Nah i was talking about the scammer guy lol
#help-28
analog sand
rough tundra
idk specifics—but it's kind of in the name 😅
haughty ember
Ok well
That’s it rn
But does it make it clear?
Cause it could have just wanted n
steel solar
But does it make it clear?
Yes
haughty ember
Oh yeah every row has n+3
steel solar
"how many students are there in each row"
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haughty ember
If u need translation lmk
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haughty ember
It wants me to solve the equation; by completing the square
There’s so many similar things
Like
Idk which strategy to use
I learned the 2nd degree equation
I also learned this Steven trinomial factoration
Idk what completing squares are
gritty rose
Idk what completing squares are
,tex .cts
glossy valveBOT
riemann
terse echo
This is confusing
I'm gonna explain u easier
haughty ember
Ty
gritty rose
This is confusing
gotta be more clear about what you're confused by
haughty ember
The letter and stuff
gritty rose
what letters
gritty rose
gotta be more clear about what you're confused by
.
haughty ember
Can u give an example with my question
gritty rose
then write here when you don't understand something
haughty ember
What’s b
gritty rose
b is a placeholder number
in here, bx means b times x
just like 6x means 6 times x
gritty rose
dunno what your teacher taught you ¯_(ツ)_/¯
haughty ember
Yeah imma leave this for when he does it in clsss
Cause yeah idk
.clsoe
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jolly wraith
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jolly wraith
it said i got it wrong but i cant figure out how i did get it wrong
jolly wraith
do i have to use desmos?
young night
do i have to use desmos?
i mean its a easy way to test your asnwers
hearty kite
your line is parallell instead of perpendicular to the given line
jolly wraith
your line is parallell instead of perpendicular to the given line
yeah just found that out thanks
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oak creek
I'm making exercises for a math exam I have next week and this one was in the 'extra', I didn't know where to begin so Ilooked at the answer and I still don't understand what I'm looking at could somoene tell me: (I used Chatgpt to translate the question)
carmine minnow
I'm making exercises for a math exam I have next week and this one was in the 'e...
the solution writing is a bit sloppy due to these equal signs
Not sure what the issue is
oak creek
Not sure what the issue is
I just don't know what all of that is
carmine minnow
okay, do you know how to use the binomial theorem to expand $(a+b)^n$?
glossy valveBOT
Element118
oak creek
I looked up the binomium of newton and I don't understand how we went from the question to the answer
oak creek
okay, do you know how to use the binomial theorem to expand $(a+b)^n$?
that's what I'm trying to understand
lime ether
I'm making exercises for a math exam I have next week and this one was in the 'e...
you mirin brah
carmine minnow
okay, so basically, whenever you see $(a+b)^n$, you can replace with $\sum_{k=0}^n\binom{n}{k}a^kb^{n-k}$
glossy valveBOT
Element118
carmine minnow
so in the first line, we are applying it for $n=10,a=3x^2,b=\frac{1}{x}$
glossy valveBOT
Element118
oak creek
ok
and then we get this? :
I understand that now but what about the rest?, how do we get to our answer?
carmine minnow
I understand that now but what about the rest?, how do we get to our answer?
you want the coefficient of x^8
so you need to figure out which of the terms in the sum gives you a x^8
carmine minnow
and then we get this? :
Usually 10 choose k is written as $\binom{10}{k}$ or $C^{10}k$ or ${}^{10}C_k$, not $C^k{10}$.
glossy valveBOT
Element118
oak creek
Usually 10 choose k is written as $\binom{10}{k}$ or $C^{10}_k$ or ${}^{10}C_k$,...
weird... it's the solution I got given
carmine minnow
weird... it's the solution I got given
do you have other solutions that also use this notation?
if it's consistently used like this, and it's what the teachers gave you, then follow it
oak creek
if it's consistently used like this, and it's what the teachers gave you, then f...
this is another one but a bit different, quesion is: Calculate 1, 01^5 with the binomial of Newton:
carmine minnow
this is another one but a bit different, quesion is: Calculate 1, 01^5 with the ...
yeah, then just follow the notation if this is what the teachers gave you
oak creek
yeah, then just follow the notation if this is what the teachers gave you
the notation doesn't really bother me lol, I just don't understand any of it, like... idk what I'm looking at
I want to understand step by step what is going on in the equation
carmine minnow
I want to understand step by step what is going on in the equation
okay, so are you fine with binomial theorem, like you can write
$(3x^2+1/x)(3x^2+1/x)...(3x^2+1/x)$ and basically the number of ways to form a term with 6 $3x^2$ and 4 $1/x$ is to choose 6 out of 10 of the brackets to give you $3x^2$
glossy valveBOT
Element118
oak creek
okay, so are you fine with binomial theorem, like you can write
$(3x^2+1/x)(3x^2...
I don't understand what you mean with 'to choose 6 out of 10 of the brackets'
carmine minnow
I don't understand what you mean with 'to choose 6 out of 10 of the brackets'
Let's say you expand $(3x^2+1/x)(3x^2+1/x)...(3x^2+1/x)$ by hand
glossy valveBOT
Element118
oak creek
yes
carmine minnow
it's not easy but you have to go through every term
let's make things simpler ig
$(a+b)(a+b)(a+b)$
maybe expand the first two
$(aa + ab + ba + bb)(a+b)$
then expand out the third
$aaa + aba + baa + bba + aab + abb + bab + bbb$
glossy valveBOT
Element118
carmine minnow
there's quite a lot of repeated terms coming from the product - we can see there are 3 terms with "1 b and 2 a" for instance
oak creek
mhm
but does that have to do with this?
nvm I actually solved it myself lol thanks!
