spark otter
One or the other, they are equivalent
#help-33
dawn grotto
ok ty
marsh citrusBOT
@dawn grotto Has your question been resolved?
pseudo dock
i have a general question for all type of questions like this. When do i multiply it by the variable and when do i put the variable to the exponenet when creating the equation.
marsh citrusBOT
i have a general question for all type of questions like this. When do i multipl...
outer lodge
i have a general question for all type of questions like this. When do i multipl...
If you are multiplying multiple times, exponent.
If you are adding multiple times, multiplication
E.g. blue jeans lose 1% per wash
losing 1% means you subtract 1% every wash
Colour% = 100% - 1% - 1% - 1%… = 100% - (1% + 1% …) = 100% - n%
marsh citrusBOT
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valid cape
why is the scalar multiplicativity of linear map called "homogeneity"?
valid cape
what exactly is homogeneous here?
fervent rampart
a map with that property is called homogenous regardless of whether it's linear
valid cape
yeah, but i mean, homogeneous means "of the same kind" right? if we call a map homogeneous, what about the map is of the same kind?
fervent rampart
the motivating example is a homogeneous polynomial, where all terms have the same degree
valid cape
hm
fervent rampart
similarly "linear" originally meant things relating to straight lines, but has been generalized to the point where the connection is no longer obvious
marsh citrusBOT
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lyric bay
Hello
lyric bay
Why can't we just integrate both sides from the given equation?
Why are we moving the ys and the constants to one side?
dry prawn
y is a function of x
marsh citrusBOT
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shy shard
Am I doing this correct so far? It looks messy so that’s why I ask
marsh citrusBOT
Am I doing this correct so far? It looks messy so that’s why I ask
brave marsh
Do you have to use IBP?
brave marsh
In general, you use IBP when you have a product of things that differentiate/integrate to something simpler.
brave marsh
IBP is integration by parts
shy shard
IBP is integration by parts
I see
brave marsh
In general, you use IBP when you have a product of things that differentiate/int...
And in this case neither choise of u and dv help much
shy shard
Well I would assume I would have to do IBP since the section is just about it
shy shard
And in this case neither choise of u and dv help much
What would be the best course of option then?
Cause I have no clue
brave marsh
I think substitution would probably work better
shy shard
I think substitution would probably work better
Could you give an example by any chance?
Like
Just use normal u sub?
brave marsh
Yeah
shy shard
Ah
static quarry
tbh, you do a lot of problems and start to get an instinct for what will work and what won't
there is not really a simple rule for when to use IBP or not
here you can see that if you let dv = r^3 then you're gonna end up with r^4 in your new integral, which makes things worse
and if you do it the other way then even if the square root behaves (it won't) then it's gonna take three iterations to get rid of the r^3
shy shard
i see
still temple
id just like a hint: how many distinct pairs of positive integers have product less than 10,000?
marsh citrusBOT
id just like a hint: how many distinct pairs of positive integers have product l...
pale urchin
let the numbers be (x) and (x+1)
we know that x * (x+1)=10,000
thus x(x+1)=10000
x^2 + x -10000 = 0
the roots are ( (-100.50124999219 ), (99.501249992188))
we take 99.5
stray hound
I'm not sure how to do this off the top of my head, but I'd just go integer by integer and try to detect a pattern
still temple
let the numbers be (x) and (x+1)
so is this just for one case or smth
because the numbers can be more than 1 apart
still temple
I'm not sure how to do this off the top of my head, but I'd just go integer by i...
ohh okay - yeah i was initially doing casework
but it was taking quite a while and i was pretty sure there was a faster way
pale urchin
well i just took the basic integer method cause they are supposed to be consecutive
still temple
i dont think the two numbers necessarily have to be consecutive in the problem
cobalt sedge
for every integer n, you can form pairs (n,k) where k is in [n+1, ceil(10000/n)-1 ] for n in 1, 100
stray hound
well if the first number is 1, you have 10,000 possible pairs. if the first number is 2, you have 5,000 - 1 possible pairs, etc
cobalt sedge
those are all admissible
still temple
well if the first number is 1, you have 10,000 possible pairs. if the first numb...
yuh i was taking that appraoch
still temple
for every integer n, you can form pairs (n,k) where k is in [n+1, ceil(10000/n)-...
