So the point where it reaches 0 must be at a positive x
#help-28
torn jolt
Well you could just use the first method I mentionned
torn jolt
Everything after that is just me desperately trying to find a trick
Which failed
@tacit haven Don't forget to .close
tacit haven
.close
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normal spoke
Is there a number, n, where n^2 equals n, but n does not equal 1 or zero?
amber sapphire
Try solving the equation n^2-n = 0 and see if there are any other solutions
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tame bobcat
Hey guys I have a question on the proof that there are infintely many primes congruent to 3 mod 4 so of the form 4n +3
My question: I found two proof techniques. The one I find quite bizarre. The other is kind of like the proof for euclids theorem.
Suppose we have only finitely many of them.
We observe that a number x that is the product of odd numbers is congruent to 3 mod 4 if and only if the product has an odd amount of factors that are congruent to 3 mod 4
That means if there were only finitely many of those primes, p1 through pn, then every number thats congruent to 3 mod 4 had to be divisible by one of these primes, pi. But we find that either x is congruent to 1 or to 3. In the first case, if we add 2 to x, so x+2 that number is congruent to 3 again. And else if x is congruent to 3 then we do x+4 which is again congruent to 3. But we find that neither x+2 nor x+4 would be divisible by any of those pi.
I thought a lot about it and cant quite figure out why exactly x+2 and x+4are both not divisible by an pi. I did some examples and it always was the case. But can someone please provide a "proof" that neither is ever divisible by any of those pi?
tame bobcat
I know the theorem that says that for any number x the bcd(x, x+1)=1
But thats not been helpful to me here
torn jolt
you can show that gcd(x,x+2) = gcd(x,2) via euclidean algorithm
tame bobcat
And also for x and 4?
torn jolt
yeah
torn jolt
yup
tame bobcat
yup
You cant imagine how thankful I am right now lol
In some books they just assume everything is obvious
tame bobcat
yup
Theres this lemma here that says gcd (x, y) = gcd(y mod x, x)
Is that what I would wanna use to prove it?
torn jolt
Guys
ebon grove
some function of the form
f(x) = ax^2 + bx + c
real moat
Usually just something with x²
real moat
Multiplying x twice
torn jolt
ohhh thank u
real moat
Np
pure sandal
they are two different things
torn jolt
Can I send a photo of a problem that I need help with
dark shore
yes
weak iron
ok the length is increasing by 1 starting from 2 and the height is increasing by 1 starting from 1
weak iron
and then you can find the number of squares by new length * new height
weak iron
have a go given the information i provided
weak iron
(n)(n + 1) where n is the figure…?
torn jolt
what is the formula for the height of figure n
Adding one each time
pure sandal
that is a term to term formula
torn jolt
Wait
pure sandal
do you know an nth term formula?
torn jolt
Like the recipe or sum
weak iron
(n)(n + 1) where n is the figure…?
this is your answer maybe someone can explain this i have to go to sleep now sorry
torn jolt
OHH ok thanks 😭
weak iron
figure = iteration
torn jolt
Whatever that means yes
wide sundial
How many boxes is in the first one (and how wide and tall is the shape)
torn jolt
2+3+2.?
celest fulcrum
Quick question, if x y and z are all negitive what octant would it be in?
real moat
7 i think
onyx glen
the all-negative octant
celest fulcrum
Ok so
real moat
real moat
Yea i think so
celest fulcrum
damn alr
ill leave this up incase its not but its legit day 1 of the class and the ans key is wrong lmao
real moat
I learned it with 7th being the all negative one but I've seen some with it being six, never with eight tho
onyx glen
does it like... matter which octants get which numbers
like is that something you are expected to know and use frequently
celest fulcrum
Yeah
we are just starting 3d spaces @onyx glen
so we have to discribe stuff like this on the test and quizzes
kinda lost tho
onyx glen
and do you have any reference on hand for how the octants are numbered?
or are you expected to just Know™️
celest fulcrum
Nope
just know
its a class i just started and he shot a bunch of vids at us which are answer and question
no work nothing
im just watching ochem tutor vids to learn lol
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onyx glen
Nope
then ask your professor for a reference or cheat sheet or whatever that tells you how the octants are numbered
brazen hound
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cloud shore
Write z in terms of y
brazen hound
so like z=
quaint prawn
yes
brazen hound
make z the subject
chrome idol
whats the difference between linearization and linear approximation?
fast peak
as a formula, yes
the linearization is just some function L you get from f
while f(x,y) is supposed to actually approximate f near (a,b)
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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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
unborn quarry
I have graphed it out but don't know how to proceed
Do you know the disk/washer method when you have an upper bound function and a lower bound?
