Why ?
#help-26
wispy pier
obsidian sparrow
well the math is right
i just messed up with the hypotenuse
its -1 not 1
but the math is right hooray
grand trellis
What π
grand trellis
its -1 not 1
Length of hypotenuse can't be negative π
obsidian sparrow
quadrant 4 so its negative
obsidian sparrow
Length of hypotenuse can't be negative π
what
obsidian sparrow
π
wispy pier
Also
grand trellis
Nevermind that
wispy pier
I must mention
obsidian sparrow
Like it's a side of a traingle
hows the answer neg
grand trellis
hows the answer neg
Sin is negative in the 4th quadrant
obsidian sparrow
ah
obsidian sparrow
Sin is negative in the 4th quadrant
yes because sin is y-xis
so because its sin its negative?
grand trellis
First of all
wispy pier
Which makes cos(-45Β°) = (-1/β2)/(-1)
obsidian sparrow
since its in quadr 4
grand trellis
a is side length π
obsidian sparrow
yes
grand trellis
Wait
obsidian sparrow
grand trellis
Seems correct
obsidian sparrow
but the answer is negative so idk
grand trellis
Sin will be 1/β2/1
obsidian sparrow
what
grand trellis
Opposite side / hypotenuse
wispy pier
Sin will be 1/β2/1
Sin will be , (1/β2)/-1
obsidian sparrow
this guy said it cant be negative tho
grand trellis
Sin will be , (1/β2)/-1
Isn't 1/β2 also negative π
grand trellis
this guy said it cant be negative tho
I confused your method with a different one
obsidian sparrow
oh
grand trellis
Proving it in general is easier for me
obsidian sparrow
wait so its (-β1/2)/-1
grand trellis
Idk man
grand trellis
Sides of a triangle shouldnt be negative π
obsidian sparrow
Sides of a triangle shouldnt be negative π
broo
obsidian sparrow
WHAT OD U MEAN NO
grand trellis
Imaginary number
obsidian sparrow
ahhhh
dnvbiuvweoiwfefhbovwnm
okay
how is the answer negative then
when does it turn negative
grand trellis
a^2 + a^2 = x^2
a= 1/β2 x
Sin(-45) = - ( 1/β2)(x) / x
sin (-45) = - 1/β2
obsidian sparrow
thats the only way it would make sense
grand trellis
a^2 + a^2 = x^2
a= 1/β2 x
Sin(-45) = - ( 1/β2)(x) / x
sin (-45) = - 1/β2
This is what I'd generally do
grand trellis
thats the only way it would make sense
Maybe
obsidian sparrow
oh my god bro im gonna kms
ok whatever next question
Isnt this just the y value of point A?
sqrt3/2
teacher doesnt put the answers on the slides
kinda dumb
raven field
obsidian sparrow
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
digital meteor
topaz sinewBOT
digital meteor
How would i know what x is when sin x is 3/5
wispy pier
How would i know what x is when sin x is 3/5
General solution?
near bolt
How would i know what x is when sin x is 3/5
as y = sinx
just put values
you need to know inverse trignometry to understand it
as a = arc sin(3/5)
thorny flameBOT
Result:
0.60181502315205
ivory sorrel
,w sin(37deg)
thorny flameBOT
prisma mesa
Arcsin(3/5) is 37 deg if I remember right
approximately, but yes
hardy crater
but note that for the question that you showed, getting the exact value of angle a is not necessary to answer the question
topaz sinewBOT
@digital meteor Has your question been resolved?
round pagoda
ok so
round pagoda
you see the second deravtive graph
how does that show the maxima and minima
its just a straight line
near bolt
you see the second deravtive graph
actually first derivative graph determines point of max or min
second derivative graph is for concavity
low nymph
look at the first derivative graph and see how the roots line up with the original graph's local min/max
topaz sinewBOT
@round pagoda Has your question been resolved?
rocky crane
help me
rocky crane
i multiplied the recpricol of -1/20
neon iron
There is multiple ways of doing it.
rocky crane
and then once i did that i got (x+75)^3/2 = -200
and then powered both sides by ^2/3
so i got x+75 = -200^2/3
neon iron
but if you looked closely, the right side is -200
thorny flameBOT
.
neon iron
yeah
rocky crane
yea -200 to the poewr of 2/3
which can be written like sqrt(-200^3)
$sqrt(-200^3)$
neon iron
there is no solutions from what i got
rocky crane
ok same
cuz i got -10 and 10i when i plugged x back into the equation
so that means no solutions correct?
cuz for me i got X = -75 Β± β40000
neon iron
because you got (x+75)^3/2 so its not negative but left side is negative
neon iron
yes
rocky crane
so then for this eqation: sqrt(5x+6) = 1+ sqrt(3x+3 or 2 i forgot)
i got 2 and -1
but when i plugged -1 back into the eqatuon i got 2 different decimal ansewrs
but when i plugged in 2 i got 4 on both sides
neon iron
is it 3 or 2
rocky crane
yea wait its 3
i might have plugged in -1 wrong in the calculator tho cuz i was running out of time
it might have been -1 aswell
rocky crane
sqrt
thorny flameBOT
SWR
neon iron
Step 1: Start with the equation
neon iron
we doing sqrt(5x+6) = 1+ sqrt(3x+3 now?
rocky crane
yes
neon iron
Five x plus six equals one plus the square root of three x plus three.
rocky crane
i got 2 and -1 but when i plugged in -1 it didnt give me the same answer on both sides
but i feel like i might have plugged it in wrong on the calculator
rigid ivy
but it gives cuberoot (40000)
You forgot the minus
rigid ivy
which gives Β± 43.something
No. There is no plus or minus
rigid ivy
Oh
neon iron
400
rigid ivy
You are right
rigid ivy
I be mixing them up too 
rigid ivy
No. There is no plus or minus
But this still stands I'm quite sure
neon iron
cuberoot 40000
neon iron
34.19 is what i got
rigid ivy
negative and postive number works
This is incorrect
neon iron
but is it an irrational?
rocky crane
wait but in my math test am i suuposed to know if there are parenthesis when it says -200^2
rigid ivy
wait but in my math test am i suuposed to know if there are parenthesis when it ...