ohh
do you know where i could go from there if i took that approach... 😭
pale urchin
so is this just for one case or smth
Yeah
stray hound
that's exactly what I was saying, just in a more compact form
still temple
that's exactly what I was saying, just in a more compact form
ye
ohh
💀
yeah i was just writing it out like
1x(1~9,999), 2x(2~5000), etc.
cobalt sedge
You can have an integral over that fhat pair range
This is like a lower riemann sum
still temple
oh i dont think ive learned that yet... 😭
cobalt sedge
Ohh, then idk any faster way
still temple
awrr
cobalt sedge
just do casework ig
still temple
thank you sm tho !!
still temple
just do casework ig
ye
still temple
You can have an integral over that fhat pair range
is this in like calc
cobalt sedge
is this in like calc
ye
still temple
i gotta lock in 😭 💀
eh ill eventually know what you mean by lower riemann sum
thank you though !!!
.close
marsh citrusBOT
cobalt sedge
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cobalt sedge
ye
At best you can simplify the casework is by taking steps factors of 10000
Like sum over [10000/n]
where [] is floor function
pale urchin
can you simplify the riemann sum to get this without doing a case by case analysis?.
cobalt sedge
I only made the condition for ordered pairs
cobalt sedge
can you simplify the riemann sum to get this without doing a case by case analys...
I mean, if you wanna simplify to case by case, you dont need a riemann sum anyways
.close
marsh citrusBOT
still temple
At best you can simplify the casework is by taking steps factors of 10000
ohhh okay
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terse basin
When we say light is a wave, do we mean just a wave or an EM wave?
marsh citrusBOT
When we say light is a wave, do we mean just a wave or an EM wave?
terse basin
yea ik that
so when we talk about the wave theory of light we're discussing how light is an EM wave correct?
rancid geode
yes
still temple
so when we talk about the wave theory of light we're discussing how light is an ...
yeah
terse basin
ight thanks guys
with the particle theory, is it correct to say: "light is made up of little packages of energy called photons. The energy of each package is determined by E = 6.626 x 10^-34 f"?
correct me if im wrong but 6.626 x 10^-34 is planck's constant 'h' right?
marsh citrusBOT
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marsh citrusBOT
@terse basin Has your question been resolved?
terse basin
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sour umbra
Hello can anyone please help me understand how to evaluate limits from a graph when there's an asymptote? I'm looking at a limit as x approaches -3, and I'm having trouble understanding how to compare these functions.
marsh citrusBOT
Hello can anyone please help me understand how to evaluate limits from a graph w...
sour umbra
Oh ok and why are we able to just ignore this over here? Is it just because it will never reach -3?
Also when I look at the limit as x approaches 1, I see the left side goes to 1 and the right side goes to -4. does this mean the limit doesn't exist because the sides approach different values?
meager badger
Also when I look at the limit as x approaches 1, I see the left side goes to 1 a...
Yes
That’s the def of a limit, in order for it to exist the right hand limit and the left hand limit have to equal the same value
meager badger
Oh ok and why are we able to just ignore this over here? Is it just because it w...
Wdym?
meager badger
Also just to check my understand as x approaches -1 the limit is 3?
Yep
marsh citrusBOT
sour umbra
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sour umbra
@meager badger What does the solid black dot mean do we have to include that in our limits?
meager badger
<@711469140736475146> What does the solid black dot mean do we have to include ...
That’s a removable discontinuity
As long as the limit doesn’t go towards the solid black dot, you don’t have to include it
sour umbra
Ok snd for both infinities - or positive both limits will be negative 2 because both ends of the graphs approach -2?
edgy sundial
bridget total fish = 5
dad total fish = 4
assume the weight of bridget fish = x
fish 1 = x
fish 2 = 2x
fish 3 = 2 + 3x
fish 4 = 1/2x
fish 5 = 3/5x
assume the weight of dad fish = y
fish 1 = y = 2 + 3x
fish 2 = y = 4/5x
fish 3 = y = 4 + 2x
fish 4 = y = 1/2x
x + 2x + (2 + 3x) + 1/2x + 3/5x = (2 + 3x) + 4/5x + (4 + 2x) + 1/2x
2 + 6x + 1/2x + 3/5x = 6 + 5x + 4/5x + 1/2x
2 + 6x + 5/10x + 6/10x = 6 + 5x + 8/10x + 5/10x
2 + 6x + 16/10x = 6 + 5x + 13/10x
2 + 60/10x + 16/10x = 6 + 50/10x + 13/10x
2 + 76/10x = 6 + 63/10x
2 - 2 + 76 / 10x = 6 - 2 + 63/10x
76/10x = 4 + 63/10x - 63/10x
76/10x - 63/10x = 4
13/10x = 4
13 = 40x
x = 13/40 ounch
y = 2 + 3x
y = 2 + 3 * 13/40
y = 2 + 39/40
y = 2 + 1 = 3 ounch
something wrong i guess...
