y1 = y0 + pa where p is an integer
#help-42
jolly charm
yea sure
main spade
jolly charm
yes perfect!
main spade
tysm, can i also ask about c) i was struggling for which method to use for that too
jolly charm
for c) try to guess a solution for x, y through naked eye..just put random values of x and y and see when the resulting expression becomes 4
then after that use the results obtained in b)
main spade
is this like a linear diaphantione eq
main spade
?
jolly charm
forget about diophantine equations
38x+17y = 4
tell me a solution of this equation..just plug in some numbers and check
x and y must be integers
jolly charm
great!
now what is the final result of b)
use that.. you have a and b as 38 and 17 respectively..you just found out x0 and y0 as 1 and -2
plug in those values and then solve for k
main spade
but arent there multiple
main spade
in the question it says all italicized so i was worried
jolly charm
no wait!
you are right
let me think
got it now
see number of solutions will be infinite
you need to mention the general form of the solution in part c)
main spade
wdym? like based on b)?
jolly charm
yes
so from b) you can write x1 = 1 + 17k and y1 = -2 - 38k
so for any integer values k the solution of the equation 38x+17y=4 is x1 = 1 + 17k and y1 = -2 - 38k
jolly charm
yes it will
you can check
hey I gotta go..if you still have doubts just call others or you can msg me.. I can help you later on
bye
main spade
tysm!
jolly charm
pleasure!
main spade
.close
fringe reef
suppose $R=\bZ[x]$ is a ring. what conditions apply to $R$ so that the polynomial remainder theorem is true?
potent lotusBOT
artemetra
fringe reef
not a textbook question, just wondering
from this question https://math.stackexchange.com/questions/2309994/polynomial-remainder-theorem-for-polynomials-over-ring
i assume R must be a Euclidean domain but can someone confirm this
mortal orbit
Z[x] is a subring of R[x]...
fringe reef
from this question https://math.stackexchange.com/questions/2309994/polynomial-r...
actually no, i misread the answer, doesn't seem to a euclidean domain
mortal orbit
Wait
oblique current
yeah Z[X] is not a euclidean domain
fringe reef
yeah my bad
mortal orbit
The thing is when you divide by x-a, a must be an integer for you to be allowed to divide in Z[x]
Then you can expand Z[x] as a polynomial in R[x] where you know the remainder theorem to be true
oblique current
you can just try and divide X by any non-unit and run into problems
fringe reef
i see
mortal orbit
Yes, it's not dividing by X-a that will cause problems
fringe reef
from this question https://math.stackexchange.com/questions/2309994/polynomial-r...
i'm very confused by the answer
so polynomial remainder theorem applies to any ring? only a commutative one? no other conditions?
oblique current
The polynomial division algorithm works over any ring as long as you are dividing by a monic polynomial.
fringe reef
yeah i'm confused by that line specifically
oblique current
its just saying there is a division algorithm for polynomials over any R is you are dividing by monics
but not in general for any ring if you're trying to divide by non monics
calm coralBOT
@fringe reef Has your question been resolved?
fringe reef
but not in general for any ring if you're trying to divide by non monics
okay i see
thanks
calm coralBOT
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Available help channel!
white atlas
does stars and bars assume a case of x_i >= 0?
calm coralBOT
does stars and bars assume a case of x_i >= 0?
calm coralBOT
@white atlas Has your question been resolved?
calm coralBOT
@white atlas Has your question been resolved?
olive wigeon
calm coralBOT
olive wigeon
I don’t understand what this quesiton means
fierce raft
it means that the vertical height and radius of cylinder and cone is same
olive wigeon
.close
simple badge
i need a polynomal to do polynomal division with i dont know how to make one that i can solve
graceful dust
wdym “that i can solve”
simple badge
wdym “that i can solve”
none of them goes up cuz cant factorize or somthing
graceful dust
@leaden thunder 
graceful dust
none of them goes up cuz cant factorize or somthing
you want ones that “divide nicely”?
leave no remainder?
simple badge
yeah
graceful dust
take any two polynomials and multiply them together
simple badge
leave no remainder?
cuz idk how to do that with the rest with polynomal
simple badge
take any two polynomials and multiply them together
how
graceful dust
just take something like (x^3 + 2x + 2)(x - 8) and multiply it out
simple badge
just take something like (x^3 + 2x + 2)(x - 8) and multiply it out
i only know division
graceful dust
then if you divide it by x - 8 it should divide nicely
you know polynomial division but not how to expand polynomial multiplication?
simple badge
you know polynomial division but not how to expand polynomial multiplication?
no
not supost to know anyways
graceful dust
alright i’m lost on what you’re asking for
simple badge
me to my brain is melted
halcyon rock
alright i’m lost on what you’re asking for
me too ^
halcyon rock
me to my brain is melted
come back later
simple badge
come back later
k
runic jewel
No excuses
halcyon rock
^^
simple badge
No excuses
bro
runic jewel
Never too late for a good nights walk
simple badge
Never too late for a good nights walk
mabye but idk i want to do some math or somthing
runic jewel
Where are you from if I may ask?
simple badge
and i am still bad at the polynomal stuff
simple badge
Where are you from if I may ask?