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fathom trench
Hello
fathom trench
,rccw
glossy valveBOT
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rare pine
(60 + 3)^6 i guess?
fathom trench
I found b tho
rare pine
But I will have to expand it
yeah but youd only have to choose some terms that contribute to the 1st digit
violet haven
b is just the last digit of 3^6? ||i dont have an answer im just invested 😭 ||
rare pine
yeah but youd only have to choose some terms that contribute to the 1st digit
for example like first 3-4 terms and check if theyre > 10^11
rare pine
first 2-3 terms dont take that long
violet haven
maybe logs would help?
steel raptor
I assume you dont have a calculator?
rare pine
if i had less time if just take the first digit of 60^6 lmao
violet haven
6log10(9)+6log10(7)
onyx glen
6log10(9)+6log10(7)
you would need to find the fractional part of that
how are you gonna do that without a calculator
and in 3 minutes
violet haven
you would need to find the fractional part of that
wdym?
rare pine
log9 and 7 are irrational
violet haven
yh but you js remember the first few digits
violet haven
ive seen people do it in amc before
onyx glen
and then compare it against log(2), log(3), ..., log(9) to figure out what your first digit is
violet haven
and then compare it against log(2), log(3), ..., log(9) to figure out what your ...
i think if you somehow can memorize it, it shouldnt take too long
onyx glen
i think there isn't really any "smart" way to do it
maybe like, working out 63^2 and then cubing that but that may take too long
violet haven
||i have so much respect for people in math competitions|| i got c but i dunno 😭
wait nvm e
rare pine
maybe like, working out 63^2 and then cubing that but that may take too long
well you could just skip the last few digits
fathom trench
rare pine
probably faster
onyx glen
if you can work these out both in under a minute, knock yourself out...
fathom trench
…
violet haven
i still think memorizing logs is the way to go
fathom trench
Estimation ig
steel raptor
Ok I think i got it. First, you should memorize log(2-9), then you want to find the number of digits from taking log(63) turning it into log(7) + log(9), next take your appx log(63) and multiply it by 6 since you are going 63^6. You get ~ 10.8. Then keep the decimal and do 10^.8 (which you have also memorized) which will give you the first digit
rare pine
9^3 and 7^3 are memorized ig
steel raptor
Then obv a*b is your answer
rare pine
just take them squared
violet haven
Ok I think i got it. First, you should memorize log(2-9), then you want to find ...
why 10^0.8??
steel raptor
why 10^0.8??
You keep the decimal portion of 10.8
violet haven
mhm and compare to the logs?
steel raptor
wdym compare to the logs
steel raptor
Wait nvm
violet haven
yh no, you have to compare it
my amc teacher taught me it but lwk got lazy and didnt memorize the logs
steel raptor
yes
violet haven
so e is the answer
steel raptor
yes
violet haven
i dont think theres any other faster way tho 😔
violet haven
how tho, 60^6?
steel raptor
If you really dont want to solve for the first digit there is another way to get an answer
but its not 100% accurate
fathom trench
Since 63^2
3969
And it’s rlly close to 4000
If we can Juz multiply 4000 3 times
And it gives a rough estimation of 64 smth
steel raptor
Thats also very fair
violet haven
yh thatd work, but it wont work all the time
steel raptor
I mean in the end, its not like you need 100% accuracy
fathom trench
And the ans is a252350220 b
fathom trench
I mean in the end, its not like you need 100% accuracy
Exactly
steel raptor
You already know there are only 3 answer choices c, d, and e
violet haven
6
steel raptor
If you can estimate that the first digit is 6, you can practically eliminate c and d since 6 is so far from 2 or 3
fathom trench
And 9 x5 and 9 x 6
It’s not in the multiple choice so
E it is
Ig this would b pretty fast
steel raptor
If you saw it again fs
violet haven
good luck on your math comp 🙏
fathom trench
good luck on your math comp 🙏
Thx
carmine minnow
mod 10 gives you b, mod 9 gives you a
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night scarab
how do u approach this question
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how do u approach this question
night scarab
this has been my work till now
i thought if the left/right and top/bottom were equal that means the limit exists
buoyant wadi
Hmm, well all the powers of y are even, but x has one odd power on top
Maybe you could get results by trying something like taking the limit as a->0 of (a,a) and (-a,a)?
night scarab
wdym
buoyant wadi
Well you're trying to find two different approaches that give different limits, right?
night scarab
yea coz they are not continous right
buoyant wadi
Yeah
So instead of trying the lines where y=0 and x=0, what about along the lines y=x and y=-x?
night scarab
y=x and y=-x covers the top/bottom part
what about the left/right part
should i go with x=0
im getting this
doesnt seem right
i came up with this instead
does this work make sense
or is this gibberish
buoyant wadi
Hmm
So you understand how doing y->0 and x->0 are like lines approaching (0,0) coming from the y-axis and the x-axis respectively, right?
night scarab
yea we are basically finding out the limits at those axis if im not wrong
buoyant wadi
Yeah, well you can also try finding limits coming from different directions
(Oh hey I got a role)
Like from 45 degrees or 135 degrees!