For this case
elder cape
Well I know the formula, yes.
I don't know what my upper and lower radii here are becuause the questions I have dealt with so far the region is always touching the x-axis, here it is floating.
ember shadow
floating?
unborn quarry
There are 2 functions here, the upper semicircle and the lower one. You can find both of them with the equation given
elder cape
There are 2 functions here, the upper semicircle and the lower one. You can find...
yes
but we only care about the right side
Will the limits of integration with 0 being the lower and 4 being the upper take care of that?
unborn quarry
Yes, but you restrict that with the x values
unborn quarry
Will the limits of integration with 0 being the lower and 4 being the upper take...
Yes that's for x
elder cape
hmmm
elder cape
There are 2 functions here, the upper semicircle and the lower one. You can find...
Is my upper: y = sqrt(16-x^2) + 4
lower: y = -sqrt(16-x^2) + 4
unborn quarry
Is my upper: y = sqrt(16-x^2) + 4
lower: y = -sqrt(16-x^2) + 4
Upper bound should be above the lower one (xd), look at those bounds carefully
elder cape
Upper bound should be above the lower one (xd), look at those bounds carefully
🤣 fixed it
what is the next step?
is this region not important to us?
because this is what creates the hole no?
ember shadow
yes
elder cape
Is my upper: y = sqrt(16-x^2) + 4
lower: y = -sqrt(16-x^2) + 4
how does this help us with that region?
elder cape
Upper bound should be above the lower one (xd), look at those bounds carefully
How should I proceed?
unborn quarry
how does this help us with that region?
Maybe watch a video about the disk/washer method, it would be useless if I just tell you the integral, and there are many videos that explain it really well
elder cape
Maybe watch a video about the disk/washer method, it would be useless if I just ...
I understand the disk/washer method. It's just that I havn't come across a problem like this with it. I am not sure what to do with this region:
.
unborn quarry
You should just care about what is inside the circle
That region doesn't matter, you've already stated lower and upper bounds. That region is not between them
elder cape
Yes but
if we let our upper radius be:
y = sqrt(16-x^2) + 4
and our lower be:
-sqrt(16-x^2) + 4
And we apply the disk method as:
V = [0 to 4] integral (pi (sqrt(16-x^2) + 4)^2) dx - [0 to 4] integral (pi (-sqrt(16-x^2) + 4)^2) dx
unborn quarry
Yes that's correct
No, you lower bound is not y=4
It's y = 4-....
which is the lower part of the circle
elder cape
So the final answer is indeed this?
unborn quarry
So the final answer is indeed this?
Yes, but you can try to put that all in the same integral
elder cape
I am still confused as to why we didn't have to calculate the volume of revolution under this curve:
.
and substract that from the volume of the semi circle being rotated along the x-axis
spiral vigil
that's what your second integral is doing
spiral vigil
and the first integral is doing this region's volume
elder cape
and the first integral is doing this region's volume
I'm not able to wrap my head around how the first integral gives this region's volume
unborn quarry
elder cape
bro I just finished watching that lol
but all the examples in that
the region was glued to the x-axis
here the region had this area:
spiral vigil
the green region is glued to the x axis
elder cape
ohhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
because here the outter radius is: y = sqrt(16-x^2) + 4 and lower radius does not exist so we get: [0 to 4] integral (pi (sqrt(16-x^2) + 4)^2) dx
and then this region:
the outer radius is: y = -sqrt(16-x^2) + 4 and the lower radius does not exist so we get [0 to 4] integral (pi (-sqrt(16-x^2) + 4)^2) dx
and that's why we subtract the 2 volumes
to get:
this.
Is my reasoning correct?
elder cape
the green region is glued to the x axis
wait hmm... but how does the first integral give us that entire region?
because the graph is not fixed to the x axis:
spiral vigil
the limits of integration form vertical lines
elder cape
https://youtu.be/SAHSVg7Jw_A?t=682
the volume of my outer radius is given by: [0 to 4] integral (pi (sqrt(16-x^2) + 4)^2) dx
and the volume of my inner radius is given by: [0 to 4] integral (pi (-sqrt(16-x^2) + 4)^2) dx
Then based off the disk method/washer method we just have to subtract the outer radius by the inner which is why that is the final answer?
elder cape
ok cool
elder cape
we subtract the outer volume from the inner volume
I had another quick question if u don't mind
spiral vigil
I'm about to drive for like thirty minutes 
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wanton scroll
Hello! I'm trying to solve an exercice and it asks me to identify the quadric represented in this equation:
4x²-2y²+z²=1.