You deduce this based on your own workings.
neon iron
π π π π π
rocky crane
wait im so confused
neon iron
Stop spamming please
ok
rocky crane
ok but how do i know if there are brackets if i get to -200^2 from evaluating
is that given or something
rocky crane
would either one of them give me 10
neon iron
from what
rocky crane
cuz it was x -75 Β± cuberoot(40000) or is tehre a different number given if it was cuberoot(-40000_
rigid ivy
wait im so confused
The parentheses must be there. Because at one point, you alegraically got to $(x+75)^{3/2}=-200$. So when you raise both sides to power $2/3$, that exponent is applying to all of $-200$, which you represent through parentheses
thorny flameBOT
SWR
rigid ivy
cuz it was x -75 Β± cuberoot(40000) or is tehre a different number given if it w...
Again. There is no $\pm$
rocky crane
oh i see
thorny flameBOT
SWR
rocky crane
so if you subsituted X into the equation with 40000 cuberoot would it equal to 10
rigid ivy
would either one of them give me 10
No
neon iron
there is no solutions
rigid ivy
so if you subsituted X into the equation with 40000 cuberoot would it equal to 1...
No. You forgot the 75
rocky crane
cuz wehn i plugged x in i got -10 and 10i
neon iron
misclick?
rocky crane
No. You forgot the 75
and the -75 i was about to add
rigid ivy
there is no solutions
Incorrect. There is one real (irrational) solution
neon iron
Incorrect. There is one real (irrational) solution
what is it?
rigid ivy
cuz wehn i plugged x in i got -10 and 10i
Ah you are right
rocky crane
really?
rigid ivy
I forgot the square root nonsense
rocky crane
wait the problem is on the test i wrote x = -75 Β± cuberoot(-40000)
neon iron
Step 1: Start with the equation
Five x plus six equals one plus the square root of three x plus three.
Step 2: Isolate the square root
Move one to the left side by subtracting it from both sides:
Five x plus six minus one equals the square root of three x plus three.
This simplifies to:
Five x plus five equals the square root of three x plus three.
rocky crane
i showed work though
rigid ivy
Algebraically, you would arrive at $x=\sqrt[3]{40000}-75$, but this would not work. You would end up with $-10=10$
neon iron
Step 3: Square both sides to remove the square root
Square both sides of the equation:
The left side becomes the square of five x plus five.
The right side becomes three x plus three.
Step 4: Expand the square
The square of five x plus five equals twenty-five x squared plus fifty x plus twenty-five.
So the equation becomes:
Twenty-five x squared plus fifty x plus twenty-five equals three x plus three.
Step 5: Move all terms to one side
Subtract three x and three from both sides:
Twenty-five x squared plus fifty x minus three x plus twenty-five minus three equals zero.
This simplifies to:
Twenty-five x squared plus forty-seven x plus twenty-two equals zero.
Step 6: Solve the quadratic equation
Use the quadratic formula to solve for x. The formula is:
x equals negative b plus or minus the square root of b squared minus four a c divided by two a.
Here, a is twenty-five, b is forty-seven, and c is twenty-two.
thorny flameBOT
SWR
neon iron
oh shit wrong question
rigid ivy
i showed work though
Consider a simpler problem: $\sqrt{x}=-1$. If you square both sides, you get $x=1$, but that clearly is not a solution.
thorny flameBOT
SWR
neon iron
nah this is on my top 10 dumbest moments
rigid ivy
nah this is on my top 10 dumbest moments
Me too
rocky crane
ok what about the second question
rigid ivy
Half awake on mobile one eye open. Not at my best
rocky crane
i got 2 and -1 but i said -1 was an extranueos solution
rigid ivy
so then for this eqation: sqrt(5x+6) = 1+ sqrt(3x+3 or 2 i forgot)
This second question?
rigid ivy
?
neon iron
Start with
negative one over twenty times the quantity x plus seventy five to the power of three over two equals ten.
Step one. Eliminate the fraction.
Multiply both sides by negative twenty to cancel out negative one over twenty.
x plus seventy five to the power of three over two equals ten times negative twenty.
This simplifies to
x plus seventy five to the power of three over two equals negative two hundred.
Step two. Check the left side.
The power three over two means taking the square root first and then cubing the result. The square root of any number is always zero or positive. This means x plus seventy five to the power of three over two cannot equal a negative number.
Final answer. There are no real solutions to this equation.
this is what i did
neon iron
you gotta eliminate the sqrt first
thorny flameBOT
.
$\sqrt(5x+6)\ = 1+ \sqrt(3x+3)\$
```Compilation error:```! Missing $ inserted.
<inserted text>
$
l.50 \end{document}
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.```
rigid ivy
Start with
negative one over twenty times the quantity x plus seventy five to th...
