf difference does it make whether i is sqrt(-1) or -sqrt(-1)?

Anonymous

Anonymous Sun 27 Mar 2022 15:11:05 No.14340771 Report

Quoted By:

>>14339310
Rotate vector [1 0] by pi radians and you get [-1 0]

Anonymous

Anonymous Sun 27 Mar 2022 18:25:06 No.14341372 Report

Quoted By:

>>14340650
what?

Anonymous

Anonymous Sun 27 Mar 2022 18:29:03 No.14341377 Report

Quoted By:

>>14340657
for simplification and to avoid multiple value functions

Anonymous

Anonymous Sun 27 Mar 2022 18:57:58 No.14341461 Report

Quoted By: >>14341503 >>14341559 >>14343184

>>14339310
Allow me to explain. Exponentiation means repeated multiplication. You can't multiply e with itself ipi times. The equation is nonsense. But there is a deeper reason behind this. Mathematicians discovered the polynomial 1+x+x^2/2+x^3/3+... was equal to e^x for all real numbers x. Likewise sin and cosine had were equal to infinite polynomials. If you plug in ix into the polynomial e^x you get the polynomial of cos(x) plus the polynomial of sin(x). But mathematicians only proved e^x is equal to that polynomial for real inputs. Everyone knows e^x isn't equal to it's polynomial series for complex values because you can't multiply e with itself a complex number of times. What they proved was that the polynomial of e^x, which we will call exp(x), exp(ix)=cos(x)+isin(x), not that e^(ix)=cos(x)+isin(x). But mathematicians write e^ix instead of exp(ix) as an abuse of notation for convenience. Euler's identity is false. Exp(x) is a continuation of e^x it's not equivalent to e^x. If you accept e^ix=exp(ix) then you should as well accept 1+2+3+4+5+6+...=-1/12 because of the analytic continuation of the Zeta function being -1/12 at s=-1. e^ipi+1=0 has no meaning. It's false.

Anonymous

Anonymous Sun 27 Mar 2022 19:05:39 No.14341485 Report

Quoted By:

>>14339310
e^sqrt(-1)pi + 1 = 0

Anonymous

Anonymous Sun 27 Mar 2022 19:08:58 No.14341503 Report

Quoted By: >>14341598

>>14341461
>But mathematicians only proved e^x is equal to that polynomial for real inputs
No they didn't. $\sum x^n/n!$ converges ABSOLUTELY. This is precisely why it is okay plugging in the imaginary unit.

Anonymous

Anonymous Sun 27 Mar 2022 19:20:32 No.14341544 Report

Quoted By: >>14341626

>>14339504
>e^x is its own derivative, and sin(x) is its own 4th derivative, and that's how it makes sense.
i think the only thing this summarization is missing is why imaginary numbers are required. sin(x) is its own 4th derivative, but it is also the additive inverse of its 2nd derivative, so if you make a function that is the sum of sin(x) and its first three derivatives, you get a function whos derivative is itself, but the function is also equal to 0 for all x which is not the exponential function. however, sin(ix) is its own second derivative, and it is not the additive inverse of its first derivative, so a function that is the sum of sin(ix) and its first derivative, -icos(ix), is a function whos derivative is itself.
>>14340566
repeated multiplication is still exponentiation. its just that the function defined as the nonzero function whos derivative is itself is also the repeated multiplication function if you are working in just the integers. we don't distinguish between repeated multiplication and exponentiation because they are actually the same thing.

Anonymous

Anonymous Sun 27 Mar 2022 19:25:01 No.14341559 Report

Quoted By: >>14341788

>>14341461
thats like saying the gamma function doesn't exist simply because factorialization doesn't make sense for non-integers. you're fucking stupid.

Anonymous

Anonymous Sun 27 Mar 2022 19:30:10 No.14341580 Report

Quoted By:

if you take the derivative it says i pie.
What did Euler mean by this....