In the first moment, I thought it was a one sheet hyperboloid, but when I tried it in Matlab, using the following code (I've never used Matlab before, so it can have a mistake), it looks a lot more like a hyperbolic paraboloid. Here's the code:
x=linspace(-10,10,40);
y=linspace(-10,10,40);
[x,y] = meshgrid(x,y);
4x.^2 - 2y.^2 + z.^2 == 1;
mesh (x,y,z);
view ([130,30]);
wanton scroll
and here's the image matlab gave ne
wanton scroll
The equation follows the form of the equation of a one-sheeted hyperboloid, but the image says another thing
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lime dawn
if a,b,c are nonzero, how do you solve for (x,y,z) in terms of a,b,c
fast peak
row reduce
lime dawn
well i wanted to do it without a calculator using something like substitution
i think i figured it out now tho ty
.close
covert wolf
For the k value I thought it was just the reciprocal but it’s not how do u find the k value like how do u transform it. For ex these 2 ones I don’t get
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For the k value I thought it was just the reciprocal but it’s not how do u find ...
@covert wolf Has your question been resolved?
covert wolf
<@&286206848099549185>
somber kindle
For the k value I thought it was just the reciprocal but it’s not how do u find ...
How has k-value been defined in your class? That's not a standard term.
balmy tusk
Katie has 52 playing cards, she wants to give one to Bill, one to fred and one to jill. How many different ways could she do this?
would this be 52C3 or 52P3?
nova basin
P3, order matters because people are different
balmy tusk
so if katie wanted to give 3 cards to one person, we would use 52C3?
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dense vector
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soft agate
where do I start for this question
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where do I start for this question
ocean geyser
Nancy's book is given as x, find the equation of Braveens books in terms of x, then find Cal's equation in terms of x and sum the three equations
soft agate
Nancy's book is given as x, find the equation of Braveens books in terms of x, t...
how would I find the equation of braveens without knowing
the x
ocean geyser
since x is the books, we can create braveens equation based on the number of books
read the question carefully
trying substituting x for where the question says "number of books nancy has"
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soft agate
trying substituting x for where the question says "number of books nancy has"
WAIT IS IT 3?
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WAIT IS IT 3?
stiff summit
WAIT IS IT 3?
No
stiff summit
WAIT IS IT 3?
How did you get that?
stiff summit
I'm still confused 😭
If you show your working, we can help you find where you went wrong
soft agate
okay I'll show you what I did
soft agate
If you show your working, we can help you find where you went wrong
honestly I'm sure that i did something wrong as this isn't making sense
stiff summit
honestly I'm sure that i did something wrong as this isn't making sense
How did you get from “nancy has x books” to “nancy has 3 books”? Because Nancy doesn’t have 3 books
soft agate
because this person
soft agate
trying substituting x for where the question says "number of books nancy has"
^
I thought they were referring to triple, so I thought 3
what am I doing wrong
@stiff summit sorry for ping
stiff summit
what am I doing wrong
when you triple x and add 5, what do you get?
soft agate
when you triple x and add 5, what do you get?
stiff summit
Do you know what triple means?
soft agate
Where did the x go?
x15
stiff summit
What do you mean
soft agate
X15
stiff summit
I don’t understand, can you say it using a sentence?
stiff summit
you said to triple x which would = 15x
no, when you triple x, you multiply it by 3(which is the same thing as tripling), so it becomes 3x
stiff summit
when you triple x and add 5, what do you get?
soft agate
when you triple x and add 5, what do you get?
we don't know what x is tho?
or would it turn into an exponent?
stiff summit
we don't know what x is tho?
Just now, I said what happens if you triple x
soft agate
3x
stiff summit
So what happens if you triple x, and then add 5?
stiff summit
This is what happens when you triple 3 and add 5.
Now what happens if you triple x and add 5?
soft agate
Now what happens if you triple x and add 5?
x3 + 5
soft agate
yay
stiff summit
So how many books does Braveen have, in terms of x?
soft agate
3
stiff summit
How did you get that?
stiff summit
It says:
Nancy has x books.
Braveen has triple the number of books Nancy has, plus 5 more books.
We can replace “the number of books Nancy has” with x, since we know Nancy has x books
So Braveen has triple x plus 5 more books
How many books does Braveen have?
stiff summit
It says: Nancy has x books
stiff summit
we figured out that ancy has 3 books?