Norway i hate when it is so dark in the winter
oblique current
do you currently have a polynomial in front of you that you are trying to factor
simple badge
Norway must be cold rn
not really
simple badge
Norway must be cold rn
it is just 3 degrees celcius rn not that cold
runic jewel
Dang okay
simple badge
do you currently have a polynomial in front of you that you are trying to factor
no i need one
runic jewel
I thought it would be colder since it’s 5 degrees in Germany rn
simple badge
I thought it would be colder since it’s 5 degrees in Germany rn
k
simple badge
I thought it would be colder since it’s 5 degrees in Germany rn
i am not in the coldest part
oblique current
take any polynomial that you know has a root, and divide it by (x-that root)
simple badge
take any polynomial that you know has a root, and divide it by (x-that root)
tf i only know how to do the devide i know nothing else i am not supost to learn it in my grade
runic jewel
(x^{4}+4x^{3}-7x^{2}-22x+24)
halcyon rock
tf i only know how to do the devide i know nothing else i am not supost to learn...
watch the language
potent lotusBOT
FirstNameLastName
simple badge
watch the language
💀 bro
oblique current
then you really have no other option than just trying if you literally dont know anything else about polynomials
simple badge
then you really have no other option than just trying if you literally dont know...
yeah
i wil just try
or i can start with my grade math or somthing
oblique current
i suggest you look up how to multiply polynomials
that way you can just multiply two together and then divide by one of them
simple badge
ok
oblique current
generally multiplying is easier than division
dawn idol
**FirstNameLastName**
whats this?
simple badge
whats this?
polynomals or somthing
simple badge
generally multiplying is easier than division
so it aint that hard?
dawn idol
polynomals or somthing
whats the problem in it?
oblique current
simple badge
whats the problem in it?
idk
ivory vapor
Compare $(2n)^{ln(n)}$ and $(ln(n))^n$ for $n > 1000$.
Someone in the forum suggested to divide one by the other and compute for n -> infity, but that produced a different result than WolframAlpha and I couldn't understand the equation transformations, so I'm unsure what to do next.
#1172829725748629555 message
I apologize if it's not allowed to post in both places.
potent lotusBOT
improbablep
calm coralBOT
@ivory vapor Has your question been resolved?
calm coralBOT
@ivory vapor Has your question been resolved?
calm coralBOT
@ivory vapor Has your question been resolved?
calm coralBOT
@ivory vapor Has your question been resolved?
calm coralBOT
@ivory vapor Has your question been resolved?
calm coralBOT
halcyon rock
snopes
solar pewter
hi
solar pewter
wdym
halcyon rock
you can't ask us to do every problen here
solar pewter
why
halcyon rock
come back in 30 minutes
solar pewter
why
halcyon rock
snopes is a friend of mine
solar pewter
okok
calm coralBOT
Available help channel!
bright hawk
I'm very confused about how to solve this without setting up a differential equation
bright hawk
do you understand why?
Not really
pure kayak
(gal/min)*min=gal, the total number gallons that flowed out are given by the area under that line
dont even really need to set up an integral
bright hawk
(gal/min)*min=gal, the total number gallons that flowed out are given by the are...
Oh I see
Thank you
.close
hollow sphinx
calm coralBOT
hollow sphinx
I know that f'(x) dne at x= 3.5
and critical points at x=2.5 and x=4.5
but idk what to do after that
im guessing since the graph decreases before 2.5 aand increases after it its a minimum and a max at x=4.5 since it increases before 4.5 and then decreases
calm coralBOT
@hollow sphinx Has your question been resolved?
hollow sphinx
<@&286206848099549185>
calm coralBOT
@hollow sphinx Has your question been resolved?
old falcon
I know that f'(x) dne at x= 3.5
i could be wrong but it looks like you have a lot of maximums and minimums
isnt f' just 4 constant functions
pine schooner
is there a specific technical term for the part of the definition of an ideal about for a in I and b in R that ab and ba are in I?
pine schooner
My prof has been colloquiually referring to it as the "sticky" property, but I don't want to put that in a proof and I was hoping there was a shorter phrasing than just saying that whole part of the definition
frozen ibex
I is stable by multiplication with an element of R
mortal orbit
Outer product stability?
slate field
is there a specific technical term for the part of the definition of an ideal ab...
absorption (used by wikipedia, but i have not seen a name for the property outside of that)
calm coralBOT
calm coralBOT
Available help channel!
outer oak
how can i show this combinatorially?
calm coralBOT
how can i show this combinatorially?