marsh citrusBOT
Available help channel!
edgy sundial
2 + 6x + 11/10x = 6 + 5x + 13/10x
2 + 60/10x + 11/10x = 6 + 50/10x + 13/10x
2 + 71/10x = 6 + 63/10x
2 - 2 + 71 / 10x = 6 - 2 + 63/10x
71/10x = 4 + 63/10x - 63/10x
71/10x - 63/10x = 4
8/10x = 4
8 = 40x
x = 5 ounch
y = 2 + 3x
y = 2 + 3 * 5
y = 2 + 15
y = 2 + 15 = 17 ounch
yeah wrong on 5/10x + 6/10x supposely 11/10x not 16/10x 😂
marsh citrusBOT
2 + 6x + 11/10x = 6 + 5x + 13/10x
2 + 60/10x + 11/10x = 6 + 50/10x + 13/10x
2 + ...
tulip idol
!original
marsh citrusBOT
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.
tulip idol
yeah I'm asking my pet ai to comprehend that 
edgy sundial
anyway how to see the list of command on that bot ?
tulip idol
F + 2F + (3F + 2) + F/2 + (3F/5) = (3F + 2) + (4F/5) + (2F + 4) + F/2
edgy sundial
yes i'm wrong in addition of the left side
and also do you know website for practicing math question from beginner to pro, like leetcode but for math ?
tulip idol
F = (F/5) + 4 => F = 5
tulip idol
and also do you know website for practicing math question from beginner to pro, ...
no, there's only books for math
edgy sundial
yo how can u so fast counting it ?
2 + 6x + 11/10x = 6 + 5x + 13/10x
2 + 60/10x + 11/10x = 6 + 50/10x + 13/10x
2 + 71/10x = 6 + 63/10x
2 - 2 + 71 / 10x = 6 - 2 + 63/10x
71/10x = 4 + 63/10x - 63/10x
71/10x - 63/10x = 4
8/10x = 4
8 = 40x
x = 5 ounch
i need this long process 😂
edgy sundial
no, there's only books for math
owh ok
tulip idol
2 + 6x + 11/10x = 6 + 5x + 13/10x
2 + 60/10x + 11/10x = 6 + 50/10x + 13/10x
2 + ...
T-T I just cancelled stuff I didn't need
$F +\cancel{2F} + \cancel{(3F + 2)} + \cancel{F/2} + \cancel{(3F/5)} = \cancel{(3F + 2)} + \cancel{(3F/5)} + F/5 + (\cancel{2F} + 4) + \cancel{F/2}$
elfin berryBOT
Arya
edgy sundial
ahh i see
x +~~ 2x~~ + (2 + 3x) + 1/2x + 3/5x =~~ (2 + 3x)~~ + 4/5x + (4 + 2x) + 1/2x
8/5x - 4/5x = 4
4/5x = 4
x = 5
.close
marsh citrusBOT
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woven condor
Question (translated):
“In a survey conducted by the Consumer Association, people were asked how significant they thought individual agricultural products were for the environment. The Consumer Association made mistakes in presenting the results with misleading visuals. In what ways were the mistakes made?”
Title of the chart:
“Are the differences significant enough?”
Labels on the chart (translated):
• Y-axis (from top to bottom): Milk, Eggs, Vegetables, Pork, Chicken, Beef, Lamb, Potatoes
marsh citrusBOT
Question (translated):
“In a survey conducted by the Consumer Association, peopl...
modest hawk
Question (translated):
“In a survey conducted by the Consumer Association, peopl...
