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1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:05:59
so then
0
1
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1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:06:26
$\text{div} \vec{F} = \text{grad} \vec{F} \cdot \vec{F}$
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
510,789,298,321,096,700
TeXit
04/15/2022 22:06:28
**EndTimes**
https://cdn.discordapp.c…104800104449.png
551269104800104449.png
0
1
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null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:06:33
?
0
1
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null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:06:34
No
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:06:44
because the gradient vector is the vector of partial derivatives
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:06:46
which is nabla
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:07:23
I'm on phone, so it'll take a while until I tex the following
0
1
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null
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1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
510,789,298,321,096,700
TeXit
04/15/2022 22:10:17
**Zanarcane**
https://cdn.discordapp.c…051127099396.png
299974051127099396.png
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
510,789,298,321,096,700
TeXit
04/15/2022 22:11:44
**Zanarcane**
https://cdn.discordapp.c…051127099396.png
299974051127099396.png
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:13:32
ok now i'm even more confused hold on a sec
0
1
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null
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null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:13:49
https://cdn.discordapp.c…8718/unknown.png
unknown.png
0
1
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null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:13:55
Yes
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:14:03
how can you have just d/dx
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:14:09
of no function
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:14:14
Because it's just a notation
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:14:24
Not an actual vector
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:14:32
huh
0
1
551,269,104,800,104,450
EndTimes
8053
EndTimes
false
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:14:51
i guess
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:15:06
$\text{div} \vec{F} = \text{grad} \vec{F} \cdot <1,1,1>$
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
510,789,298,321,096,700
TeXit
04/15/2022 22:15:08
**EndTimes**
https://cdn.discordapp.c…104800104449.png
551269104800104449.png
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:15:19
does that work?
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:15:29
No
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:16:03
grad is only defined for scalar fields in this context
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:16:11
but the gradient vector is <dF/dx, dF/dy, dF/dz>
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:16:39
Divergence is the trace of the Jacobian matrix
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:16:51
oh i see it's different because there's actually three different functions
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:17:19
Yeah
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:17:32
right
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:17:36
that makes sense
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:18:43
ok
0
1
null
null
null
null
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1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:18:53
it's still kind of weird seeing d/dx by itself
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:19:08
so it's not really a vector
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:19:15
Some mathematicians are not particularly fond of it
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:19:29
But I think it's quite useful
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:19:33
but an operation vector
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:19:40
Yeah
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:19:54
so derivatives are fractions then
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:20:01
Uhh what?
0
1
null
null
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null
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1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:20:53
yeah. <d/dx, d/dy, d/dz> . <Fx, Fy, Fz> = dFx/dx + dFy/dy + dFz/dz
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:21:17
It's not a fraction
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:22:17
then how does multiplication make any sense
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:22:20
If you continue your studies, maybe you will come across the fact that d/dx can be seen as a derivation / partial differential operator
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:23:05
Again, this is only a notation
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:24:36
The differential operator <d/dx, d/dy, d/dz> • maps a vector field F to the scalar field dFx/dx + dFy/dy + dFz/dz
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:25:14
Right ok
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:25:34
So it's not a fraction but merely something we treat as a fraction in every conceivable context
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:25:46
Checks out
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:26:37
It does not behave like a fraction
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:26:44
But it's linear
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:28:58
$\frac{\dd}{\dd x}\colon C^1(\mathbb{R})\to C(\mathbb{R}),\, f\mapsto f'$
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
510,789,298,321,096,700
TeXit
04/15/2022 22:29:39
**Zanarcane**
https://cdn.discordapp.c…051127099396.png
299974051127099396.png
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:30:22
wdym by C
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:30:56
Maps a continuously differentiable function f to a continuous function f' (its derivative)
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:31:51
So, you can see d/dx as a mapping between function spaces
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:32:01
Better than seeing it as fractions
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:32:37
C1(R) is the space of continuously differentiable functions on R
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:32:50
C(R) is the space of continuous functions on R
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:32:54
right yeah that checks out
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:33:16
so if there is such an issue with treating it as a fraction though then how would you 'formally' write the divergence operation
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:33:39
:this:
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:34:08
ok so yeah i guess that makes sense. the dot product is a useful mnemonic i guess
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:34:14
Yeah
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:34:16
but it doesn't really apply
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:35:04
alright that clears things up
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:35:32
Good
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:36:42
oh shit green's theorem
0
1
551,269,104,800,104,450
EndTimes
8053
EndTimes
false
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:36:54
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:37:04
Literally all of calculus 3
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:37:23
😫
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:38:00
tangential vector form
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:38:05
kill me
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:38:21
What about it?