Never seen someone here write math in all words like this 
neon iron
from ELA
rigid ivy
$\sqrt{5x+6}=1+\sqrt{3x+3}$
thorny flameBOT
SWR
neon iron
i think imma go now
rigid ivy
i got 2 and -1 but i said -1 was an extranueos solution
Why is -1 extraneous?
rocky crane
cuz when i plugged it in the calc i got on the left side a 3.1irrational and the other side i got a 3.4 irrational
but i think i may have plugged it in wrong in the calc
neon iron
cuz when i plugged it in the calc i got on the left side a 3.1irrational and the...
ig not
it not
rocky crane
granted when i was checking the answers the class had ended like 5 seconds ago and i had to give the paper
rigid ivy
but i think i may have plugged it in wrong in the calc
Yes. You may have
neon iron
nooo
neon iron
what was wrong
rocky crane
for the first question i providede i said x = -75 +- cuberoot(-40000)
neon iron
all these silly mistakes i swear
use your time to double check, heck, even triple
rocky crane
when it was no soliution
neon iron
bru
rocky crane
and the second one i said x = 2 and not x = -1
rocky crane
and then i dont know if i needed to rationalize the denominator for one of them
neon iron
but its ok.
rocky crane
cuz i had to find the inverse of 3x^8 = y
neon iron
there is always another day
rigid ivy
and then i dont know if i needed to rationalize the denominator for one of them
Hard doubt
rocky crane
cuz the next question was asking to see if they were inverese and it was f(x) = 2x^5 and g(x) = 5throot(x/2)
and the fifth root was not rationalized
and also is the rationalized verison of 8throot(x/3) 8throot(3x)/3
?
neon iron
quick question, whos your fav mathmatician
rocky crane
Nobody
neon iron
welp
rocky crane
Maybe Sal khan
neon iron
yeahhhhh
rocky crane
lol
rigid ivy
$\sqrt[8]{\frac x3}=\frac{\sqrt[8]{3^7x}}3$
thorny flameBOT
SWR
rocky crane
guys
i might have read the question wrong
my freinds are sayig it was 60 - (1/20) * (x+75)^3/2 = 10
and they got 25
i swear to GOD on my paper there was no 06
60
it must have gotten cutten off in the printing no way
rigid ivy
topaz sinewBOT
@rocky crane Has your question been resolved?
neon iron
NAHHHH
south peak
Am I correct in thinking that what the author means by "the same" is really "proportional"?
topaz sinewBOT
Am I correct in thinking that what the author means by "the same" is really "pro...
fallow heart
Nope, the same is correct
grim pike
By "the same linear combination", I guess the author just means that the coefficients that multiply $\mathbf{x_1}$ and $\mathbf{x_2}$ are the same as the coefficients that multiply $f(\mathbf{x_1})$ and $f(\mathbf{x_2})$, respectively.
thorny flameBOT
ucheo
neon iron
Am I correct in thinking that what the author means by "the same" is really "pro...
omg i love that font
whats the font pls?
south peak
omg i love that font
You and me both. I love it, too. I specifially thought... "I love this font, I wonder which one it is". I tried to find the name, but couldn't. However, the font I'm using for my own notes (by coincident) is either very similar or the same.
I can get you the name of this font I'm using...
Font: Cambria
I'm also using "Cambria Math" for the math symbols.
neon iron
i remember i have an alt in this server with that latex font, let me check
dont ask me why
south peak
By "the same linear combination", I guess the author just means that the coeffic...
Hmm, you're saying the same a and b multiply both x and f(x), but he's talking about the function itself.
Not the expression before the function is resolved.
Or maybe you mean the multiplication by the constant inside the function?
neon iron
i think its eulervm font
south peak
i think its eulervm font
Thanks for the name. Am gonna save it π
grim pike
Hmm, you're saying the same `a` and `b` multiply both `x` and `f(x)`, but he's t...
A linear combination of two vectors $\mathbf{x_1}$,$\mathbf{x_2} \in V$ is an expression of the form $a\mathbf{x_1} + b\mathbf{x_2}$, where $a$ and $b$ are elements of the field over which $V$ is defined. So a linear combination can be thought of as an operation uniquely determined by the coefficients $a$ and $b$. If $f$ is a linear transformation, then $f(a\mathbf{x_1} + b\mathbf{x_2}) = af(\mathbf{x_1}) + bf(\mathbf{x_2})$. In this sense, the linear combination determined by $a$ and $b$ remains unchanged under $f$.
thorny flameBOT
ucheo
topaz sinewBOT
welp after searching in google it seems to be nazanin
neon iron
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
static kettle
How do i find the greatest common divisor and least common multiple of (xΒ³-3xΒ²+4x-12) and (xβ΄-16)?
I just don't know what to do with x
topaz sinewBOT
How do i find the greatest common divisor and least common multiple of (xΒ³-3xΒ²+4...
neon iron
it GCD x^+4?
sweet shard
How do i find the greatest common divisor and least common multiple of (xΒ³-3xΒ²+4...
Factor both terms first
static kettle
Factor both terms first
I know how to factor but I don't know what exactly i should do with the x, where should i put it?
neon iron
Th x represents the variable and stays within the polynomial during calculations. To find the GCD and LCM, factor both polynomials, identify common factors for the GCD, and combine all unique factors for the LCM.
sweet shard
I know how to factor but I don't know what exactly i should do with the x, where...
Show your factors
neon iron
are you sure you factored correctly?
static kettle
Th x represents the variable and stays within the polynomial during calculations...
Soo for the 1st one i should tuen it into (x-3)(xΒ²+4) and the 2nd one into (xΒ²-4)(xΒ²+4)?
static kettle
Ohhh okayyy
twin grotto
Is there anyone willing to help me on a large amount of geometry π i canβt understand it at all and itβs the only class i have left
static kettle
From (xΒ²-4)(xΒ²+4) to (x-2)(x+2)(xΒ²+4)?
neon iron
Is there anyone willing to help me on a large amount of geometry π i canβt unde...
ill do a portion. claim your own channel
static kettle
Sorry if I interrupted this one i did not know.