Do you see what I mean?
night scarab
(Oh hey I got a role)
congo
buoyant wadi
No, like (a,a) with a->0
So like (1,1), (0.5, 0.5), (0.1, 0.1), (0.05, 0.05), (0.01, 0.01), ...
night scarab
cant we do l'hopital rule over here
cos its techinically 0/0
and its only 1 variable
that wont work on another thought
its gonna be 2/2a
and 1/a is und
buoyant wadi
If you're doing the reduced one, 2a/(1+a^2) at the end there, that doesn't go to 0/0
It goes to 0/1
Maybe try modelling it in desmos? That could help you find some paths where this function behaves strangely
night scarab
wait ur right
mb im tweaking
i havent slept properly recently
ima sleep
thanks for ur help
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naive monolith
this is a past paper, how does horizontal translation equal vertical dilation
naive monolith
like
i get the e^2 thing
but the effect that it has isnt the same as dilation right
or is it that its just dilating by e^2 is just whoosh weird effect different from normal constant so technically theyre the same
nvm
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lament compass
Earlier, @void nova showed me how to solve telescoping series like the attached image. But how come when I try for some series it gives me a different answer than expanding it?
lament compass
like for example when I tried to do the infinite sum of ln(n/n+1) i solved it = to 0
but when I expand it and cancel ou tthe first and last few terms manually and then find the lim of Sn then it gives me infinity
so like one way says its convergent but the other divergent
is it a mistake in my algebra ?
<@&286206848099549185>
void nova
is it a mistake in my algebra ?
I can't tell it if you don't show it 😬
lament compass
I can't tell it if you don't show it 😬
hi
ok
1 sec
I cant really send a picture im on my pc but I split Sum of ln(n/n+1) into sum ln n - sum ln n+1 then changed sum ln n+1 into sum_k=2 (ln k) then split that into sum_k=1 (lnk) - sum_k=1^1 (lnk) then the sum ln n cancel out right? so all thats left is sum_k=1^1 (lnk) = 0
or heres it shown better but its m nistead of k
but the other way gave me infinity and divergent
lament compass
ok ty
i tried ur way on another question tho and it worked sooo im hoping I can pull it off on the test in like 5 hours
cuz its 10x quicker
void nova
In short the problem is that the two infinities (the upper bounds) are not the same, but they're 1 unit apart
And this doesn't allow you to cancel the tail of your summation
lament compass
In short the problem is that the two infinities (the upper bounds) are not the s...
can u explain what this means
so I think ur saying that they dont appraoch the same infinity ? but i dont understand the 1 unit apart thing
void nova
You need to use this fact: $$\sum_{n = n_0}^\infty \ a_n =\lim_{N\ \to \ +\infty }\ \sum_{n = n_0}^N \ a_n$$
glossy valveBOT
Alberto Z.
lament compass
**Alberto Z.**
isnt that just saying the lim n--> infinity of the partial sums = the infinite sum?
or is that something else
lament compass
yeah the way I know to do it is u take the partial sums and simplify it by telescoping thru observing what cancles in test cases then take that lim as n --> infinity
but why doesnt that apply for the thing I showed u?
or why doesnt ur way apply*
void nova
yeah the way I know to do it is u take the partial sums and simplify it by teles...
Exactly, the important thing is to take the limit only at the end of all the simplifications
void nova
but why doesnt that apply for the thing I showed u?
Because the limit at +∞ was 0
lament compass
Because the limit at +∞ was 0
so
wouldnt that mean tho that the series converges to 0?
not infinity
or divergent*
wait is it because you cant split the sums unless both sequences are convergent?
cuz thats what chatgpt telling me and I guess that would make sense? but idk how u could tell that before you started doing the work
lament compass
Because the limit at +∞ was 0
also does this mean then its kinda like harmonic where the sequence converges but not the actual series
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@lament compass Has your question been resolved?
void nova
wouldnt that mean tho that the series converges to 0?
Not the series, just the general term $a_n$
glossy valveBOT
Alberto Z.
lament compass
Not the series, just the general term $a_n$
im still kinda confused as to why it doesnt work tho
so i get that the Sequence converges to 0
since lim n--> infinity of an = 0
but then to find the Series dont u just take lim n--> infinity of Sn
void nova
Let me do it on paper so that I can show you better
Here you are @lament compass
lament compass
Here you are <@1151694055542112327>
why is there N+1 on top of the sum?
void nova
Because of the change of variable
lament compass
Because of the change of variable
so because k = n+1 N --> N+1?
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void nova
so because k = n+1 N --> N+1?
Let k = n+1
So when n = 1, k = 1 + 1, so k starts from 2
When n = N, k = N + 1, so k will stop at N + 1
lament compass
Let k = n+1
So when n = 1, k = 1 + 1, so k starts from 2
When n = N, k = N + 1,...
ok one last question for a p - series can u find the infinite sum? or only if it diverges or converges
because like for geometric theres the formula and telescoping terms will cancel out
lament compass
ok one last question for a p - series can u find the infinite sum? or only if i...
like because the teacher only said about p series that it converges if p > 1 and diverges otherwise
<@&286206848099549185>
primal horizon
I am getting A,C,D but the answer is given as only D (multiple right answer)|
analog shale
I am getting A,C,D but the answer is given as only D (multiple right answer)|
show the work pls
primal horizon
show the work pls
1 min
analog shale
remember that denominator gives you a condition
waxen mauve
I am getting A,C,D but the answer is given as only D (multiple right answer)|
C is wrong because denom=0
uncut nebula
B cannot be true ever
primal horizon
uncut nebula
1/a + 1/b + 1/c = 0
primal horizon
Got it
uncut nebula
requires all of them to be inf
primal horizon
No need
uncut nebula
or negative
primal horizon
Yes
uncut nebula
included
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analog shale
or negative
so yeah you can't simply conclude B is impossible from this
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brisk folio
i got the resultant sum of all the x components to equal -350.736 is this correct?
unreal wren
i got the resultant sum of all the x components to equal -350.736 is this correc...
