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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:18:05
cubic roots are you talking about principle roots?
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Kiretori
11/12/2021 16:18:50
I'm talking about the nth roots of unity in complex numbers
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:18:58
3rd roots in this case
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Deleted User
11/12/2021 16:19:53
im sorry i lost you
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:20:07
whats an nth roots of unity
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:20:37
We call Z a nth root of unity if Z^n = 1
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:20:58
3rd roots of unity would be written as Z^3 = 1
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Deleted User
11/12/2021 16:21:20
Z is a + bi?
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:21:33
Yeah it's a complex number
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Deleted User
11/12/2021 16:21:51
oh okay
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:22:20
Are you familiar with the exponential notation for complex numbers?
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:22:33
no it wasn't mentioned in our course
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Kiretori
11/12/2021 16:22:54
I see
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Kiretori
11/12/2021 16:24:29
Well it's known that the 3rd roots of unity are 1, -1/2 + i√3/2 and -1/2 - i√3/2
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:24:36
They're not equal
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Kiretori
11/12/2021 16:24:37
But
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Kiretori
11/12/2021 16:25:23
1³ = (-1/2 + i√3/2)³ = (-1/2 - i√3/2)³ = 1
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:25:59
Even though they're not equal their cubes are equal
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:26:03
And that's a counter example
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Kiretori
11/12/2021 16:26:34
When trying to find a counter example for these kinds of questions, just try to think of something simple
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:26:37
Of small numbers
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:26:41
Like 1 in this case
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Deleted User
11/12/2021 16:35:47
interesting but i didn't know about nth roots of unity
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Deleted User
11/12/2021 16:35:59
do you think that was what i would have been missing to find such example?
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:36:29
i mean when i am asked about a complex number i straightaway think about a+bi form
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:40:25
oh looking at the script now i see they did mention z = re^{i alpha}
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:41:22
saying that alpha is an angle and r is the magnitude
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:41:24
thats all
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:47:16
Yes that's the exponential notation I was talking about
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:47:29
Learning it is really crucial when working with complex numbers
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help-10
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Kiretori
11/12/2021 16:47:33
It makes things way easier
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:48:02
You should also learn the trigonometric form of a complex number
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:48:10
but how do I deduce from that z = re^{i alpha} that nth roots of unity is 1
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:48:22
thing is, they just gave us a very small intro about it
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:48:29
The nth roots of unity aren't 1
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:48:37
The nth roots of unity are the nth roots of 1
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:48:42
Unity means 1
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:51:13
It's not straight forward for you since you don't know a lot about exponential and trigonometric forms of complex numbers
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:51:22
yeah
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:51:25
im still confused tbh
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 16:51:35
Just look it up on youtube
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:51:38
i skipped the question and just worked on the other assignments
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
351,449,458,136,842,240
Kiretori
11/12/2021 16:51:41
It'll help you a lot
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:51:44
but def im going to look this up
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:52:23
you suggest to look up complex numbers in general or specific points ?
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:52:33
because you mentioned this "You should also learn the trigonometric form of a complex number"
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:53:08
im lowkey triggered the fact i wasn't able to find the solution straight up
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:53:56
like through the whole day i've spent in total 6 hours thinking about a solution and absolutely no clue
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:54:55
i thought that z = a+bi is a direction to a point and every time you ^n it rotates counter clockwise so whats the number that when you set it ^3 will get the same value at the end
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 16:55:41
it def came to me this angle stuff but nothing was mentioned in my course that talks about it so if start writing stuff outside the script it will be sus
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Kiretori
11/12/2021 16:58:52
Some problems can't be solved just by using the algebraic expression of a complex number which is a +bi
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 17:06:16
They told you a complex number is written as re^theta
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Deleted User
0000
Deleted User
false
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 17:06:36
re^theta is the same as r^theta+2π
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:07:31
wait a sec i think they mentioned another way to find out about this 1^3
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:07:43
they gave a lemma with
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:08:06
$g: \mathbb{R} \rightarrow \mathbb{C}, \ g(a) = a+0i$
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1bab00acaf8c4f69821f6cc76dc6d5c3
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TeXit
11/12/2021 17:08:08
**mhD**
https://cdn.discordapp.c…330214293544.png
878046330214293544.png
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help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:09:03
mentioned some operation to show that we can use real numbers to make it seems like we using real numbers but instead its a complex number
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:09:17
also it says that the function is injective
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:10:00
next lemma after this one talks about
https://cdn.discordapp.c…4504/unknown.png
unknown.png
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 17:10:27
Just work with e theta they talked about
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:10:53
nah there is no theta
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:10:53
https://cdn.discordapp.c…5084/unknown.png
unknown.png
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Deleted User
11/12/2021 17:11:00
this is what they have written into
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:11:14
saying that alpha is an angle and it is irrelevant when |z| = 0
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Deleted User
11/12/2021 17:14:23
thank you very much for the help though @Kiretori
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Kiretori
11/12/2021 17:15:24
Give a sec
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:15:33
yeah im still around
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Kiretori
11/12/2021 17:20:08
these are the different ways to express a complex number
https://cdn.discordapp.c…11112_231915.jpg
IMG_20211112_231915.jpg
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Kiretori
11/12/2021 17:21:53
If you know trignometry you'll understand why a = rcos(alpha) and b= r×isin(alpha)
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 17:25:11
And then from this we can deduce that re^i(alpha) is the same as re^i(alpha + 2π)
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Kiretori
11/12/2021 17:25:44
Because we would just turn counter clock wise and reach the angle alpha again
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Kiretori
11/12/2021 17:25:51
We would have the same number Z
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:26:03
like I might get why (-1/2 + i√3/2)³ = 1^3
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:26:34
because if we ^3 the left side then it will counter clockwise and probably the angle is so small that it becomes flat on the a line
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1bab00acaf8c4f69821f6cc76dc6d5c3
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Deleted User
11/12/2021 17:26:53
but how is this supposed to be equal (-1/2 - i√3/2)³ = 1^3?