exotic falcon
Think of what each side is counting
you want to create a scenario, where it is obvious that each side is counting the same thing and hence they must be equal
thinking of what that 'thing' is though, is the difficult part
outer oak
i was thinking about 2^(n-1) as "how many ways there are to colour n vertices using 2 colours, but be don't care which colour is which"
but i don't really know how to interpret multiplying it by n
hmmm
unless
n might be (n choose 1)
graceful dust
powers of 2 and n choose k both have to do with sets and subsets
so try relating them that way
outer oak
n might be (n choose 1)
what would mean that we choose n-1 vertices from the set of n vertices
and colour them using 2 colours
but i have no idea how to interpret right hand side multiplication 🤔
fluid flame
what would mean that we choose n-1 vertices from the set of n vertices
this is close but not quite there
outer oak
so i guess i should interpret it a bit differently 🤔
maybe we choose 1 vertex out of n and colour the rest using n-1 colours? 🤔
but it still doesn't feel right
i am not allowed to transform this equation, right?
because i've already notices that after dividing whole thing by n it becames trivial
ok, i've got it B)
thanks
.close
calm coralBOT
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Available help channel!
obtuse rivet
what is 359/442 in base 35
calm coralBOT
what is 359/442 in base 35
obtuse rivet
probably shouldnt be wasting math helpers time with it though
my friend asked me this
what do you mean?
next hound
my friend asked me this
its a joke question
obtuse rivet
can u explain why
next hound
,w 359/442 in base 35
potent lotusBOT
obtuse rivet
wtf lol
@next hound can u explain what is 359/442 in base 35 tho?
just any fraction base smth
shadow tendon
Lol
next hound
<@153882568322711553> can u explain what is 359/442 in base 35 tho?
works the same as fractions in any base?
obtuse rivet
works the same as fractions in any base?
yeh like can u explain that in general
next hound
convert the numerator and denominator into the relevent base, then do long division (in that base)
probably the easiest way
obtuse rivet
i see
how can i convert
a number into base smth
then
and why does it output letters
next hound
well, base 10 has 10 digits, namely {0 1 2 3 4 5 6 7 8 9} - if we want the base to go up, we need more symbols
what symbols make sense to add? the alphabet of course!
obtuse rivet
oh i see
next hound
so base 16 (a common one), includes A-F as well
obtuse rivet
why cant we have more numbers
obtuse rivet
well, base 10 has 10 digits, namely {0 1 2 3 4 5 6 7 8 9} - if we want the base ...
base 20 has 20 digits
next hound
well, do we have more digits? Im only aware of 10 of them
next hound
base 20 has 20 digits
as usually written, it has the 10 digits and then the first 10 characters from the alphabet
next hound
are you familiar with binary by chance?
obtuse rivet
like the word means 2 choices?
next hound
no
okay hmm
well lets use base 16 as an example
lets say we want to convert 10 from base 10 to base 16
how can we do this?
next hound
yeah thats exactly right
but wait we have a problem
what if we want to write 16? F is only 15
what do we do?
next hound
no, we still need to use base 16
next hound
hint: its like going from 9 (one digit) to 10 (2 digits)
next hound
what is this?
do you agree in base 10, if we have a number like 56, its really just 5 * 10 + 6 * 1?
next hound
thats exactly right
obtuse rivet
so to make 16 as
next hound
thats how every base works
next hound
yes $16_{10} = 10_{16}$
potent lotusBOT
next hound
reads as "16 in base 10 is 10 in base 16"
for a challenge, try doing 254 in base 16 and let me know what you get
obtuse rivet
ok
hollow totem
iron
obtuse rivet
for a challenge, try doing 254 in base 16 and let me know what you get
ok i was gonna say
,w 254/16
potent lotusBOT
obtuse rivet
,w 15*16
potent lotusBOT
hollow totem
,w 254 in base 16
obtuse rivet
you obtain a quotinet of 15 and a remainder of 4
the remainder 4 yields d?
@next hound
wait 14
is remainder
bruh
next hound
you obtain a quotinet of 15 and a remainder of 4
remainder of 4 is wrong
yeah
14 is not D though, one letter off
remember, 10 is A
obtuse rivet
ok so it repeats i suppose every 6 digits
A, B, C, D, E, F
A, B, C, D, E, F
A, B
next hound
ok so it repeats i suppose every 6 digits
what?
can you give me the result of 254 in base 16 though to make sure you are right?
obtuse rivet
ohoh
hollow totem
write as many numbers as you can in base 16 in order before you don't know
next hound
so what is the whole number?
obtuse rivet
the quotient needs to be converted as wel right
next hound
yes
next hound
yep!
next hound
this is how it works for all bases, just powers of that number instead of 10 for base
hollow totem
next hound
,w 254 in base 16
potent lotusBOT
calm coralBOT
calm coralBOT
Available help channel!
sturdy hamlet
Did i do this correctly?
calm coralBOT
Did i do this correctly?
calm coralBOT
@sturdy hamlet Has your question been resolved?