What made you think that?
stiff summit
How about this, then:
N has 1000000 books.
B has triple the amount of books N has, plus 5. How many books does B have?
remember, triple means “three times”
soft agate
1000000 × 3 + 5
stiff summit
Yes
soft agate
= 3000,005
stiff summit
It says:
Nancy has x books.
Braveen has triple the number of books Nancy has, plus 5 more books.
We can replace “the number of books Nancy has” with x, since we know Nancy has x books
So Braveen has triple x plus 5 more books
How many books does Braveen have?
stiff summit
alright so X × 3 = X3 + 5?
Do you mean that Braveen has 3x + 5 books?
soft agate
yes
stiff summit
How many books do they all have together?
stiff summit
Did you mean that they have 10x + 15 books in total?
soft agate
so it can also be as 1x?
that's what I was confused on about I thought we had to figure what x was but I didn't know it already had its value
stiff summit
Yes, but if you want to save time you can also write x
stiff summit
that's what I was confused on about I thought we had to figure what x was but I ...
we don’t know its value, could be 4 or 5 or 15
soft agate
oh
stiff summit
how does it = 245?
the equation you now have to solve is: 10x + 15 = 245
soft agate
the equation you now have to solve is: 10x + 15 = 245
× = 23
stiff summit
WAIT IS IT 3?
So can you answer the question now?
stiff summit
Is your problem resolved?
stiff summit
so am I done
try inputting your answer into the question
soft agate
10(23) + 15 = 245
I have a question
so if nancy has 23 books
wouldn't the triple be 23 x 3?
+5
stiff summit
wouldn't the triple be 23 x 3?
triple 23 is 69 but triple 23 and then plus 5 is 74
primary school teachers are underrated
sweet rain
so I have a 94% in my class
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so I have a 94% in my class
sweet rain
final goes in as a normal test
my current average for tests is 94.79
there'd like 10 assignments in there already
what would be like rhe range I need to get to have a 92 final grade
light sonnet
frosty raven
frosty raven
try the second
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torn jolt
can anyone help me with math
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can anyone help me with math
onyx glen
we can if you
(a) stop spamming "can anyone help me with math"
(b) post your question
spice orchid
literally every single channel lmao
slender onyx
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viscid bluff
can you use generating functions to derive a formula for this: We roll
n dice and sum the highest 3. What is the probability that the sum is
18
18?
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can you use generating functions to derive a formula for this: We roll
n dice a...
@viscid bluff Has your question been resolved?
viscid bluff
<@&286206848099549185>
normal tree
hmmmm I don't think so
I think the problem here would be that your highest 3 random variables have a dependence on one another
normal tree
but I think this particular problem is tractable to solve
this is basically the same thing as asking the probability that you get at least 3 sixes in n throws
viscid bluff
yea
normal tree
that's just a binomial thing
or if you really want to use generating functions, you can envision that as the sum of independent Bernoulli random variables lol
viscid bluff
hmmmm
no im just wondering
like if i cant derive a summation formula
if i can just use generating functions
each time
normal tree
order statistics are really messy I think
viscid bluff
thats annyoing
normal tree
like the order statistics of a uniform distributed thing are disgusting
viscid bluff
hmmmmmmm
normal tree
you can get the distributions of most of them I think
just by doing the whole cdf thing
viscid bluff
yeah true
i gues
guess
what about a problem like this Player A and Player B take turns rolling two standard dice, first player A, then player B. If Player A rolls a sum of
6
6, A wins. If Player B rolls a sum of
7
7, B wins. What is the probability that Player A will be the victorious
can we use generating functions for this problem
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crisp fjord
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crisp fjord
Is this correct
kind jay
,rotate
glossy valveBOT
knotty marlin
Is this correct
yeah
scenic badge
How can i find third side if i know two of them and area of a triangle
wide sundial
cosine rule
scenic badge
i don't know a degree
wide sundial
you could try heron's formula
scenic badge
this exam is very stupid
wide sundial
heron's formula you dont need the angle
scenic badge
how is this 1 point question
wide sundial
if you need the angle you go area = 1/2 absinc
scenic badge
and all the answers are in the root
wide sundial
it's multiple choice?
scenic badge
yes
wide sundial
...