Dw i have time
twin grotto
Ok thank you
neon iron
you can make the first one (x-2)(x-2)
static kettle
LCM=(xΒ²+4)(x+2)(x-2)(x-3)?
static kettle
you can make the first one (x-2)(x-2)
Oohhh okay thanks
weary sage
you can make the first one (x-2)(x-2)
Wait no then wouldnβt you be left with -4x
neon iron
LCM=(xΒ²+4)(x+2)(x-2)(x-3)?
yes
neon iron
Wait no then wouldnβt you be left with -4x
? lemme see
weary sage
(x-2) (x-2)
x * x = x^2
-2 * x = -2x
-2 * x = -2x
-2 * -2 = 4
x^2-4x+4
static kettle
(x-2) (x-2)
x * x = x^2
-2 * x = -2x
-2 * x = -2x
-2 * -2 = 4
x^2-4x+4
So it can just stay as (xΒ²+4)
?
weary sage
Yeah I mean can u think of any other way to simplify
static kettle
No I can't
Thanks for helping me on this one
I also don't know what exactly to do for another exercise tho
GCD(a,b)=48
a+b=576 is another exercise I don't understand
static kettle
GCD(a,b)=48
a+b=576 is another exercise I don't understand
Lemme fix it rq
weary sage
And LCM= (xΒ²+4)(xΒ²+4)(x-2)(x+3)?
You should be fine but also you could also do (x^2+4)^2
weary sage
Wait show what u mean
heady robin
Thatβs when signs alternate
heady robin
i put * not x
Well, discord aint
static kettle
GCD(a,b)=48
a+b=576 is another exercise I don't understand
Soooo what do i do with this thingy
weary sage
GCD(a,b)=48
a+b=576 is another exercise I don't understand
What is the context
Is this a linear algebra question ?
static kettle
What is the context
What values can a and b be
neon iron
be back in 5 min
static kettle
Is this a linear algebra question ?
No idea π
weary sage
Oh I was thinking when you originally wrote D (a,b) = 48 that it was the distance between 2 points
static kettle
Oh I was thinking when you originally wrote D (a,b) = 48 that it was the distanc...
Yes i accidentally did that
weary sage
So what is GCD mean ?
static kettle
I meant to write the greatest common divisor
static kettle
Like GCD(6,2)=2 if im not mistaken
weary sage
Itβs saying a and b are both multiples of 48
And you just have to find what two values that are multiples of 48 = 576
Wait
I got that wrong
Or wait it may still apply
weary sage
Yeah show just in case Iβm misunderstanding
weary sage
Or reversed values for a and b
Yeah itβs pretty much just the explanation I gave
weary sage
And you just have to find what two values that are multiples of 48 = 576
This one
static kettle
Umm lemme try
a= 48x, b=48y
a+b=576=
=48x+48y=
=48(x+y)
576Γ·48=12
x+y=12
Then i find the values?
x could be 1, y could be 11
Anddd
X could be 5 and y 7 i think
So then a=48Γ1 , b=48Γ11 or a=48Γ5, b=48Γ7
I think?
neon iron
a= 48x, b=48y
a+b=576=
=48x+48y=
=48(x+y)
576Γ·48=12
x+y=12
Then i find the va...
my brain aint braining this
heady robin
She got convincing results
heady robin
my brain aint braining this
Since both numbers have 48 as GCD, you can just write them as 48x and 48y, and you have the info of a+b=576
Equate, 576=48x+48y
Factor 48 and divide 576 by it
neon iron
aww man i hate algebra 2
heady robin
Are you to get specific values for them or not
neon iron
oh wait this is algebra 1
static kettle
Are you to get specific values for them or not
I mean if i multiply the x and y that i get with 48 it matches the answer
heady robin
Any expression for 12 will be the answer
Even negative numbers
Smth like -18 for a and 30 for b will fit
static kettle
I maybe forgot to say it says that they have to be natural numbers
heady robin
My question is, is that ok or do you need specific numbers
heady robin
I maybe forgot to say it says that they have to be natural numbers
You can then exclude
heady robin
My question is, is that ok or do you need specific numbers
For a and b
Like does a need to be a specific value
static kettle
My question is, is that ok or do you need specific numbers
I think that they have to be composite numbers
Idk how it correctly translates into English
heady robin
Does the question say smth like: find βall possible valuesβ of a and b, or find βthe valuesβ of a and b
heady robin
I think that they have to be composite numbers
Non prime?
static kettle
Non prime?
Like those who don't share a common divisor apart from 1
Like 1 and 11
And 5 and 7
static kettle
Ohhh
heady robin
They only divide by 1 and themselves
But
1 is not prime
1 is just excluded
So you only get 5 and 7
static kettle
1 is just excluded
Because it only has 1 divisor?
static kettle
So you only get 5 and 7
Ohhh okay
heady robin
Anytime
topaz sinewBOT
static kettle
Are the prime numbers values of a and b or x and y
Should be x and y
topaz sinewBOT
Available help channel!
neon iron
topaz sinewBOT
neon iron
I keep getting C but its supposed to be E?
I use a u substitution
where u is x-3
and du is 1
so its the integral from -1 to 1 of 1/u^2
which i then get as finding from -1 to 1 -1/u
which is just -1 - (-1) or 0
young geyser
in the interval [2,4] there is a point where the function is not defined
x=3
Then you must calculate the integral from 2 to 3, and from 3 to 4
neon iron
oh so if you take the definite integral of a function which contains an undifiend point, is it always N/A
*DNE
young geyser
oh so if you take the definite integral of a function which contains an undifien...
probably
silent siren
did you change the bounds of integration?
silent siren
so its the integral from -1 to 1 of 1/u^2
yes you did, sorry
neon iron
yeah i did
young geyser
yeah DNE
neon iron
probably
wait ur right
theres an asymptote that goes to infinity
thank you!