forest ocean
calm coralBOT
steel mason
Lcm then divide
forest ocean
What does lcm mean
steel mason
Put in same denominator
forest ocean
Lowest common multiple ??
steel mason
Yeah dog
royal raft
Yep now divide that by 2/3
forest ocean
How
forest ocean
Flip it
royal raft
Yeah so it’d be 3/2 and since division is the same as multiplying by the reciprocal we can multiply 9/30 by 3/2
forest ocean
Do you only do the numerator
royal raft
When multiplying fractions you multiply the numerators together and you also multiply the denominators together
royal raft
21/30 -12/30
This should be 21/30 - 8/30
Because we multiplied the 4/15 by 2 to make the denominator 30
So we multiply 4 by 2
We multiply 7/10 by 3 to make the denominator 30 so we multiply 7 by 3
forest ocean
Oh Yh
forest ocean
26/60
royal raft
Well since the reciprocal is 3/2 it’s going to be 13x3/2x30
forest ocean
Is that the answer
calm coralBOT
calm coralBOT
Available help channel!
hybrid hawk
I, uh, don't have a clue on where to begin
calm coralBOT
I, uh, don't have a clue on where to begin
fierce raft
u can use distance formula to get the distance btw them as a function of time and then differentiate
brazen obsidian
Bruh
polar jetty
Do you have a question to ask? If not, please close the channel.
calm coralBOT
@brazen obsidian Has your question been resolved?
main spade
is there an easy way to remember the gram schmidt algorithm
calm coralBOT
is there an easy way to remember the gram schmidt algorithm
trail yarrow
I mean if you can remember what it's doing it becomes easier
main spade
okay, could we go over that
trail yarrow
it just goes through the basis vectors in turn, and removes the components parallel to the ones before
I've tried find an animation or something online, wikipedia has a decentish one
neat gorge
I'm reading The Theoretical Minimum by Leonard Susskind, and he has a section that gives you a tiny overview of trigonometry needed for the book.
I have a few questions.
-
In his unit circle explanation, he has hypotenuse as side c, and he has opposite as side a. He then says that sine = c sin(θ). Is there a reason for the multiplication of c here or is it just a distinction without a difference? Usually I see it as just sine = sin(θ). Since it's just a/c which is obviously a. Multiplying by c gives the same answer since c = 1, but just wondering if there's a pedagogical reason for it.
-
He says: if θ is the sum of α + β, then we may write: sin(α + β) = sin(α)cos(β) + cos(α)sin(β). It's not immediately obvious to me why this works.
-
Lastly, with vectors. Everything made intuitive sense aside from multiplication using the dot product. He says that multiplying vectors is multiplying their magnitudes and then multiplying that by the cosine between them.
My only confusion is that a vector has magnitude and direction. If we multiply their magnitudes, what happens to the direction? As an example, if we have two vectors with magnitude 3 and we multiply them (ignoring the cosine for now), we get 9. But if these two vectors are pointing in opposite directions, what direction is "9" pointing toward?
trail yarrow
- I honestly don't know
- It shouldn't be immediately obvious, but there is a proof using circle trigonometry you can probably find online
oblique current
for 2, its not immediately obvious to anyone why that works when seeing it for the first time either, there are geometric proofs you can find ^
trail yarrow
Now the long one: unlike multiplication of numbers where you get another number, the dot product does not give you another vector, and you lose (most) direction information
just about the only thing you can read out of the result without knowing the initial magnitudes is whether they point in roughly (up to 180 degrees / pi radians) the same direction or not, and this comes from the sign
it's still helpful to have an operation like this, for reasons I can't explain other than 'it is this way'
neat gorge
Okay, thanks folks. Glad I wasn't just hopeless. It makes sense that we wouldn't be able to retain the direction and I was partially hoping that we couldn't (because I couldn't imagine any way to make it work).
What exactly should I look up for those geometryic proofs? I tried googling something like a + b = theta, but I'm not sure that yieled the right results.
trail yarrow
"angle sum proof" should be good
neat gorge
it's still helpful to have an operation like this, for reasons I can't explain o...
This is fine with me. I don't mind it being just a definition.
Okay, t hanks.
trail yarrow
*sine angle sum formula proof would probably work better
neat gorge
The first one got me where I needed. Thanks again. This was very helpful 🙂
hollow totem
My favorite way for 2 is to find the formula for rotating an arbitrary point (x,y) by an angle θ about the origin
which you can rotate (x,0) and (0,y) and then add the results
oblique current
i would add that given this was a "tiny overview of trig" and you presumably haven't learnt trig then these are pretty natural things to be confused about
neat gorge
i would add that given this was a "tiny overview of trig" and you presumably hav...