scenic badge
4 choice
wide sundial
show the choices
scenic badge
my teacher solved it with heron's formula
amber radish
pretty much cos or heron
well you could find the angle
but that'll js take more time probs
oh wait A=1/2 absinC for an SAS triangle
worn basin
just pythagorean finds cosC
amber radish
Using A=1/2 ab sin C we have sin C = 2/b
nah wait nvm
A=1/2 bh
Find h=2A/b, h is altitude
pythagorean theorem to find the missing length?
amber radish
well let's say it's orientated this way and we have p+q, and r and the area
well this is an orthogonal so it'll be easier
i mean duh
let me draw a scalene triangle
ye nvm fg Law of sines, law of cosines, heron's formula would do the job well done
worn basin
was it stated the triangle is right?
worn basin
generalize what, you cant generalize anyting by assuming the triangle is right
amber radish
no duh i meant in the sense of whatever triangle
worn basin
oh ok
amber radish
i mean the only think that generalises what we're looking for is heron's
you could use Mollweide's but you'll need angles for those ...
worn basin
yeah
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tame bobcat
in my book it says this (first picture)
Now lets say I have a random variable that is pareto distributed with parameters a = 3 and b = 5
Since the pdf for pareto variables are defined as in the second picture I dont understand how to integrate this function now ?
Do I only integrate the first case or how does that work ?
tame bobcat
or even more generally : How do I determine the probability P(X in [3,5]) for a pareto distributed random variable X ?
.close
torn bane
just want to check if i'm understanding this correctly, the dim(row(A)) is = dim(col(A))= rank of the matrix and null(A) is = # of free variables in a matrix
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just want to check if i'm understanding this correctly, the dim(row(A)) is = dim...
@torn bane Has your question been resolved?
torn bane
also, how is this true for all invertble matrices?
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celest fulcrum
Hi, I got this right but I want to know where they got the 1/3sqrt(19) from for part 2, I guessed based on the direction that I already know
rocky vale
They divided by the magnitude in order to normalize the vector (make it a unit vector)
3sqrt(19) = sqrt(191), they just simplified the radical a bit more
celest fulcrum
Yeah I realize that
but its supposed to be one perpendicular
so is multiplying it by the resprical make it perpendicular?
rocky vale
No, the cross product is always perpendicular to the plane formed by the original two vectors
Multiplying by the reciprocal makes it a unit vector
celest fulcrum
oh
ok one sec i need to look over this again
OHHHH
Ok I relixed it now
thank you
the wording got me
rocky vale
Awesome, np 👍
celest fulcrum
so the cross product is always orthogonal to the other two lines
and divinging it by the magnitude just gives its its unit vector
rocky vale
right, unit vector just means a vector with magnitude 1
So if you divide a vector by its own magnitude, you get a unit vector pointing in the same direction
cedar monolith
Let $p$ and $q$ be two prime numbers. \ Show that $pq$ divides $p^{q-1} + q^{p-1} - 1$
glossy valveBOT
lilisworld
cedar monolith
do i use fermat's little theorem?
Fermat's theorm: if p is prime, p divides $a^p - a$
glossy valveBOT
lilisworld
cedar monolith
<@&286206848099549185>
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@cedar monolith Has your question been resolved?
stark sky
Thinking, this seems a fair bit harder than your previous ones.
cedar monolith
Thinking, this seems a fair bit harder than your previous ones.
i can also use gcd and lcm and probably Bézout
thorn zenith
gosh I think I remember using this formula in number theory at some point proving this exact same thing lemme try to figure it out again
cedar monolith
ok
astral crown
i feel like if you multiply everything by pq
cedar monolith
since we learnt about gcd, lcm, bézout's lemma and fermat's little theorem maybe it would help idk
astral crown
you get a nice form
thorn zenith
^ you do
astral crown
$qp^q + pq^p -pq$
glossy valveBOT
*-algebra
thorn zenith
that's a good hint I would start there
astral crown
that just plain looks nice
cedar monolith
ok
astral crown
hm
stark sky
Tried this, I'm not sure it gets anywhere but by all means give it a go.
thorn zenith
yeah it also sort of makes the proof that pq divides it trivial because proving pq divides this form of it is obvious since we multiplied the whole thing by pq
stark sky
Yeah exactly the issue I ran into.
astral crown
that's a neat idea
thorn zenith
maybe something like $(p^q+q)(q^p+p)$ ?
glossy valveBOT
Blue Guilmon
thorn zenith
but you'd need to fiddle with it
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cedar monolith
.reopen
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thorn zenith
$(p^q+q)(q^p+p)=p^qq^p+p^{q+1}+q^{p+1}+p^qq+pq^p+pq$ so $qp^q + pq^p -pq=(p^q+q)(q^p+p)-(p^qq^p+p^{q+1}+q^{p+1})$
glossy valveBOT
Blue Guilmon
cedar monolith
**Blue Guilmon**
what is this
stark sky
So, FLT was the right shout.
thorn zenith
xD
I basically just expanded that polynomial that looked nice and subtracted the terms I didn't want 😛
but yeah probably overcomplicated it
cedar monolith
maybe something like $(p^q+q)(q^p+p)$ ?
why would we do this?
twilit leaf
What the hell is happening here
stark sky
why would we do this?