Yes it is correct
brisk folio
Yes it is correct
thank you so much and is the y force -1534.549?
green merlin
<@&268886789983436800>
unreal wren
thank you so much and is the y force -1534.549?
I got a slightly different answer
In the decimal places
It is probably just a calculator problem
brisk folio
It is probably just a calculator problem
most likely thanks again
unreal wren
Close the help channel
brisk folio
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low crystal
<@&268886789983436800>
glossy fern
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gritty flax
Whats D and S ?
wide sundial
What have you tried
glossy fern
not much
wide sundial
What are you stuck on
glossy fern
yes
wide sundial
Okay so what are you stuck on
glossy fern
how to solve after that
wide sundial
Well what have you put down on paper
glossy fern
nothing much, i dont know how to start
wide sundial
Mate I’m not here to do the problem for you, you have to show some initiative
glossy fern
yeh but i dont know what to do
wide sundial
Okay, what does it mean to be isomorphic?
glossy fern
theres a bijective function going from one to the other
i dont know how i would define the bijective function for rotation and reflection to permutations
wide sundial
This isn’t enough
You want group isomorphism
A bijective function is only a set isomorphism
glossy fern
the other condition comes after
wide sundial
Okay
wide sundial
i dont know how i would define the bijective function for rotation and reflectio...
Write out all the elements of each group
glossy fern
i have to come up with a bijective fn first right
glossy fern
And match them to each other
there has to be an analogy right
wide sundial
Yes, the analogy is the bijective function
glossy fern
no like an analogy for the rotation and reflection to permutation
wide sundial
So write it out and try it, D₆ and S₃ are small enough you can just write them out
glossy fern
ok
how would i check the second condition on the permutation group
its like phi(aB)=phi(A)phi(b)
but what operation is on the permutation group
wide sundial
Composition?
I means S₃ is a group
And there’s a group operation on that
It’s to do 1 permutations after another
glossy fern
like whats the operation
glossy fern
Composition?
the permutations themselves are the elements of the set right
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@glossy fern Has your question been resolved?
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pls help me im very poor indian🙏
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
analog sand
nm
i thought it was someone else's help channel lmao.
What's the original question @glossy fern ?
glossy fern
.
analog sand
You can use the helpers ping by typing @Helpers into the chat, I can't help because I'm not familiar with the concept.
glossy fern
<@&286206848099549185>
carmine minnow
For $D_6\cong S_3$, you can explicitly exhibit such a mapping.
For $D_{24}\not\cong S_4$, consider computing stuff about the elements in the groups such that you get different results.
glossy valveBOT
Element118
glossy fern
For $D_6\cong S_3$, you can explicitly exhibit such a mapping.
For $D_{24}\not\...
i dont understand how you can do the mapping
glossy fern
like what exactly is the permutation group
carmine minnow
you can notate the 6 elements of D6 and the 6 elements of S3 right?
glossy fern
whats the operation in it
carmine minnow
just say, for each element of D6, what it corresponds to in S3
another way is conceptually
glossy fern
like what exactly is the permutation group
?
is the set composed of all permutations of 123
then whats the operation in it
carmine minnow
or you can think of them as bijective automorphisms on ${1,2,3}$ operated by composition.
glossy valveBOT
Element118
carmine minnow
bijective maps ${1,2,3}\to{1,2,3}$, that is
glossy valveBOT
Element118
glossy fern
like arrows okay
so the arrows are the operation
thats exactly what the problem was as to how they relate conceptually
like reflections are what kind of arrows
or rotations
waxen flax
What do you mean by arrows?
glossy fern
What do you mean by arrows?
bijective mapping
the arrows would give a new permutation
waxen flax
where are the arrows going?
carmine minnow
this represents a function $f$ where $f(1)=3,f(2)=1,f(3)=2$
glossy valveBOT
Element118
carmine minnow
i think
glossy fern
yeh
waxen flax
How have these groups been introduced to you?
carmine minnow
perhaps an important question
waxen flax
Okay, do you know what the dihedral group D₆ and symmetric group S₃ are?
carmine minnow
glossy fern
yeh
carmine minnow
this is how it is defined in napkin
but basically D6 is the group of symmetries of an equilateral triangle
also we have this for symmetric groups
One trick here is to think what parts of an equilateral triangle a permutation is moving around
glossy fern
so thinking in terms of rotation and reflection doesnt work
waxen flax
so thinking in terms of rotation and reflection doesnt work
if you follow the suggestion it does
waxen flax
Yes
glossy fern
for the next part the hint talks about the orders of the elements
carmine minnow
for the next part the hint talks about the orders of the elements
ah
like consider how many times you need to operate an element with itself to reach the identity
you can consider that for each element of D24 and each element of S4
glossy fern
you can consider that for each element of D24 and each element of S4
dang thats a lot
like its kinda easy for d24
carmine minnow
okay what can you say for D24
glossy fern
you can just take the powers for elements for d24 directly
in s and r notation
but how do you quantify for s4
like its arrows
carmine minnow
you can just take the powers for elements for d24 directly
think of them as symmetries of a 12-gon
flipping over: well if you do that again, you get back where you started - order 2
rotating some amount, could have order 2, 3, 4, 6 or 12
identity: order 1
glossy fern
flipping over: well if you do that again, you get back where you started - order...
yeh thats what i was saying with the s and r notation
it directly does it with taking powers
but the s4 is arrows
how do you do it for arrows
carmine minnow
have you considered disjoint cycle notation
carmine minnow
glossy fern
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torpid perch
\begin{enumerate}
\item Represent on the plane the set ${z \in \mathbb{C} \mid |z - 1| < |z + 3 - 2i|}$.