Yeah, I've not learned trig yet and anything about vectors. I'm happy to learn them though. It's been very fun so far.
oblique current
definitely stuff work checking out, as you see they crop up everywhere:)
obsidian geode
Can somebody help pls
calm coralBOT
Can somebody help pls
leaden thunder
Can somebody help pls
go one by one and factor primes and prime powers out of both numbers
(ab + ac) = a(b + c)
applies to all numbers a, b, c
calm coralBOT
@obsidian geode Has your question been resolved?
calm coralBOT
Available help channel!
red locust
calm coralBOT
late edge
Hey there
red locust
sup
leaden thunder
am i right with D here?
!show
calm coralBOT
am i right with D here?
Show your work, and if possible, explain where you are stuck.
late edge
Think about it, that's a cubic equation representing volume. It has three roots. You have the length and the width. From this, how do you think you can get the height?
@red locust
red locust
i did this
leaden thunder
,w expand (x-5)(x+1)(x+3)
potent lotusBOT
leaden thunder
yes you're right
late edge
You don't really need to do algebraic long division here though @red locust
red locust
You don't really need to do algebraic long division here though <@56347237742647...
yeah i know
late edge
You can do it by observation, I find it's a little less complex
But yes, x+3 is correct
red locust
You can do it by observation, I find it's a little less complex
your right i just find the other way easier for some reason idk
late edge
That's fine, you're doing great!!
limber ridge
calm coralBOT
limber ridge
Find the t and p value as well
calm coralBOT
@limber ridge Has your question been resolved?
calm coralBOT
@limber ridge Has your question been resolved?
cedar shell
if a number gets increased everyday for a year (365 days) by how much % did it increase
calm coralBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
next hound
eng:
a) a size that increases by 1% each day has increased by what percent overall (over a year)?
b) same but decrease 1% each day
hasty fiber
Aha, suddenly it's not so vague
next hound
consider the case of 1 day, then 2 days, then see how you can extrapolate
cedar shell
eng:
a) a size that increases by 1% each day has increased by what percent overa...
idk if my german is shi but isnt it each day 1% not each year?
next hound
would it be f(1)=m*a^1
what is m and a
for simplicity, you can just use n for number of days, and 1 for starting amount
calm coralBOT
@cedar shell Has your question been resolved?
coral dew
I don’t understand why it’s showing me this is incorrect. I put the equations into a graphing calculator and what I picked was true
coral dew
.close
remote mural
calm coralBOT
remote mural
How to do
pure kayak
looks very fun
remote mural
can you please do it and show me the steps
and then explain
😭
we didnt learn this we're supposed to research it
remote mural
gx^2
pure kayak
honestly it may be better to get g(x) first itself
in the expression for g(x), replace all the f(x)'s with what f(x) actually is
the exponent of the 2 makes it look worse than it is
remote mural
.close
mortal geyser
can someone help me with geomtery😭
calm coralBOT
can someone help me with geomtery😭
mortal geyser
how to find the range for both of these?
i know the answer bc of t he answer key but i need to know how to find teh answer😞
guys please help im struggling
burnt robin
draw pictures and see what the minimum and maximum angles could be
drowsy widget
calm coralBOT
drowsy widget
I'm pretty sure I have to divide both sides by dy
and then reciprocate both sides
to get dx = (1/4y) dy
but when I take the integral
I get e^x = e^(ln(4y))
which makes y equal to .25e^x
which isn't one of the choices
well
I do the e's after
but
its x = ln(4y)
oh
hmm
OH
IM DOING IT BACKWARDS
hmm
no im not
upper sparrow
its x = ln(4y)
I mean, I guess technically you could make some argument based on integration constants 
But differentiate ln(4y) please? (wrt y and don't forget ⛓️)
upper sparrow
You've basically claimed the integral of 1/(4y) wrt y is ln(4y)
So I'm asking you to differentiate ln(4y) with respect to y
(and strongly hinting you'd need to make use of chain rule!)
drowsy widget
oh
it would be
4/4y
so you are saying I need to divide by 4 when doing the integral
or
oh
it would be
ln(y)/(4) + c
irght
@upper sparrow
upper sparrow
ln(y)/(4) + c
And correct 
drowsy widget
oh god
how do i remember that
why do i not know taht
I guess just separate constants first
upper sparrow
I guess just separate constants first
Yea a very good idea, cause it's somewhat "hidden" in a way because it's in the denominator
Something like 2/y might be quite obvious you're multiplying by a constant but then it's easy to get tripped up on something like 1/(4y)
drowsy widget
yes
also wait
next step is use the given parameters
to find c
but how do i do that with ln(4)/4
do I just subtract itself to equal 0?
c = -ln(4)/4
upper sparrow
Yep that's fine to do 
So then you basically get to x = ln(y)/4 - ln(4)/4
Which shouldn't be too painful to rearrange for y now
drowsy widget
I put as exponent of e like before right?
drowsy widget
multiply both sides by 4
upper sparrow
Yep, makes your life that tiny bit easier 
It would work out in the end if you didn't though, but a tiny bit more manipulation you'd need to do!