Ignore this, there is a simpler way.
thorn zenith
^
cedar monolith
do we use polynomials?
thorn zenith
nah don't
twilit leaf
This is an sfft problem
cedar monolith
fermat's theorem?
stark sky
Consider the expression $q^p - q$. By FLT, $p$ divides this expression, yes?
glossy valveBOT
kimbo7
stark sky
fermat's theorem?
Yeah
thorn zenith
oh I see
cedar monolith
i dont know what flt is
stark sky
Fermats theorem
stark sky
(It is often called Fermats little theorem to distinguish from Fermats last theorem, which I presume you wont be needing too soon)
cedar monolith
ok so p divides that
stark sky
The expression I wrote just now also has the form $q(q^{p-1}-1)$, can you say anything interesting?
glossy valveBOT
kimbo7
cedar monolith
q divides it
stark sky
Sure but that's not what I meant. Stick with $p$ first.
glossy valveBOT
kimbo7
cedar monolith
p divides it
stark sky
$p$ divides the whole thing, can it divide both factors?
glossy valveBOT
kimbo7
cedar monolith
it cant devide q but the other one then
stark sky
Exactly, so $p$ divides $q^{p-1}-1$. A completely analgous argument will show that $q$ divides $p^{q-1} - 1$, do you agree?
glossy valveBOT
kimbo7
cedar monolith
yes
stark sky
Now we can write the thing you care about in two suggestive ways. As $p^{q-1} + (q^{p-1}-1)$ or alternatively as $q^{p-1} + (p^{q-1} - 1)$, do you see how to progress from here?
glossy valveBOT
kimbo7
thorn zenith
**kimbo7**
keep in mind what you proved here
thorn zenith
you just proved that $p$ divides $q^{p-1}-1$ and that $q$ has to divide $p^{q-1}-1$
glossy valveBOT
Blue Guilmon
thorn zenith
use those facts
stark sky
$p$ divides the first one and $q$ divides the second, yeah. But they're both equal to what you care about, so...
glossy valveBOT
kimbo7
thorn zenith
yes
cedar monolith
$q | q^{p-1} + (p^{q-1} - 1)$
glossy valveBOT
lilisworld
thorn zenith
similarly q divides the other expression (which is the same expression)
cedar monolith
yes
stark sky
Altogether we have that $p$ and $q$ both divide $p^{q-1} + q^{p-1} -1$.
glossy valveBOT
kimbo7
thorn zenith
because p and q are prime otherwise you could run into issues but yeah
stark sky
In other words, both appear in the unique prime factorisation of that number.
stark sky
because p and q are prime otherwise you could run into issues but yeah
Only issue with primality so far is being able to use FLT no?
(Also much like an earlier question these should both be distinct primes, if they aren't this doesnt hold)
cedar monolith
In other words, both appear in the unique prime factorisation of that number.
so it's done?
thorn zenith
yeah we wouldn't have gotten this far without FLT either way but if for some reason you were given that p is just a number that happens to divide that expression and q is also a number that just happens to divide it certainly this wouldn't be necessarily true, as they could share common factors
I'm just sort of reiterating this
stark sky
so it's done?
It can be if you conclude properly!
thorn zenith
but it's not necessary
stark sky
In other words, both appear in the unique prime factorisation of that number.
Use this fact.
stark sky
yeah we wouldn't have gotten this far without FLT either way but if for some rea...
True, but we havent done that step yet!
cedar monolith
ok thanks you explained things well
stark sky
ok thanks you explained things well
Wait
I'd like to see you conclude that $pq$ divides that expression you care about
glossy valveBOT
kimbo7
stark sky
So far we have just said that $p$ and $q$ do
glossy valveBOT
kimbo7
thorn zenith
FTA baby
cedar monolith
because they both appear in the unique prime factor so in the factorisation there's at least a "qp"
uh
stark sky
because they both appear in the unique prime factor so in the factorisation ther...