0
1
null
null
null
null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
551,269,104,800,104,450
EndTimes
04/15/2022 22:38:44
it's fine lol i'll figure it out
0
1
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null
null
1,910
help-10
0365ce51dbd64474befc4d2bca51ca8a
299,974,051,127,099,400
Zanarcane
04/15/2022 22:38:48
Alr
0
1
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null
null
null
null
1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
952,268,248,248,897,700
Noneuclidian
04/15/2022 23:11:26
Hello. I'm working through some factoring examples in my college algebra book and I'm confused about an example.
1
0
null
null
null
null
null
1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
952,268,248,248,897,700
Noneuclidian
04/15/2022 23:11:29
,tex \begin{math} x^2 + 6x = 0\\ x(x+6) = 0\\ x = 0 \text{ or } x + 6 = 0\\ x = 0 \text{ or } x = -6 \end{math}
1
0
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null
null
null
null
1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
510,789,298,321,096,700
TeXit
04/15/2022 23:11:32
**Noneuclidian**
https://cdn.discordapp.c…248248897627.png
952268248248897627.png
0
1
null
null
null
null
null
1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
952,268,248,248,897,700
Noneuclidian
04/15/2022 23:12:23
How were they able to remove the x outside the parens? I should mention that at this part in the book they haven't introduced us yet to radicals.
1
0
null
null
null
null
null
1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
299,974,051,127,099,400
Zanarcane
04/15/2022 23:14:57
It's called factoring out
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
299,974,051,127,099,400
Zanarcane
04/15/2022 23:15:35
Follows from the distributive law
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
299,974,051,127,099,400
Zanarcane
04/15/2022 23:15:47
And you don't need radicals for that
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
952,268,248,248,897,700
Noneuclidian
04/15/2022 23:17:14
I understand factoring `x^2 + 6x` into `x(x+6)`, but I don't understand how they can just drop that `x` and be left with `x+6`. Don't you have to divide both sides by`(x+6)` to get that x on its own? (or is taht what's going on?)
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0
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help-10
b13d7e50b4a644e9929fea2ce1d6d505
200,013,547,491,688,450
a disappointing son
04/15/2022 23:17:58
the next step is just using the zero product property
0
1
510,789,298,321,096,700
TeXit
0796
TeXit
true
1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
200,013,547,491,688,450
a disappointing son
04/15/2022 23:18:10
if ab=0, then a=0 or b=0
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
952,268,248,248,897,700
Noneuclidian
04/15/2022 23:20:21
What about getting `x+6` on its own without the other `x` in front of it? How does that work?
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help-10
b13d7e50b4a644e9929fea2ce1d6d505
200,013,547,491,688,450
a disappointing son
04/15/2022 23:20:48
are you talking about line 2 -> line 3
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
952,268,248,248,897,700
Noneuclidian
04/15/2022 23:22:12
The last two lines basically
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0
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
200,013,547,491,688,450
a disappointing son
04/15/2022 23:22:42
as i mentioned, that's the zero product property
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
200,013,547,491,688,450
a disappointing son
04/15/2022 23:23:34
if you're multiplying multiple terms together and that product is equal to zero, that can only happen if at least one of the terms is equal to zero
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
200,013,547,491,688,450
a disappointing son
04/15/2022 23:24:04
for example, if you had $x\cdot y\cdot z = 0$, this can only be true if $x, y,$ or $z$ is equal to zero.
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help-10
b13d7e50b4a644e9929fea2ce1d6d505
510,789,298,321,096,700
TeXit
04/15/2022 23:24:07
**a disappointing son**
https://cdn.discordapp.c…547491688448.png
200013547491688448.png
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1
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
200,013,547,491,688,450
a disappointing son
04/15/2022 23:24:16
since that would make the entire product zero
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
551,269,104,800,104,450
EndTimes
04/15/2022 23:26:06
Right
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
952,268,248,248,897,700
Noneuclidian
04/15/2022 23:26:12
I understand anything multiplied by anything 0 is 0
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0
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
551,269,104,800,104,450
EndTimes
04/15/2022 23:26:21
No
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1
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
952,268,248,248,897,700
Noneuclidian
04/15/2022 23:26:27
?
1
0
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null
1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
551,269,104,800,104,450
EndTimes
04/15/2022 23:26:35
3×2 is not 0
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1,911
help-10
b13d7e50b4a644e9929fea2ce1d6d505
551,269,104,800,104,450
EndTimes
04/15/2022 23:26:44
Ok yeah
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help-10
b13d7e50b4a644e9929fea2ce1d6d505
551,269,104,800,104,450
EndTimes
04/15/2022 23:26:48
That is true
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