.close
topaz sinewBOT
topaz sinewBOT
Available help channel!
supple heart
how to find the gcd (n, 209)
topaz sinewBOT
how to find the gcd (n, 209)
supple heart
those three informations are given , forget about the french
i know that the the $gcd(n,209) \ | \ n$ and $gcd(n,209) \ | \ 209$
thorny flameBOT
<rajel />
wraith iron
how to find the gcd (n, 209)
does this mean n ~ 4 mod 19
supple heart
yes
wraith iron
so final one says n ~ 137 mod 209
that means n don't divide 209
so we have gcd < 209
merry hill
i know that the the $gcd(n,209) \ | \ n$ and $gcd(n,209) \ | \ 209$
idk this but from n ~ 137 mod 209 you get n = 346 which satisfies the first two conditions too
supple heart
that means n don't divide 209
how did you know
supple heart
oh make sens
wraith iron
idk this but from n ~ 137 mod 209 you get n = 346 which satisfies the first two ...
appears that the first two eqs lead to the mod 209 one
209 = 11 x 19 if that helps
supple heart
n leaves a remainder of 137 when divided by 209
so the gcd is bigger than 209
wraith iron
so gcd can only be 1, 11, 19
wraith iron
so the gcd is bigger than 209
how does that follow?
wraith iron
how to find the gcd (n, 209)
first two eliminates gcd being 11 or 19
so I think it easily equals 1
supple heart
how does that follow?
ah my bad got confused a bit
wraith iron
ah my bad got confused a bit
gcd is the greatest common divisor
so like it must divide both n and 209
does a number bigger than 209 divides 209?
supple heart
does a number bigger than 209 divides 209?
no
wraith iron
so we can say gcd<209
supple heart
given that the gcd can only be {1,19,11} , after verifying it seem to be only 1
because none of the other values divides n
takin 19 , which have remainder of 4 so it doesnt divide , and takin 11 it has a remainder of 5 non of them have a remainder of 0
wraith iron
yep
grand trellis
how to find the gcd (n, 209)
I'm 90% sure that it's gotta do something with the Chinese remainder theorem
supple heart
I'm 90% sure that it's gotta do something with the Chinese remainder theorem
i'll search for it thx too
.close
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empty spoke
do not ping us for help or when there's nothing going on please
bright trail
The helpers are volunteers
empty spoke
yes, he's not obligated to help you at all 
bright trail
They don't have an obligation to help you within some time frame
Depends on a lot of things how fast this happens
It is just random internet strangers after all
empty spoke
unrealistic to expect this given that people here are volunteers, please be empathetic
gleaming phoenix
use a calc?
bright trail
Why not just check on a calculator?
acoustic pecan
you have access to the internet
bright trail
You have an internet connection
rigid ivy
Nice job, admitting this is a ban-evasion account
torpid sparrow
You can use the command ,calc
bright trail
Yeah fair lmao
torpid sparrow
Anyways, stop trolling please.
inner topaz
Let $a_n$ be a sequence such that ${a_n : n \in \mathbb{N}} \subseteq [a, b]$. Let $\epsilon > 0$, a sequence $a_n$ is epsilon-dense if for each $x \in [a, b]$ and for each $N \in \mathbb{N}$ there exists an $n \ge N$ such that $|a_n - x| < \epsilon$
topaz sinewBOT
Let $a_n$ be a sequence such that ${a_n : n \in \mathbb{N}} \subseteq [a, b]$. L...
thorny flameBOT
jason2D
inner topaz
so if i understood that correctly, that just means that every number in the interval is a sub limit right?
this channel is occupied
empty spoke
please post in an available help channel:
inner topaz
so if i understood that correctly, that just means that every number in the inte...
but this is impossible isnt it?
shut orchid
so if i understood that correctly, that just means that every number in the inte...
sounds right
empty spoke
so if i understood that correctly, that just means that every number in the inte...
I don't think so since this is for one specific epsilon
no
someone else will be able to surely
inner topaz
I don't think so since this is for one specific epsilon
well the questions continues to say "given a sequence a_n that is epsilon-dense for any epsilon > 0..."
but how can every number in an interval be a sublimit?
the cardinality of an interval over R is larger than the cardinality of N
empty spoke
but how can every number in an interval be a sublimit?
this is not true of an epsilon-dense sequence
inner topaz
what does it mean then
empty spoke
here's a crude drawing of a (b-a)/4 dense sequence for instance
inner topaz
well the questions continues to say "given a sequence a_n that is epsilon-dense ...
but the sequence i have is epsilon dense for any epsilon
empty spoke
it means that for any point "x" in the interval, you can always find arbitrarily large n such that a_n is at a distance < epsilon from x
empty spoke
but the sequence i have is epsilon dense for any epsilon
okay that makes more sense then
inner topaz
how so
empty spoke
I didn't read that bit lol
because then what you can do is build a subsequence $x_{n_k}$ such that $|x-x_{n_k}|<1/k$ for each $k\in \bN$
thorny flameBOT
derivada.schwarziana
inner topaz
but thats just all of the rationals
empty spoke
by picking $\varepsilon=1$ to get $x_{n_1}$, $\varepsilon=1/2$ to get $x_{n_2}$ and so on
thorny flameBOT
derivada.schwarziana
empty spoke
but thats just all of the rationals
not necessarily, another example of a sequence that's epsilon-dense for any epsilon is all rationals that can be written as $a/2^m$ for some $a\in \bZ$ and $m\in \bN$
thorny flameBOT
derivada.schwarziana
empty spoke
this is a proper subset of the rationals that is dense on $\bR$
thorny flameBOT
derivada.schwarziana
empty spoke
well the questions continues to say "given a sequence a_n that is epsilon-dense ...