upper sparrow
y = e^(4x) + 4
Show your steps, think you've broken the law along the way 
upper sparrow
Yep yep 
upper sparrow
Also if you did it rules of exponents way, you get $e^{\ln(y) - \ln(4)} = \frac{e^{\ln(y)}}{e^{\ln(4)}} = \frac{y}{4}$
potent lotusBOT
@upper sparrow
upper sparrow
answer is A
Yep
drowsy widget
thank you
thank you lord chartbit
what honorary title would you prefer in the land of the mathematics
lord? squire?
elite?
drowsy widget
the honor is all mine tysm
I may have more questions to be discovered soon
have a great day
.close
frosty mica
How do I go about proving this?
calm coralBOT
How do I go about proving this?
frosty mica
I would imagine it has something to do with the fact that every nth set requires double the amount of subsets of the previous one, but I have to use induction.
upper sparrow
Assuming that a set with n elements has 2^n subsets, and then you get a set with an extra element, think about removing it
Then you know of course the assumption, and think about what you can do with that element you removed...
frosty mica
How does one remove the extra set?
Am I able to say "If a set with n elements contains 2^n subsets, then a set with n+1 elements contains 2*P(n) subsets"?
I'm not really sure what that gets me, but I'm just tryign to take some steps.
upper sparrow
So say you assumed the induction hypothesis, that a set with $n$ elements has $2^n$ subsets, and then I give you the set
[
{ x_1, \ldots, x_n, x_{n+1} }
]
with $(n + 1)$ elements. You can take out that $x_{n+1}$ and you're left with a set with $n$ elements
potent lotusBOT
@upper sparrow
upper sparrow
Am I able to say "If a set with n elements contains 2^n subsets, then a set with...
Well P(n) is a statement rather than a number, and if you meant 2 * 2^n, you'd need to justify that
frosty mica
Right. We're supposed to denote the hypotheses as P(n) and P(n+1)
upper sparrow
Yea, so you assume $P(n)$, which then tells you that a set with $n$ elements (such as ${ x_1, \ldots, x_n}$) has $2^n$ subsets
potent lotusBOT
@upper sparrow
upper sparrow
Does the x_(n+1) subset contain 2 subsets?
Well the subset {x_{n+1}} does have two subsets of it sure, but the idea is mainly that...
upper sparrow
<@788085606483361802>
...for each of these subsets, we can then think about that $x_{n+1}$, and there are of course, two choices when it comes to $x_{n+1}$ and the subset
potent lotusBOT
@upper sparrow
frosty mica
I think I know what you're saying, I'm just trying to figure out how to write it out.
upper sparrow
Yeee it's mostly about how to word it 
wicked igloo
I'm computing for an oblique asymptote
calm coralBOT
I'm computing for an oblique asymptote
calm coralBOT
@wicked igloo Has your question been resolved?
calm coralBOT
@wicked igloo Has your question been resolved?
wooden osprey
calm coralBOT
wooden osprey
i can do the u substitution for the bottom but i dont get it for the top i still have an x in the integral
i haev symbolab open and it tells me that the top is u-2 but i have no idea where the -2 is coming from
<@&286206848099549185>
calm coralBOT
@wooden osprey Has your question been resolved?
calm coralBOT
@wooden osprey Has your question been resolved?
calm coralBOT
calm coralBOT
Available help channel!
fierce elbow
deduce whether this congruence is true or false:
[6]x[38]=-2
My solution:
6mod(5) = 1
38mod(5) = 3
1x3=3
3=[-2]mod(5)
So true
is this correct?
remote mural
Hello
fierce elbow
Hello
remote mural
Hmm
That should be correct
Unless I’m missing something
Does what ever your doing say it’s wrong?
fierce elbow
Does what ever your doing say it’s wrong?
wym
cloud topaz
Yes
remote mural
Unless I’m missing something in the question
thorny stump
5a + 6*38 = 5b -2, yes
fierce elbow
Thanks
cloud topaz
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calm coralBOT
fierce elbow
.reopen
calm coralBOT
✅
fierce elbow
For [24]x[32] = [2]x6
would solution be
24mod7 = 3
32mod7 = 4
3x4=12
2mod7=2
6mod7=6
2x6=12
12=12mod7
So false?
potent lotusBOT
TTV_Tripzy
remote mural
Srry wrong person
😂
Didn’t mean to send that to you
Helping 2 people at once
Rn
If you ask pure
He can help
fierce elbow
ok ty
fierce elbow
Because 12mod7 = 5?
cloud topaz
Which is obv true
fierce elbow
How ?
cloud topaz
When we way a = b mod n
It means n divides b - a
That’s all it means
From 12 = 12 mod 7 you can see that 7 divides 12 - 12
And you can keep reducing it mod 7
fierce elbow
Ahh ok
cloud topaz
12 = 12 mod 7 is the same as saying 5 = 5 mod 7
fierce elbow
so n=n mod (x) where n and x are just any integer is always true ?