Yes
But here you do need what Blue mentioned
If they weren't both prime, they could be, say, 2 and 4, then one divides the other and you run into issues
I.e. that wouldnt imply that 8 divides the number
thorn zenith
right it's enough to be divisble by 4 in order to be divisible by both 2 and 4
stark sky
Exactly
cedar monolith
so i need to say, according to the FTA ...
thorn zenith
FTA says two things
stark sky
FTA baby
As someone who uses algebra a lot i object to the appropration of this acronym for Fundamental theorem of arithmetic
thorn zenith
haha I know it's just a speacial case
of a much larger theorem too
in abstract algebra
but anywho the fundamental theorem of arithmetic says two things:
woops
stark sky
Lol
thorn zenith
but they're the same theorem!
stark sky
my point exactly
thorn zenith
XD
just over different rings
anywho
it says two things:
first and this might sort of an obvious statement: every number has a factorization into a product of primes
second: that factorization is unique up to multiplication by a unit
so use the first part
if every number has a factorization as a product of primes, just write one out for this expression
say that there are primes $\alpha_i^{\gamma_j}$ such that $\alpha_1^{\gamma_1}\cdot....\cdot\alpha_n^{\gamma_n}$ form a prime factorization of your expression in question
glossy valveBOT
Blue Guilmon
thorn zenith
then for some choice of i $\alpha_i=p$ and for another choice of i $\alpha_i=q$ and we just showed that $\gamma_i>0$ for those
glossy valveBOT
Blue Guilmon
thorn zenith
rearrange this factorization to show that $pq$ times the rest of this is equal to the expression
glossy valveBOT
Blue Guilmon
thorn zenith
that should be enough to show that $pq$ divides it
glossy valveBOT
Blue Guilmon
thorn zenith
I think you need to check that $p\neq q$ ?
glossy valveBOT
Blue Guilmon
thorn zenith
I don't think we assumed that
cedar monolith
no it was in the assignment, it said two distinct prime numbers but i didnt write it mb
thorn zenith
oh ok
yeah you need that otherwise it wouldn't be enough to show that the power is >0 we'd need >1
and that's a whole other proof altogether
cedar monolith
indeed
thorn zenith
sorry about the insane complication btw, I haven't done number theory in years lmfao
vapid crystal
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vapid crystal
Im not sure how to do this
humble steppe
Start by multiplying (1+cos x)/(1+cos x) to the fraction and then simplify 🙂
stiff musk
other possibilities, depending on what tools you have available: taylor series for cos(x), or l'hopital's rule
stiff musk
(1-cos x)(1 + cos x) is not 1 - cos x
vapid crystal
Yeah it is
stiff musk
i bet you $5 that it is not
glossy valveBOT
Bungo
vapid crystal
= 1
stiff musk
yep
vapid crystal
1/2
stiff musk
and so what's this limit? $\lim_{x \to 0}\frac{\sin^2(x)}{x^2}$
glossy valveBOT
Bungo
vapid crystal
1
vapid crystal
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umbral obsidian
You know that if y=f(x)g(x)
Then y'= f'(x)g(x)+f(x)g'(x), right?
vapid crystal
But its respect to y
umbral obsidian
You know that if y=f(x)g(x)
Then y'= f'(x)g(x)+f(x)g'(x), right?
This in leibnitz form equals to if y=u×v
dy/dx= u dv/dx +v du/dx
umbral obsidian
x^2 1/x^3 + 2xln(x^3)
ln(x³) differentiated is not 1/x³
There are multiple ways to prove wrong how it is
Which method would you like to hear, by not using chain rule or using chain rule?
vapid crystal
so then x^2 3/x +2xln(x^3)
umbral obsidian
Yes
That is dy/dx
dy/dx means to differentiate y with x
Which is what y' means in this context
vapid crystal
Ohh okay
umbral obsidian
Nothing really special
vapid crystal
And then d/dx is the same thing?
umbral obsidian
????
umbral obsidian
How does one even get 3x for differentiating this
vapid crystal
3x(1+2lnx)
umbral obsidian
so then x^2 3/x +2xln(x^3)
That
umbral obsidian
3x(1+2lnx)
Is equal to that
vapid crystal
Okay I get it now thanks
umbral obsidian
Simple modification
vapid crystal
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torn jolt
Hey
static bramble
Can someone please tell me how they derived this equation?