Anonymous

Anonymous Sun 27 Mar 2022 19:33:39 No.14341598 Report

Quoted By:

>>14341503
he has a point. the fact that the taylor series converges doesn't magically make a function's domain larger than its definition. the taylor series are just how euler was able to make the connection between trig and exponentials because you can sum up coefficients of trig functions until they look like coefficients of exponentials. at some point you have to say "exponentiation's domain is all imaginary numbers", and the reason we do that has nothing to do with the taylor series

Anonymous

Anonymous Sun 27 Mar 2022 19:36:55 No.14341618 Report

Quoted By:

>>14339326
>>14339351
was kinda interesting seeing this actually come up in my physics class earlier last week talking about SHM, it's kinda fucked they just gloss over the complex plane in algebra courses

I still hate physics

Anonymous

Anonymous Sun 27 Mar 2022 19:38:02 No.14341626 Report

Quoted By:

>>14341544
> repeated multiplication and exponentiation because are actually the same thing.

dude woah t: derivative guy

it's like holomorphism and complex numbers are ALL about derivatives

I'm beginning to understand everything little by little. Some day I will understand holomorphic dynamics. I was so fascinated by the intro to holomorphic dynamics video on YouTube.

Anonymous

Anonymous Sun 27 Mar 2022 19:47:39 No.14341658 Report

Quoted By:

>>14340638
In the case above they have a different phase. The cool thing about that function is that depending on the plane you are looking at it it can give you a circle, a cosine wave, a sine wave, or a spiral.

Anonymous

Anonymous Sun 27 Mar 2022 20:23:56 No.14341788 Report

Quoted By: >>14341804

>>14341559
No it's like saying the factorial isn't magically defined for all values just because the gamma function exists.

Anonymous

Anonymous Sun 27 Mar 2022 20:29:46 No.14341804 Report

Quoted By: >>14341979

>>14341788
thats an autistic and useless thing to say

Anonymous

Anonymous Sun 27 Mar 2022 21:32:24 No.14341975 Report

Quoted By:

>>14339310
It’s part of a Jewish trick to solve differential equations. Other than some schizo Kabbalah numerology it doesn’t mean shit.

Anonymous

Anonymous Sun 27 Mar 2022 21:34:12 No.14341979 Report

Quoted By:

>>14341804
Not him, but I think one specifies a function with a different domain.

Anonymous

Anonymous Mon 28 Mar 2022 06:18:26 No.14343184 Report

Quoted By: >>14343348

>>14341461
you also cannot multiply e with itself pi times, or 5/6 times, or -3 times. "repeated multiplication" makes sense only for natural numbers. you have been indoctrinated into believing that the definition of e^x for real numbers is somehow repeated multiplication while it clearly isn't. I might as well define e^x = e^(floor(x)) for x > 0 and e^x = 1 otherwise. go on, convince me that your definition is more "repeated multiplication" than my definition.

Anonymous

Anonymous Mon 28 Mar 2022 07:02:34 No.14343247 Report

Quoted By:

>>14339310
>the identity also links five fundamental mathematical constants

>the number 0
Not a number. Could be considered a representation of infinity

>the number 1
A number, but there are arguments it dosent have to be one. Also can be a representation of infinity.

>the number pi
Pi is not a number. But could be a representation of infinity.

>the number e
e is not a number but could be a representation of infinity

>the number i
'i' is not a number. But could be a representation of infinity.

All this "equation" is saying is that infinity, to the power of infinity infinity plus infinity equals infinity. Come on, fuck this brainlet shit

Anonymous

Anonymous Mon 28 Mar 2022 07:06:20 No.14343251 Report

Quoted By:

>>14339310
They are defined in such a way that the statement is true...Euclidean geometry is an obvious choice.

Anonymous

Anonymous Mon 28 Mar 2022 08:12:01 No.14343348 Report

Quoted By: >>14343365

>>14343184
Brainlet. The only logical definition of doing something half a time is that if you do that twice you've done it once. 1/2*2=1. 1/2+1/2=1. a^(1/2)a^(1/2)=a. Any rational works. Irrationals only make sense as limits of rational sequences so they make sense too.

Anonymous

Anonymous Mon 28 Mar 2022 08:21:38 No.14343365 Report

Quoted By: >>14343376

>>14343348
The only logical definition of doing something half a time is that you haven't done it at all.

Anonymous

Anonymous Mon 28 Mar 2022 08:27:40 No.14343376 Report

Quoted By:

>>14343365
1*0.5=0
woah crazy

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