cloud topaz
Correct
calm coralBOT
cloud topaz
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dusky verge
hi can you help me with sig figs
calm coralBOT
hi can you help me with sig figs
dusky verge
is this just one sig fig so 8.0?
this is the sig figs i did for pka using the formula above
please check if its correct
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vague path
How do you prove pqrt c?
calm coralBOT
How do you prove pqrt c?
calm coralBOT
@vague path Has your question been resolved?
leaden thunder
you just need to see if it equals (2 * u_n - 1)/3
it seems easier to start on the right side and then manipulate it to equal u_{n+1}
calm coralBOT
@vague path Has your question been resolved?
grim berry
#1173091291735797841 message I made this question in the help forum due to a mistake, when I should have made it here... but please, help me.
grim berry
basically, I need to figure out the dimensions of n-gons and/or 2n-gons... that form part of a progression... the link above has more detail
with the first 21 polygons, these must all be larger in area than the previous ngon in the set, but smaller than the next ngon in the set... Then comes a ngon with 130 sides that must be larger than the ngon with 128 sides but also form a ngon of 260 sides with another ngon of 130 sides, together these two must form a ngon with 260 sides smaller than the ngon with 510 sides made of two ngons in the set with 255 sides each, this then continues the progression to a ngon with 512 and an ngon with 1024 sides, these last two are less of a headache as they continue the rules of the smaller ngons
While the basis are even sided ngons (260 and 510 sides) the visible ngon are not those (130 and 255 sided)
I found these articles that might help but dont help me...
https://planetmath.org/constructionofregular2ngonfromregularngon
The first 14 polygons have the apothem of one as the circumradius of the other, then a leaping progression starts, by the 20th polygon the thickness of the previous polygon's orthostat (boundary mark) is added to the circunradius and made the apothem... but 22nd to 25th are a mistery as they involve two consecutive polygons of the same proportions making up a larger polygon so the four fut into two...
Sets of polygons, value of n (number of sides)
3
4
5
6
7
8
15
16
17
18
19
32
33
34
35
36
37
38
39
64
128
130 and/or 260
130 and/or 260
255 and/or 510
255 and/or 510
512
1024
(Sorry but I cant be concise)
The first 14 polygons have the apothem of one as the circumradius of the other, then a leaping progression starts, by the 20th polygon the thickness of the previous polygon's orthostat (boundary mark) is added to the circunradius and made the apothem... but 22nd to 25th are a mistery as they involve two consecutive polygons of the same proportions making up a larger polygon so the four but into two...
here is the full data:
calm coralBOT
@grim berry Has your question been resolved?
grim berry
<@&286206848099549185> I know I havent formulated my question in a very concise manner but its hard to put it in words despite my advanced English skills (and in part 'cause English aint my mother tongue)
calm coralBOT
@grim berry Has your question been resolved?
calm coralBOT
@grim berry Has your question been resolved?
calm coralBOT
@grim berry Has your question been resolved?
calm coralBOT
@grim berry Has your question been resolved?
little kiln
i need help with question b
calm coralBOT
i need help with question b
little kiln
also, i have to prove it by separating both sides. so solving the right side so that it equals to the left side, or vice versa
queen igloo
Format right
little kiln
theyre not common denominator
queen igloo
R.H.S. = 1/1+cx + 1/1-cx
= 2/1-c²x
= 2csc²x
=L.H.S.
QED
cx is short hand for cosx
queen igloo
$\frac{(1-\cos x)+(1+\cos x)}{(1-\cos x)(1+\cos x)}$
little kiln
$\frac{(1-\cos x)+(1+\cos x)}{(1-\cos x)(1+\cos x)}$
is this happening for the entire rhs or for one fraction?
oh ok thanks
this makes it more clear
little kiln
wait
so we put the 1+cv on the top and 1-cx on the bottom right?
then were left with (1-cx)(1-cx) on the denominator
queen igloo
yes
little kiln
then foil?
queen igloo
its literally adding up a regular fraction
potent lotusBOT
habuki (dms open)
little kiln
its literally adding up a regular fraction
i got it now
itd be 1-cosx^2 on the bottom
which is 1-sinx
little kiln
thats ok
i'll just ask my teacher when school starts
thanks for the help regardless, since you helped on the other proof
🫡
.close
blazing locust
one last proof
Sin(x + y) / cos(x+y)
(Sinx cosy + siny cosx) / (cosx cosy - sinx siny)
By dividing cos x cos y out of the fraction i think everything clicks
(Tanx + tany) / (1 - tanx tany)
queen igloo
$\frac{\sin(x + y) }{\cos(x+y)} =
\frac{\sin x \cos y + \sin y \cos x}{ (\cos x \cos y - \sin x \sin y)}$
By dividing $\cos x \cos y$ out of the fraction i think everything clicks
$\frac{(\tan x + \tan y)} {(1 - \tan x \tan y)}$
i need latex to help me read
little kiln
Sin(x + y) / cos(x+y)
(Sinx cosy + siny cosx) / (cosx cosy - sinx siny)
By divid...