you can check that if $a_n$ is $\varepsilon$-dense in an interval $[a,b]$ for all $\varepsilon>0$, then the set ${a_n: \ n\in\bN}$ is dense in $[a,b]$
thorny flameBOT
derivada.schwarziana
empty spoke
so indeed every point of the interval is a limit point of the sequence
inner topaz
but i still dont understand
how can a sequence have uncountably infinite sublimits
when there are only countably infinite subsequences
empty spoke
the idea of "sublimits" of a sequence, or limit points of a set, is simply that they can be approximated to an arbitrary degree of error by elements of the sequence or set respectively
a priori there's no reason why a countable set cannot approximate an uncountable one
and in fact that's a thing we often desire in maths, e.g. the https://en.wikipedia.org/wiki/StoneβWeierstrass_theorem gives you a way to approximate any continuous function by polynomials, and hence particularly by polynomials with rational coefficients which form a countable set
empty spoke
when there are only countably infinite subsequences
I guess another way to think about the density of $\bQ$ in $\bR$ is by looking at decimal expansions, suppose we wanted to approximate [x=\pi=3.14159265\dots]
thorny flameBOT
derivada.schwarziana
empty spoke
one way to do this by rational numbers is by truncating the decimal expansion[x_1=3, \quad x_2=3.1, \quad x_3=3.14,]and so on
thorny flameBOT
derivada.schwarziana
empty spoke
and it turns out you can do this for all real numbers
inner topaz
but isnt this the oppsite?
empty spoke
... and that the rational numbers are countable, so we can write $\bQ=(q_n)_{n\geq 1}$ for some sequence $q_n$ (called an \textit{enumeration}). So this is an example of a sequence which has any point of $\bR$ as a sublimit
thorny flameBOT
derivada.schwarziana
empty spoke
but isnt this the oppsite?
how so?
inner topaz
this says that you can map from an uncountably infinite set to a countably infinite one
no?
empty spoke
not quite, it's saying that any real number is a limit point of the set of rational numbers which is what we want to show
it's true that in some sense we're mapping the limit (the real number) to a sequence that approximates it -- we kinda need to show how the sequence can be constructed
empty spoke
i dont understand this
basically we want to define a sequence q_1, q_2, q_3, ... that counts all rational numbers right
inner topaz
what youre saying is, for any real number you can create a sequence that approaches it
inner topaz
basically we want to define a sequence q_1, q_2, q_3, ... that counts all ration...
yeah
empty spoke
yes, a sequence of rational numbers
inner topaz
right
yeah no stil
its true that you can create a sequence of rational numbers that approaches any real number
but you still have uncountably many sequences like that
empty spoke
yes, your original question was how is it possible for one sequence to have uncountably many sublimits
inner topaz
yeah i still dont understand that
empty spoke
**derivada.schwarziana**
which is why I brought up this as an example: you can actually define this sequence q_n explicitly as shown in this diagram, from https://en.wikipedia.org/wiki/Rational_number#Countability
so q_1=1, q_2=2, q_3=1/2 and so on, following the arrows drawn in the diagram
inner topaz
yeah we used that to prove that Q is countably infinite a while back
empty spoke
it can be shown that this sequence passes through all of the positive rationals
inner topaz
right
empty spoke
**derivada.schwarziana**
and from there I think doing an approximation scheme of this sort is not hard to see
take a positive real number, suppose we know its (possibly infinite) decimal expansion
inner topaz
wdym approximation scheme
empty spoke
oh just a way to define a sequence of rationals that converges to it
empty spoke
take a positive real number, suppose we know its (possibly infinite) decimal exp...
so we can do this, and then define a subsequence q_n_k by taking the corresponding fractions of denominator 1, 10, 100, and so on, that describe the decimal expansion truncated to k digits
and since this can be done for any positive real number, the sequence q_n has uncountably many sublimits right
I haven't been too rigorous explaining this but hopefully that gives an idea of how this kind of thing can happen
inner topaz
so we can do this, and then define a subsequence q_n_k by taking the correspondi...
so for example
for the number 0.12345678...
our subsequence would be 0.1, 0.02, 0.003, 0.0004...?
empty spoke
I'd say 0.1, 0.12, 0.123, 0.1234, etc.
since you want the sequence to actually converge to 0.12345678...
inner topaz
but what if the number can be truncated to an irrational number?
empty spoke
I'd say 0.1, 0.12, 0.123, 0.1234, etc.
and this can be written as 1/10, 12/100, 123/1000 which respects the order of the sequence q_n but that's not too important -- just shows that it is a valid subsequence since q_n goes first through all fractions of denominator 1, then 2, and so on
empty spoke
but what if the number can be truncated to an irrational number?
that doesn't really matter I think, you can always take rational truncations like this and have it converge
empty spoke
inner topaz
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vocal dew
i know that when g(x) translates 1 unit to the left that it becomes g(x+1) but how would i use that to create an equation of the new curve?
topaz sinewBOT
i know that when g(x) translates 1 unit to the left that it becomes g(x+1) but h...
acoustic pecan
do exactly what you said
replace x with x+1
then expand and simplify if you want to do so
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naive carbon
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naive carbon
oh correction S1 wont be there
for obvious reasons π₯²
,tex
$S_k$ is the sum of the infinite G.P. where first term $a_k = k$ and common ratio $r_k = \frac{1}{k}$ $(k\in\mathbb{N})$. Find the value of $S_2^2+S_3^2+...+S_{2n-1}^2$.
thorny flameBOT
NightHaunter
naive carbon
i tried partial fractions to turn S_k into a telescoping series but didnt work
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topaz sinewBOT
Iβm not sure how to approach this problem
pseudo horizon
you could get two equations and solve for the intersection point
smoky sparrow
Question 41
the Cartesian equation first, so $y = -\frac{1}{2}x + 2$
thorny flameBOT
south
smoky sparrow
which becomes $r \sin \theta = -\frac{1}{2} r \cos \theta + 2$
thorny flameBOT
south
smoky sparrow
solve for r then use (x, y) = (r cos theta, r sin theta)
smoky sparrow
How do you know the radii are equal there
definition of polar form
smoky sparrow
jeez
brazen inlet
this is section 1 on polar and parametric
smoky sparrow
basically
smoky sparrow
cool
brazen inlet
Iβm saying how do you know the coefficients are equal
smoky sparrow
Iβm saying how do you know the coefficients are equal
it's a change in coordinates
this applies to any equation when you write it in polar form
brazen inlet
How do we know it isnβt an elliptical path?