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351,449,458,136,842,240
Kiretori
11/12/2021 17:51:14
https://cdn.discordapp.c…11112_235013.jpg
IMG_20211112_235013.jpg
0
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null
null
null
null
null
1,835
help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
351,449,458,136,842,240
Kiretori
11/12/2021 17:51:41
Here I showed that even tho we have z³ = z'³
0
1
null
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null
null
null
1,835
help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
351,449,458,136,842,240
Kiretori
11/12/2021 17:51:55
z doesn't equal z
0
1
null
null
null
null
null
1,835
help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
351,449,458,136,842,240
Kiretori
11/12/2021 17:52:18
Because alpha/3 and (alpha/3 + 2π/3) is not the same angle
0
1
null
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null
null
null
1,835
help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
351,449,458,136,842,240
Kiretori
11/12/2021 17:52:41
While alpha and (alpha + 2π) is the "same" angle
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1
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1,835
help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
251,854,584,597,708,800
PapaBread
11/12/2021 17:53:12
,rotate
0
1
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null
null
null
1,835
help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
510,789,298,321,096,700
TeXit
11/12/2021 17:53:29
https://cdn.discordapp.c…584597708800.png
251854584597708800.png
0
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null
null
null
null
null
1,835
help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
251,854,584,597,708,800
PapaBread
11/12/2021 17:53:34
😎
0
1
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null
null
null
null
1,835
help-10
1bab00acaf8c4f69821f6cc76dc6d5c3
351,449,458,136,842,240
Kiretori
11/12/2021 17:55:04
😱
0
1
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null
null
null
null
1,837
help-10
d2edc9fd853843fc908c336dc472611e
116,699,495,055,491,070
ihaverizzstrainingorder
04/12/2022 01:51:52
how do you find llv+wll vs llvll+llwll?
1
0
null
null
null
null
null
1,837
help-10
d2edc9fd853843fc908c336dc472611e
176,048,095,543,558,140
tushar
04/12/2022 01:56:14
depends on how the norm is defined
0
1
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null
null
null
1,837
help-10
d2edc9fd853843fc908c336dc472611e
176,048,095,543,558,140
tushar
04/12/2022 01:57:19
but the Cauchy-Schwarz inequality states that \|\|v+w\|\| <= \|\|v\|\| \|\|w\|\| when the norm is induced by an inner product
0
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null
null
1,837
help-10
d2edc9fd853843fc908c336dc472611e
176,048,095,543,558,140
tushar
04/12/2022 01:59:22
and there is equality whenever v and w and linearly dependent
0
1
351,449,458,136,842,240
Kiretori
8007
Kiretori
false
1,838
help-10
bd90451cee074abd8bb8634727a92e95
896,927,649,602,101,200
Kosta
04/12/2022 02:24:34
https://cdn.discordapp.c…3114/unknown.png
unknown.png
1
0
null
null
null
null
null
1,838
help-10
bd90451cee074abd8bb8634727a92e95
896,927,649,602,101,200
Kosta
04/12/2022 02:24:50
can someone explain how to do this?
1
0
null
null
null
null
null
1,838
help-10
bd90451cee074abd8bb8634727a92e95
591,872,343,320,821,800
Doggo
04/12/2022 02:41:35
for f(g(x)) to be defined, g(x) has to be in [1, infty) so g(x) >= 1 for g(f(x)) to be defined, f(x) has to be in (-infty, 2] so f(x) <= 2
0
1
null
null
null
null
null
1,838
help-10
bd90451cee074abd8bb8634727a92e95
896,927,649,602,101,200
Kosta
04/12/2022 02:42:24
yes?
1
0
null
null
null
null
null
1,838
help-10
bd90451cee074abd8bb8634727a92e95
591,872,343,320,821,800
Doggo
04/12/2022 02:44:13
yeah then you just solve the inequalities?
0
1
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null
null
null
1,838
help-10
bd90451cee074abd8bb8634727a92e95
591,872,343,320,821,800
Doggo
04/12/2022 02:44:39
keep in mind that for g(x) >= 1 you also have x <= 2 and for f(x) <= 2 you have x >= 1
0
1
null
null
null
null
null
1,839
help-10
c0a7618e616740f494242d90fa61dc6c
576,541,150,761,058,300
'secret
04/12/2022 08:20:29
@Helpers
1
0
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null
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help-10
c0a7618e616740f494242d90fa61dc6c
576,541,150,761,058,300
'secret
04/12/2022 08:20:41
i don’t understand this
1
0
null
null
null
null
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1,840
help-10
d27f892eba9b4045a7ec7f364507626a
246,174,820,088,152,060
99muppets
04/12/2022 09:35:18
https://cdn.discordapp.c…4808/unknown.png
unknown.png
1
0
null
null
null
null
null
1,840
help-10
d27f892eba9b4045a7ec7f364507626a
246,174,820,088,152,060
99muppets
04/12/2022 09:35:33
I cannot get B I found the forces on A = T - 2w_B forces on B = w_B - T and the accelerations of B and A are the same so (T - 2w_B) / w_A = (w_B - T) / w_B w_B(T - 2w_B) = w_A(w_B - T) and now this is already clearly super wrong so idk what to do
1
0
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null
null
null
1,840
help-10
d27f892eba9b4045a7ec7f364507626a
246,174,820,088,152,060
99muppets
04/12/2022 09:35:36
im going fucking insane
1
0
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