woah
ill definitely check this out
potent lotusBOT
habuki (dms open)
little kiln
latex ftw
queen igloo
and yes, thats exactly the intended way ;)
blazing locust
Ahh yeah, ive gotten stuck with that before
little kiln
**habuki (dms open)**
wait
but wouldnt dividing cosxcosy out of it leave the fraction with sinx+siny/-sinxsiny?
well, not really but i dont get it
blazing locust
Dividing the cos x cos y out of the bottom is cosx cosy/cosx cosy = 1, then -sinx siny/cosx cosy = tanx tany
Cause u xant remove terms of an equation with division
Then on top, u have sinx cos y / cosx cosy = sinx/cosx, because cos y cancels, and the same thing with siny cosx but cosx cancels
queen igloo
yeah...
little kiln
Then on top, u have sinx cos y / cosx cosy = sinx/cosx, because cos y cancels, a...
but wouldnt dividing it out mean that theres no more cos left? since theres 1 set of cosx and cosy on the numerator and denominator
so cancelling cosxcosy would mean 1, but i dont see how we'd still have a cosx cosy left over
if that makes sense
blazing locust
Ahh so, i think i see where ur getting caught
So when u divide things out of fractions, u apply it to all the terms on the numerator snd the denominator
To use algebra, to demonstrate
(ab + ac)/a = b+c
little kiln
yeah
blazing locust
Because we can group the a out of the numerator and then we cancel them
So it can be thought of what happened was i grouped the cosx cos u it of the numerator and denominator and then canceled
Ie
(Sinx cosy + cosx siny)
= (cosx cosy) ( (sinx cosy) /( cosx cosy) + (siny cos x)/(cosx cosy) )
And then, rhis looks really nasty, but a lot of it cancels
Ie in the first term, cosy cancels and in the second term cosx cancels
So the we have
(Cosx cosy)(sinx/cosx + siny/cosy)
Which leaves
(Cosx cosy)(tanx + tany)
little kiln
the trolls are never funny...
blazing locust
If it doest make sense, u can multiply these two together to get the original statement
Yeah…
little kiln
(Sinx cosy + cosx siny)
= (cosx cosy) ( (sinx cosy) /( cosx cosy) + (siny cos x...
so to be clear, all this division is in the numerator?
blazing locust
Yeah
Its horrid, so i usually skip the geouping step and just apply the division to the terms as are
Cause its the exact same thing but with less horrid fractions
On the denominator then, u can again group cosx cosy out of cosx cosy - sinx siny
Cosx cosy - sinx siny
= (cosx cosy) (1 - (sinx siny)/(cosx cosy)
Which can be verified by multiplying the cosx cosy back in
Then
Sinx siny / cosx cosy = tanx tany
So we have in total
( (cosx cosy) (tanx + tany) )/ ( (cosx cosy) (1 - tanx tany) )
Then dancel cosx cosy
Tanx + tany / 1 - tanx tany
Make sense?
calm coralBOT
marsh root
hi, I need some help with discrete math. there's a question on an assignment that asks to find the total number of partitions on the set [4]. does [4] in this context mean a set containing only 4 or a set containing 4 elements?
calm coralBOT
calm coralBOT
Available help channel!
vital falcon
how did they expand the denominator to that?
calm coralBOT
how did they expand the denominator to that?
leaden thunder
how did they expand the denominator to that?
Have you ever learned completing the square
vital falcon
ya
vital falcon
Have you ever learned completing the square
$(s+1)^2+4$
potent lotusBOT
LeGM
vital falcon
i still dont see how to turn this into what they did
leaden thunder
j^2 = -1
leaden thunder
$(s+1)^2+4$
4 = -(-4)
vital falcon
ohhh ok
calm coralBOT
@vital falcon Has your question been resolved?
calm coralBOT
calm coralBOT
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lost moon
how many bijective maps are there from {1, 2} to {1, 2, 3}
calm coralBOT
how many bijective maps are there from {1, 2} to {1, 2, 3}
lost moon
I think there is 1
oblique current
What do you think it is?
lost moon
1 -> 1
2 -> 2
2 - > 3
oblique current
How are you sending 2 to two different things?
oblique current
For finite sets yes
lost moon
because it has to be 1 to 1 and every eleent in the codomain must be mapped
ah
is. the answer 0 then
oblique current
Yeah
lost moon
perfect, thank you
oblique current
because it has to be 1 to 1 and every eleent in the codomain must be mapped
Yeah one of these can't happen if not the same size
lost moon
how
glass heart
it cant
delicate barn
nvm miswrote it
chilly wigeon
calm coralBOT
chilly wigeon
i dont understand the x and y intercept part howd theyd they get them
chilly wigeon
-2^x=3
chilly wigeon
whyd they do log_2(3)
hybrid vale
$-2^{x} = -3$
chilly wigeon
what rule is that idgi
potent lotusBOT
Sarro