\item Give the binomial form of all $z \in \mathbb{C}$ such that $z(\overline{z} - 2) = 7 + 4i$.
\end{enumerate}
buoyant wadi
!status
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What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
For 1., you can try using the form z=x+yi and then expanding
steel solar
But it's a little inefficient, so you might want to compare circles instead
steel solar
why cartesian
It is how I normally do it
torpid perch
circles?
steel solar
e.g. letting |z+3-2i|=|z-1|=1
Then shading the region where |z+3-2i|=1 is outside |z-1|=1
buoyant wadi
Similar to what ;( is saying, I started by plotting the points 1+0i and -3+2i, since |z-1| and |z+3-2i| are the distances from z to those points, respectively
torpid perch
Similar to what ;( is saying, I started by plotting the points 1+0i and -3+2i, s...
can you guys show the drawing
torpid perch
Then shading the region where |z+3-2i|=1 is outside |z-1|=1
do you have a rough sketch?
i dont follow
steel solar
do you have a rough sketch?
Sure ig
buoyant wadi
Blah, I have a sketch but Discord isn't letting me upload images right now...
steel solar
discordstatus.com
^
buoyant wadi
steel solar
It won't render TeXiT either I think
torpid perch
**Renato**
it rendered just fine
buoyant wadi
https://www.desmos.com/calculator/ebhf9fsg9u
|z - 1| < |z + 3 - 2i| is the same as saying z is closer to the point (1,0) than it is to the point (-3,2) on this graph
torpid perch
**Renato**
torpid perch
ok
torpid perch
`|z - 1| < |z + 3 - 2i|` is the same as saying z is closer to the point (1,0) th...
can you elaborate
torpid perch
Ok I got it to work lol
help
buoyant wadi
Do you understand how | | relates to distance?
torpid perch
https://www.desmos.com/calculator/ebhf9fsg9u
is this the argand plane
buoyant wadi
Yes
torpid perch
Do you understand how | | relates to distance?
the norm is the distance of a point (a,b) that represents a complex number wrt origin
steel solar
Did yall see that
buoyant wadi
the norm is the distance of a point (a,b) that represents a complex number wrt o...
So if |a+bi| is the distance from (a,b) to (0,0), Then |a+bi-1| is the distance from (a-1,b) to (0,0), which is the same as (a,b) to (1,0)
steel solar
I'll let NotABot handle this, I gotta go soon
buoyant wadi
Does this make sense at all?
torpid perch
So if |a+bi| is the distance from (a,b) to (0,0), Then |a+bi-1| is the distance ...
fair enough
is just the distance between two points in 2D
what u are using
buoyant wadi
Great! In general, |z-w| is the distance between z and w
torpid perch
and saying its analogous to the absolute val
buoyant wadi
Yeah!
Actually this norm on complex numbers is really the extension of the absolute val
Like, |-5| = |5i| = |5/sqrt(2) + i5/sqrt(2)| = 5
torpid perch
Actually this norm on complex numbers is really the extension of the absolute va...
so?
whats your point?
buoyant wadi
Well, if you use this idea of | | getting the distance between points, then
|z - 1| < |z + 3 - 2i|
Is the same as "z is closer to 1 than it is to -3+2i"
buoyant wadi
https://www.desmos.com/calculator/ebhf9fsg9u
Does that give you an idea of which parts of this graph will satisfy that?
@torpid perch Not to rush you, but I do have a time limit before I'll have to leave
torpid perch
Well, if you use this idea of | | getting the distance between points, then
`|z ...
what does that mean
first of all
buoyant wadi
Which part?
torpid perch
the set of complex numbers is not orderable
it does not have an order like 1,2,3,4,5, 1<2<3<4<5
buoyant wadi
Yeah, but | | takes complex numbers to the positive real numbers, which are orderable
torpid perch
|| : C -> R
buoyant wadi
Yeah
torpid perch
Yeah
Is the same as "z is closer to 1 than it is to -3+2i"
can you elaborate
@buoyant wadi
buoyant wadi
I'm thinking
waxen flax
Is the same as "z is closer to 1 than it is to -3+2i"
|z-1| is the distance between z and 1
|z+3-2i| is the distance between z and -3+2i
|z-1| < |z+3-2i| means “z is closer to 1 compared to -3+2i”
buoyant wadi
|z-1| is the distance between z and 1
|z+3-2i| is the distance between z and -3+...
This
torpid perch
|z-1| is the distance between z and 1
|z+3-2i| is the distance between z and -3+...
can you elaborate
waxen flax
what part is obscure to you?
buoyant wadi
What part needs elaboration?
torpid perch
expand the norm
buoyant wadi
So if |a+bi| is the distance from (a,b) to (0,0), Then |a+bi-1| is the distance ...
I talked about the relationship between the norm and distance earlier, up here
waxen flax
you have 3 objects, A, B, C
dist(A,B) < dist(A,C)
this is all that is happening
torpid perch
use the distance formula in 2D
waxen flax
use the distance formula in 2D
I mean, if you want to expand you can do it right?
you know what the formula is
torpid perch
I mean, if you want to expand you can do it right?
help
waxen flax
what's |z|?
waxen flax
So?
torpid perch
what's |z|?
the distnce between the (Re(z),Im(z)) and (0,0)
waxen flax
sure, then what
torpid perch
@waxen flax
waxen flax
I am right here why are you pinging, lol
torpid perch
sure, then what
i was just solving your question
i should be the one asking then what?
torpid perch
and I needed help with the expansion
buoyant wadi
So if |a+bi| is the distance from (a,b) to (0,0), Then |a+bi-1| is the distance ...