I guess it makes sense
I can draw a circle of some radius and some point on that circle to get any point on that line
Is that the idea?
smoky sparrow
I can draw a circle of some radius and some point on that circle to get any poin...
yeah!
brazen inlet
https://www.youtube.com/watch?v=-FhjwuO6Uww
Alright much appreciated
Iβll check that out after this question
smoky sparrow
like you can describe any point on the plane with
- distance to origin
- angle
that's the idea
brazen inlet
like you can describe any point on the plane with
1) distance to origin
2) angle
Got it
yeah thatβs fair
Alright back to the question
Lemme try and solve what you sent rq
I used a on accident and was too lazy to change so a = r
Is this the right idea?
@smoky sparrow
Top equations are seperate
Kind of mushed
smoky sparrow
right idea
brazen inlet
Wrong answer?
smoky sparrow
but you can just use (x, y) = (r cos theta, r sin theta) after you have found r
anyway I'm sure they're equivalent parametric forms
smoky sparrow
take r cos(t) = $\frac{2 \cos t}{\sin t + (1/2) \cos t}$ for example
thorny flameBOT
south
smoky sparrow
if you multiply top and bottom by 2/(cos t)....
smoky sparrow
also you can realise
$\frac{y}{x} = \frac{r \sin \theta}{r \cos \theta} = \tan \theta$
thorny flameBOT
south
smoky sparrow
so $x \cdot \tan \theta = y$
thorny flameBOT
south
smoky sparrow
it's no coincidence
topaz sinewBOT
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errant hamlet
topaz sinewBOT
errant hamlet
so
PC=PD then triangle PCD isosceles
M midpoint DC then PM perpendicular on DC
OC=OD then triangle OCD is isosceles
M midpoint DC then MO perpendicular on DC
so P-M-O are colliniars
here i get stuck
<@&286206848099549185>
<@&268886789983436800>
thorn briar
Pls no ping mods for math help
pearl fog
have you drawn a diagram?
errant hamlet
yes
pearl fog
could you post it here?
topaz sinewBOT
@errant hamlet Has your question been resolved?
errant hamlet
AB is radial axis for C(O,OC) and C(O', O'O)
we can proof
so AMOB inscriptible
thats what we need to proof
<@&286206848099549185>
pearl fog
btw D is not always the diameter unless im missing smth
topaz sinewBOT
@errant hamlet Has your question been resolved?
topaz sinewBOT
@errant hamlet Has your question been resolved?
topaz sinewBOT
@errant hamlet Has your question been resolved?
frosty iron
"show an example of at least one set A containing numbers such that
|A| = 4
for any x,y in A ,x^y and y^x are also in A where x !=y"
frosty iron
idk if this is valid but Z_5 = {1,2,3,4} works
im curious if theres some other solution
long dawn
I guess it depends on what x and y are
prisma mesa
is the operation ^ supposed to be the one on reals, or can you define them however you want?
long dawn
{1,2,3,4} is consistent with the rules given if we have freedom to choose what x and y are
frosty iron
I guess it depends on what x and y are
huh?
oh wait its supposed to be "for any x,y"
frosty iron
but 2, 3 are in that set, and 2^3 isn't
but its modulo 5
long dawn
The way it was initially phrased states "if x,y are in A" which doesn't imply that they are in A
frosty iron
The way it was initially phrased states "if x,y are in A" which doesn't imply th...
stuff got lost in translation, sorry
pseudo horizon
ok
long dawn
x,y could have been elements outside of A, but OP has now clarified that they are definitively in A
pseudo horizon
uhhhh but the question is talking about exponentiation, it doesn't mention exponentiation modulo 5
frosty iron
yeah it doesnt
surreal mural
just invent your own number system where you define x^y = 0 always
prisma mesa
if this is from some textbook, can you ss the original problem?
frosty iron
its in thai hold on
its from a competition
maybe google translate sucks ill have gpt translate it for me
prisma mesa
is it "x^y in A or y^x in A"?
wooden osprey
yes this is what i'm suspecting too
wooden osprey
but i don't think google translate is that dogshit it can't distinguish between "and" and "or"
frosty iron
but i don't think google translate is that dogshit it can't distinguish between ...
i got that dollar store translate π
prisma mesa
okay, so firstly, unless the question allows you so, you shouldnt just invent your own number systems to solve the question
frosty iron
okay, so firstly, unless the question allows you so, you shouldnt just invent yo...
oh alr
mb
i tried for a while with some numbers and it didnt work
frosty iron
okay thanks
pseudo horizon
seems more possible now
prisma mesa
now try experimenting with it a bit
its quite easy now
keep in mind that you dont need both x^y and y^x be in A, you only need one of them
prisma mesa
yeah, i had this one in mind
wooden osprey
that works
prisma mesa
i didnt fully verify it but i think it works
any pair with 1 obviously works
because x^1 = x which must be in that set
frosty iron
question was just hard because i translated wrong lmao
frosty iron
any pair with 1 obviously works
i think 0 should also be too
prisma mesa
i think 0 should also be too
Yep, any pair with 0 works as well
and that only leaves (-1, 2)
which works since (-1)^2 = 1 is in the set
topaz sinewBOT
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coarse bluff
hello
coarse bluff
i just left my discrete math ezam and everyone is asking about this
i got 4^2n
is it the correct answer?