Actually I can answer myself. Have you been made familiar with Itô's Lemma?
torn jolt
Nope! I am sorry, I couldn't find materital online for this.
torn jolt
It's in the research paper, I am trying to understand.
torn jolt
I see.
torn jolt
The tl;dr is that they used this
If possible, then can you please teach me on Zoom how they derived these equations? I can pay you up to $70/hr.
I sincerely apologise if this is not a reasonable request.
static bramble
We don't do that here
static bramble
All good
Actually does this paper note the difference between $\tilde{W}$ and $W$?
glossy valveBOT
steakanator
torn jolt
Nope 😞
static bramble
Suspicious
torn jolt
Yeah
static bramble
Well either way, this particular one isn't that bad, I can probably explain it pretty quickly
As long as you don't ask me to prove anything
torn jolt
Sure 🙂 Thank you 😀
static bramble
$\dd r = \left( \psi + \frac 12 \sigma^2 \right) r \dd t + \sigma^2 r \dd W$
glossy valveBOT
steakanator
static bramble
I'm dropping the subscripts to save time
Now, you want to find the SDE of f(t, r) = ln(r)
You need the first partial derivative of f with respect to t, and the first two partial derivatives of f with respect to r
Compute those, will you?
torn jolt
You need the first partial derivative of f with respect to t, and the first two ...
Sure
Will it be zero?
Sorry, I am missing something.
static bramble
Will what be 0?
static bramble
Treat r as a variable, not a function of t
torn jolt
I think I am wasting your time here. I don't know how to do that.
I sincerely apologize.
static bramble
You don't know how to compute a derivative?
torn jolt
I know. But I am not getting what you are saying.
static bramble
What about what I asked is confusing you?
torn jolt
The problem is with me, not you.
You need the first partial derivative of f with respect to t,
This one
static bramble
Does ln(r) depend on t?
torn jolt
Nope
static bramble
So then its derivative with respect to t is...?
torn jolt
Zero 🙂
torn jolt
1/r
static bramble
And again
torn jolt
-r ^ (- 2)
static bramble
Good
static bramble
The tl;dr is that they used this
now it's just a matter of plugging and chugging as directed by these two equations
static bramble
no worries, good luck
torn jolt
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torn jolt
In each of the five triangles, the sum of the numbers in the vertices must be equal to the number inside the triangle. We want the number of https://cdn.discordapp.com/attachments/1120374309416882266/1120374309563670558/image.png
I have to put 1
because 18+1+1 20
then to the 21 i have to put 2 because i can change the 1 and the 18
then in the 2 i put 18, 2 ,2 since i cant change them
then in the 23 i put 18, 3,2 that gives 23
but doesnt gives to 24
torn jolt
Photo where i put the number is where i putted the info a man talked to me
<@&286206848099549185>
silver pasture
y dont u make a bunch of system of equations?
if its not something u're gonn abe asked on a test
u can just take ur time and do it
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torn jolt
if its not something u're gonn abe asked on a test
equations?
explain me what equations i shoul ddo
silver pasture
well suppose the circles house a bunch a numbers
a b c d and so on
u know their relation
a+f+b = 24
and so on
5 equations in total
try solving them
torn jolt
i was doing them without equations
silver pasture
u want to or are needed to
it would be a bunch a pointless guessing which is exactly what maths aims to prevent
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@torn jolt Has your question been resolved?
ornate ingot
it is in portuguese but I think you can understand. how can I calculate w-z?
ornate ingot
i did this but don’t know what else can I do
stable plover
check this
ornate ingot
check this
anything wrong?
ornate ingot
oh it misses the ^2
stable plover
you have $2\sin^2 \theta - 1$, not $2(\sin^2 \theta - 1)$
glossy valveBOT
tushar
ornate ingot
you’re right thanks
swift ocean
then that simplifies to -cos(2x)
stable plover
yes
ornate ingot
how?
stable plover
double angle formula for cosine
swift ocean
ornate ingot
Im so confused
swift ocean
Im so confused
ornate ingot
how do you go from this to this
ornate ingot
thanks
swift ocean
yep
ornate ingot
never thought about that
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uncut glade
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uncut glade
part b
i have got to the point of
90-arccos(14/sqrt62xsqrt14)
i used the first vector from the line equation
and the n vector
im guessing they are qrong but
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uncut glade
<@&286206848099549185>
uncut glade
<@&286206848099549185>
gritty rose
,calc 14/sqrt(62)*sqrt(14)
glossy valveBOT
Result:
6.6526734863609
Show how you got this
uncut glade
i may have rubbed it out
but i used the cos rule
a being the (7i+3j-2k)
b being (-i-5j+3k)
uncut glade
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