Do you understand how that leads to |z-w| is the distance between (Re(z),Im(z)) and (Re(w),Im(w))?
torpid perch
a little bit of handhold if possible
buoyant wadi
Buddy we are trying
torpid perch
Do you understand how that leads to |z-w| is the distance between (Re(z),Im(z)) ...
thats what I struggle with
buoyant wadi
Can you accept that its true for the sake of the question, and worry about getting a deeper understanding later?
torpid perch
you can draw it
draw z, w and their distance and use Pythagoras theorem so see i...
in the argand plane?
buoyant wadi
Yes
torpid perch
how will i draw if Z, W are unknowns
waxen flax
You can draw an z and w
torpid perch
we dont know Re(z) we dunno Im(z)
waxen flax
pretend B is z, A is w in the picture
f is their distance
Pythagoras says it's √(h²+g²)
g is just the difference of their real parts (x-values) and h the difference of their imaginary parts (y-values)
torpid perch
Pythagoras says it's √(h²+g²)
h and g?
waxen flax
in the picture
torpid perch
in the picture
C is origin?
waxen flax
No C is just the point where the horizontal segment at B and vertical segment at A meet
I needed it to draw
torpid perch
Do you understand how that leads to |z-w| is the distance between (Re(z),Im(z)) ...
|z-w| = sqrt((Re(z-w)^2 + Im(z-w)^2) = sqrt((Re(z) - Re(w))^2 + (Im(z) - Im(w))^2)
this is the proof of your statement
waxen flax
yes, it's basically literally Pythagoras theorem
torpid perch
yes, it's basically literally Pythagoras theorem
can you elaborate with drawing
torpid perch
g is just the difference of their real parts (x-values) and h the difference of ...
?
torpid perch
yes, it's basically literally Pythagoras theorem
I dont think I get it
waxen flax
In the drawing dist(A,B) = f = √(g²+h²)
and g = |ReA - ReB|, h = |ImA - ImB|
torpid perch
whats your point?
waxen flax
That $$d(A,B) = \sqrt{(\Re A - \Re B)^2 + (\Im A - \Im B)^2} = |A-B|$$
glossy valveBOT
Luigi
torpid perch
fur A,B ∈ C
waxen flax
yes, A, B complex numbers
torpid perch
d(x,y) = sqrt((x1-x2)^2 + (y1-y2)^2)
torpid perch
yes, A, B complex numbers
A = (Re A, Im A)
B = (Re B, Im B)?
torpid perch
yes, A, B complex numbers
why is the i omitted?
waxen flax
Usually one writes A = ReA + i ImA
waxen flax
why is the i omitted?
wdym omitted?
torpid perch
yeah, you are making it analogy eith vectors in R2
waxen flax
ℂ and ℝ² are the same thing
is not really an analogy, they are literally the same thing geometrically
torpid perch
in the argand plane where x is Re(z), y is Im(z)
torpid perch
is not really an analogy, they are literally the same thing geometrically
ok
can we continue with the exercise or no?
@waxen flax @buoyant wadi
waxen flax
they are isomorphic?
yh, or literally think them as the same thing
1 = (1,0)
i = (0,1)
isomorphic is also fine
waxen flax
can we continue with the exercise or no?
yes
torpid perch
then what
waxen flax
you want to go with the geometric interpretation or expanding and solving equation?
torpid perch
can we do both?
waxen flax
okay but you gotta start with one
torpid perch
solving equations
waxen flax
okay, take your equation and expand
glossy valveBOT
Renato
torpid perch
why the norms
waxen flax
by expanding z = x+iy
You said you want to compute using the algebra
You take your equation
substitute in z = x+iy
compute the norm
solve the inequality
torpid perch
You said you want to compute using the algebra
are we doig 1 or 2
waxen flax
1?
torpid perch
You take your equation
substitute in z = x+iy
compute the norm
solve the inequal...
ok
|z-1| = sqrt((Re(z-1))^2 + (Im(z-1))^2)
|z-1|= sqrt((Re(z) - 1)^2 + (Im(z) - 0)^2)
waxen flax
Please write Rez = x and Imz = y
so it's more readable
also what's the i doing in there?
torpid perch
Im(1) = 0
waxen flax
I mean write |z-1| = |(x-1)+iy| = √( (x-1)²+y² )
torpid perch
|z-1|= sqrt((x - 1)^2 + (y)^2)
waxen flax
yes
torpid perch
@waxen flax
torpid perch
idk
torpid perch
you have computations to do
|z-1|=sqrt(x^2-2x+1+y^2)
waxen flax
you can continue until you get stuck
torpid perch
im lowjey stuck in the algebra
torpid perch
(x+3)^2 = x^2 + 6x + 9
(y-2)^2 = y^2 -4y + 4
|z-1| < |z+3-2i|
sqrt(x^2-2x+1+y^2) < sqrt(x^2 + 6x + 9 + y^2 -4y + 4)
@waxen flax
waxen flax
you can keep going
torpid perch
i am doing the heavy lifting here
torpid perch
x^2-2x+1+y^2 < x^2 + 6x + 9 + y^2 -4y + 4
-2x+1+y^2 < 6x + 9 + y^2 -4y + 4
-2x+1< 6x + 9 -4y + 4
-2x-6x+4y < 9 + 4 -1
-8x+4y < 9 + 4 -1
-8x+4y < 12
-2x+y < 3
@waxen flax
waxen flax
that it looks less ugly
torpid perch
that it looks less ugly
2x-y+3>0
2Re(z) - Im(z) + 3 > 0
waxen flax
we don't need to write Re and Im it doesn't really help
At this point we don't care that they are complex numbers, just coordinates of points in ℝ²
waxen flax
Just forget about the complex numbers
you have this inequality, 2x-y+3 > 0, can you represent points (x,y) that satisfy it?