i double checked multiple times
odd pagoda
,w \sum_{i=1}^n \binom{n}{i} 3^i 4^n
thorny flameBOT
odd pagoda
not sure what you mean with "always"
but after you move out the 4^n then the i=0 term would be 1, yes
coarse bluff
the -1
coarse bluff
but after you move out the 4^n then the i=0 term would be 1, yes
oh
fair enough
thanks for answering
.close
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vivid saffron
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neon iron
3, 5, 11, 17, ?
what's the next term?
topaz sinewBOT
3, 5, 11, 17, ?
what's the next term?
ivory sorrel
we define a polynomial $a+bx^2+cx^3+dx^4= p(x)$
thorny flameBOT
Ζ( wai ina teacup)= I don't know
neon iron
no its not. it has a pattern
desert atlas
the increments are +2 +6 +6
i dont see any oattern
ok chatgpt says its alternate prime sequence
neon iron
the increments are +2 +6 +6
3 is the 2nd prime.
5 is the 3rd prime.
11 is the 5th prime.
17 is the 7th prime.
Therefore, the next term is the 11th prime, which is 31.
HAHA GOTCHA
desert atlas
k chatgpt is stupid
neon iron
k chatgpt is stupid
did u like this pattern? it's pretty smart yk
desert atlas
did u ask this question to test us
neon iron
did u ask this question to test us
umm i just wanted to ask.. if that was a nice question.....
that's my question.
i made that up.. and its my school project
coming up with random four term that has a pattern
neon iron
fo sure
right. ty for the feedback. i really appreciate :)). if u know any interesting patterns, please tell me.
.close
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torpid sparrow
What is your question?
obsidian trout
i need help with the integral
topaz sinewBOT
obsidian trout
sweet shard
!msgdel
topaz sinewBOT
The original post of this help channel has been deleted, and it will abruptly close and possibly lock. (This is irreversible.) Please claim a new channel, and don't delete the first message of any future channel you claim.
naive lion
Make a new post in a different available channel
obsidian trout
thank you
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Available help channel!
zinc star
hello can someone please explain the weak and weak * topologies ? i wanna know why and what are they . i already have some backgorund but i cant just wrap my head around them
topaz sinewBOT
hello can someone please explain the weak and weak * topologies ? i wanna know w...
@zinc star Has your question been resolved?
rigid ivy
hello can someone please explain the weak and weak * topologies ? i wanna know w...
Not heard of them myself, but start at #point-set-topology. Someone there may have better luck helping you.
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glossy wave
A frog is travelling from point A(0,0) to point B(4,6) but each step can only be 1 unit up or 1 unit to the right. Additionally, the frog refuses to move three steps in the same direction consecutively.
Compute the number of ways the frog can move from A to B.
topaz sinewBOT
A frog is travelling from point A(0,0) to point B(4,6) but each step can only be...
glossy wave
So I boiled it down to arranging RRRRUUUUUU in a way such that that three Rs and three Us arenβt next to each other
No really sure how to proceed
topaz sinewBOT
@glossy wave Has your question been resolved?
glossy wave
<@&286206848099549185>
wet sun
Would it be simpler to calculate the number of ways to arrange it so that there are 3 Rs next to each other, and the number of ways to arrange it so that there are 3 Us next to each other, then subtract them from the total arrangements?
glossy wave
Idk
Still seems like a lot of combinations to go through
I got that the number of ways of having exactly atleast 3 Rs next to each other is 48
Not sure how to go about the Us without double counting tbh
topaz sinewBOT
@glossy wave Has your question been resolved?
long stirrup
you have 5 buckets with U
e,g, 1 1 2 1 1 for URURUURURU
0 is allowed
one 4, one 5, one 6 are disjoint cases
4 2 0 0 0: 20 ways
4 1 1 0 0: 30 ways
5 1 0 0 0: 20 ways
6 0 0 0 0: 5 ways
ok 3 is also not allowed
and 2 0 0 2 2 would have 2 consecutive 0 which is bad for the R
but 0 0 2 2 2 is fine because 0s aren't in the middle
yeah that's a lot of stuff
i get the feeling there's no neat way at all lol
so like, 2 2 2 0 0: 5β2 = 3 ways
2 2 1 1 0: 30 ways
2 1 1 1 1: 5 ways
something's missing
oh it's 10β2
yeah
long stirrup
43 is right
long stirrup
yeah
pseudo horizon
ok i'll explain my approach, maybe we can figure out where i went wrong
U's can be partitioned as 2+2+1 (3c2=3 distinct ways) or 2+1+1+1+1 (5c1=5 distinct ways)
in the former case, two R's must separate the U's, and the other R's have 8 arrangements
in the latter case, all four R's are used to separate the U's
so i got 3*8+5*1
oh i messed up the partitions
they should be:
2+2+2
2+2+1+1
2+1+1+1+1
so you get 1*8+6*5+5*1=43
it works
long stirrup
that's identical to what i said btw
long stirrup
ah maybe not exactly
it's fine
the difference is probably just that i included 0s
so it was clear that Rs usually get spread out, there's only 2 cases with two consecutive 0 in the middle
topaz sinewBOT
@glossy wave Has your question been resolved?
neon iron
How to approach this Q? The answer is 661
topaz sinewBOT
How to approach this Q? The answer is 661
pseudo horizon
<@&268886789983436800>
topaz sinewBOT
How to approach this Q? The answer is 661
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 three-digit numbers, i think it will be a pyramidal number
i mean a tetrahedral number
neon iron
I just searched about pyramidal no.s and tetrahedral but i think that's a different concept
It's a Q on PnC
neon iron
Permutations and combinations, Q-question
pseudo horizon
oh