torpid perch
idk
torpid perch
what's 2x-y+3=0?
a plane in R3
waxen flax
We are definitely not in ℝ³
torpid perch
a plane in R2 that doesnt pass through the origin then
waxen flax
How many planes have you ever seen in ℝ²
torpid perch
0
waxen flax
so it doesn't sound right
torpid perch
is a line in R2
waxen flax
yes, can you draw it?
torpid perch
ding ding ding
yes
2x-y+3=0
y = 2x + 3
slope 2, and y intercept at (0,3)
,w graph y = 2x + 3
glossy valveBOT
waxen flax
that's the solution of the inequality
do you think a line came out of the blue or we could have expected it?
and how does this line relate to the original points?
torpid perch
that's the solution of the inequality
no
waxen flax
lmao wdym no
waxen flax
oh lol mb
I forgot it was > 0
so, again, forget about Re and Im
2x-y+3>0 let's write it as y < 2x+3
y = 2x+3 is the line shown
then what's y < 2x+3?
torpid perch
waxen flax
y < [...]
torpid perch
this is y >
waxen flax
okay, there we have the solution
torpid perch
,w plot y > 2x + 3
waxen flax
do you think a line came out of the blue or we could have expected it?
and how d...
same questions above
glossy valveBOT
waxen flax
it's the other part
torpid perch
it's the other part
?
waxen flax
you plotted y > 2x+3
torpid perch
you plotted y > 2x+3
2x-y+3>0
2Re(z) - Im(z) + 3 > 0
torpid perch
which is y < 2x+3
im stupid
waxen flax
My question still stands
torpid perch
I appreciate I think I understood 1)
glossy valveBOT
torpid perch
do you think a line came out of the blue or we could have expected it?
and how d...
?
waxen flax
that's my question for you not for myself
torpid perch
im not sure what to say
the line represents when |...| = |...|
we have an inequality tho
waxen flax
the line represents when |...| = |...|
yes, and why is that
Could we have drawn the line without solving anything?
torpid perch
yes, and why is that
we did the algebra
waxen flax
Without doing the algebra
torpid perch
Could we have drawn the line without solving anything?
who the fuck knows
waxen flax
You can calm down
waxen flax
if we have to solve |z-p| = |z-q|, would we need to do the algebra again?
for p and q fixed complex numbers
torpid perch
if we have to solve |z-p| = |z-q|, would we need to do the algebra again?
i dont follow
waxen flax
The geometric interpretation is pretty simple
|z-p| is the distance from z to p
|z-q| is the distance from z to q
What points are equidistant from p and q?
Clearly one of them is the midpoint (p+q)/2, what are the others?
the midpoint lies on the line segment connecting p and q
There's a pretty simple geometric object called “the segment bisector” that has the property that every point on it is equidistant from the extrema of the segment
this is the line we found solving the equation
torpid perch
The geometric interpretation is pretty simple
|z-p| is the distance from z to p
...
wdym?
waxen flax
this is the picture
A and B are two fixed points, in our exercise they were A = 1, B = -3+2i
f (black) is the segment connecting them
and the red line is the segment bisector
every point on the red line (for example C) has the property to be equidistant from A and B
and this answers our exercise which was to draw “{z ∈ ℂ | |z-1| = |z+3-2i|}” which in words is “the set of points equidistant from A=1 and B=-3+2i”
torpid perch
?
waxen flax
you might want to sit and think about it for a bit
torpid perch
and this answers our exercise which was to draw “{z ∈ ℂ | |z-1| = |z+3-2i|}” whi...
in my exercise is an inequality
waxen flax
Yes sure, this solves the equality, then the inequality is just the part of the plane containing the point in question
that's the easy part
torpid perch
i think I prefer re doing the algebra than understanding the geometric intuition
does that make me a caveman?
waxen flax
The genius of complex numbers was putting them on a plane for us to do geometry on
I might be really biased but geometry is the heart of mathematics, algebra is just the devil's offer (quote by Atiyah? idk)
torpid perch
I might be really biased *but geometry is the heart of mathematics, algebra is j...
fair enough
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@torpid perch Has your question been resolved?
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Available help channel!
tulip jolt
Alright I have tried solving this myself as well as using chatgpt/mathway and everything i try it says is wrong. Can someone else try this and let me know what they get?
steel raptor
What have you tried/what ideas do you have?
tulip jolt
I took the derivative and set it to 0 for the critical points etc
i got increasing (-4,-1)
decreasing (-1, 4)
local minimum at (-4, -21.33)
local maximum at (-1, 7.92)
and absolute min at (4, -106.67)
and absolute max at (-1, 7.92)
chat gpt said the same thing
but its all wrong apparently
steel raptor
I would assume its formatting then because all of those are correct
I would also say make sure you are including the local mins at both -4 and 4
devout valley
(Do you happen to have more exact values for those points? It may be unhappy with that, presumably)
tulip jolt
yeah ill try that again im just not sure how else to format it
nah those are as exact as i got if i remember right
steel raptor
You also dont need to have your local min and max formatted as points
tulip jolt
oh true
this software is so picky we had an issue like this before when i asked a question in here and no one could figure out why it was wrong for like an hour lol
steel raptor
Honestly though since you got the right answers and you could always talk to your teacher and have them fix it on their end
tulip jolt
i would if i could but this is a self paced class so we have no real professor no one really answers anything hahaha
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raven mango
question on pdes
full forumBOT
question on pdes
raven mango
by minimum principle, is proving u(0,t), u(x,0), u(1,t) > 0 sufficient to proving u(x,t) > 0 for all 0<=x<=l, 0<=t<=T?
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@raven mango Has your question been resolved?
atomic cliff
Hi
buoyant wadi
!status
full forumBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
1
buoyant wadi
Well, do you know how equations look like in the form y = mx + b?
atomic cliff
Well, do you know how equations look like in the form `y = mx + b`?
yeah
buoyant wadi
Do you think you could try turning the equation 3x + 8y = -24 into that form?
atomic cliff
No
buoyant wadi
Well, do you see how I could get 8y = -3x - 24 from that equation?
atomic cliff
yes
buoyant wadi
Alright, and do you have any idea how that could be turned into y = mx + b? The only difference left is that y is being multiplied by 8
atomic cliff
What do you mean by the diffrence
buoyant wadi
Well the only thing keeping 8y = -3x - 24 from being in the form y = mx + b is that 8
buoyant wadi
That's close, you should have brought in the minus sign though
y=(-3x-24)/8
Now do you know how to split that apart into y = mx + b form?
buoyant wadi
Now, does the line on this graph match the equation y = (-3/8)x - 3 ?
atomic cliff
no
buoyant wadi
I agree, so the answer isn't 3x + 8y = -24
How about 3x - 8y = 24? Do you think it might be that one? If so, think you can turn it into y = mx + b form?
atomic cliff
No
buoyant wadi
No you don't think it's right, or no you don't think you can turn it into y = mx + b?
buoyant wadi
What's up
atomic cliff
Cant I just chose a point on the graph and plug the numbers in
buoyant wadi
Yeah that works too!
This was just a method that I thought would be more educational
atomic cliff
This was just a method that I thought would be more educational
Would the answer be diffrent?
buoyant wadi
Nope not at all
Although you'd have to make sure you checked two points with each equation
Since two different lines can intersect in one place
atomic cliff
Although you'd have to make sure you checked two points with each equation
Both of the points has to make a equation correct right?
buoyant wadi
Yeah
buoyant wadi
8x - 3y = 24
🎊
Okay... well what do you think y = x rotated 90 degrees counterclockwise looks like?
atomic cliff
Okay... well what do you think `y = x` rotated 90 degrees counterclockwise looks...
I think it gets fliped to -x
atomic cliff
Now, do you know how to move a function to the left?
you have to subtract -5
buoyant wadi
Well... kinda
Moving functions left and right is a little counterintuitive, and I'm worried if I try too hard to explain it right now I'll wind up confusing you
But in this case, since it's a straight line, moving it left or right is the same as moving it up or down
atomic cliff
Would it be y = -x-5
buoyant wadi
Yes it would
But subtracting 5 is actually more like moving it down that it is moving it left
Moving it 5 to the left is like adding 5 to x before it gets put into the equation
Like y = -(x + 5), which happens to be the same as y = -x - 5
atomic cliff
i see
buoyant wadi
It's so counter intuitive that I literally wrote it backwards anyway lmao
You might expect adding to x would move it to the right, and subtracting would move it to the left, right?
atomic cliff
You might expect adding to x would move it to the right, and subtracting would m...
Yep
buoyant wadi
Yeah, but for reasons which I sadly don't really think I can explain right now, it winds up being the other way around
But anyway
Do you have more questions?
atomic cliff
Not any moer
buoyant wadi
Great! Can you use .close to free this channel then?
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torn jolt
This feels like it’s so simple but where is he getting the + 1/2 from on his equation answer?
full forumBOT
torn jolt
Is there a reason he splits his?
gritty flax
Depends what he is using it for
If you want to graph
You make the mx + p form clearly
torn jolt
And that’s why he did it?
gritty flax
Probably
gritty flax
You're welcome
torn jolt
.close
patent valve
Hello math fellows.
I would like to get some help regarding a problem I have in Calculus II.
So using change of variable(change of system) I need to find the area of bounded region by following two hyperbolas, and two reciprocal functions.
x^2 - y^2 = 1
x^2 - y^2 = 9
y=4/x
y=2/x
So I used the following change of variable as:
u = xy, and v=x^2-y^2, then find the Jacobian as the following:
J(u,v) = 1 / sqrt(v^2 + 4u^2) .
The integral looks a little scary(at least for me), as I found it as:
\int_{v=1}^{9} \int_{u=2}^{4} \frac{1}{2\sqrt{v^2 + 4u^2}} du dv
I'm asking if I did it right? or is there any better change of variable available for such problem?
Really appreciate your time and consideration.
woeful pasture
[\int_{v=1}^{9} \int_{u=2}^{4} \frac{1}{2\sqrt{v^2 + 4u^2}} du dv]
glossy valveBOT
k
woeful pasture
u used v = x^2 - y^2 not v =x^2 + y^2, right?
patent valve
u used v = x^2 - y^2 not v =x^2 + y^2, right?
Yes, sorry, my typo. fixed that.