,rccw
elfin berryBOT
modest hawk
idk
im still thinking abt this question
i could only think of the title being useless and leading in itself
like the graph is about the percentage of people answering the survey who thinks said product affects the environment
marsh citrusBOT
@woven condor Has your question been resolved?
normal moss
I've already warned you multiple times about this stuff. You're muted for now. You can come back in a week.
.close
pallid bramble
marsh citrusBOT
pallid bramble
63, 10, 77 then let all a(n) = a(1) x a(2) x a(3)… x a(n_2) / a(n_1)
all a(n-1) is a factor of the product of all terms less than a(n-2) so is a factor of the product of all terms less then a(n-1) so a(n) is always an integer. It is also comprime to the previous term because it has take a(n-1) out of the product and is not comprime to all other numbers as it is not comprime to all numbers less than a(n-2) as a(n-1) is comprime to a(n-2) so dividing by a(n-1) doesn’t take out the factor of a(n-2) which is not coprime to all numbers less than a(n-2). Taking away a(n-1) which is coprime to a(n-2) doesn’t take away factor of a(n-2) so a(n) isn’t coprime to a(n-2). Therefore we can always add a term so we have an infinite sequence.
marsh citrusBOT
@pallid bramble Has your question been resolved?
hushed egret
63, 10, 77 then let all a(n) = a(1) x a(2) x a(3)… x a(n_2) / a(n_1)
let's try computing a_4 using your definition
,w 63 * 10 / 77
elfin berryBOT
elfin berryBOT
pallid bramble
I meant 63,10,9
elfin berryBOT
hushed egret
,w 4900/3
elfin berryBOT
hushed egret
eh that was stupid
i have a feeling this formula doesn't work at some point but maybe the starting numbers make it work
elfin berryBOT
hushed egret
lots of cancellation i suppose
cedar stone
try looking at the prime factorisation
hushed egret
this feels more complicated than it should be
there's an easy constructive solution
cedar stone
use primes here
this works
hushed egret
||2,3,2\*5,2\*3\*7,2\*3\*5\*11,...||
2*5 and 2*3*7 are not coprime
hushed egret
also the next pair
cedar stone
also the next pair
was doing for factor
hushed egret
this is not the right construction
cedar stone
yes
hushed egret
if you start with 2,3 you're screwed
hushed egret
all a(n-1) is a factor of the product of all terms less than a(n-2) so is a fact...
anyway i don't see how this justifies that a(n) and a(n-1) are coprime
hushed egret
you have to remove more than just a(n-1) in general
hushed egret
it's easier to just list out the primes and do it
you should find the construction pretty quick
pallid bramble
Ok
hushed egret
notice that it suffices to consider primes up to the first power
cedar stone
||[1,1],[0,0,1,1],[1,0,0,0,1,1],[0,1,0,1,0,0,1,1],[1,0,1,0,1,0,0,0,1,1],....||should work
pallid bramble
3x5,7x11,3x13,5x7,3x11x13,5x7x
pallid bramble
Oh
cedar stone
||[1,1],[0,0,1,1],[1,0,0,0,1,1],[0,1,0,1,0,0,1,1],[1,0,1,0,1,0,0,0,1,1],....||sh...
||let f(i) = product of first i primes, a1 = 6, a_i = f(2i)/a_(i-1)||
pallid bramble
You have to add a new prime for each term so that the other terms can share it
hushed egret
that's very vague
hushed egret
||let f(i) = product of first i primes, a1 = 6, a_i = f(2i)/a\_(i-1)||
this formula looks wrong by term 3
pallid bramble
3x5,7x11,3x7x13,5x11x17,3x7x19,5x11x13,17
pallid bramble
Basically every term you have to add a new prime number that hasn’t been mentioned yet so that if a number has to be not comprime to a term then we can use that prime number as it hasn’t been introduced before
hushed egret
i mean that's true but also i'm not convinced you can continue this pattern ad infinitum
or at least you must justify you can
cedar stone
||let f(i) = product of first i primes, a1 = 6, a_i = f(2i)/(a_(i-1)*p_(2i-4)), wher p_k is the kth prime||
pallid bramble
3x5,7x11,3x13,5x7x19,3x11x13,5x7 now there is an error because 3 and 13 is in previous term. We can take away the 13. 5x7x13,3x11x19x11
pallid bramble
or at least you must justify you can
Ok
hushed egret
you can't just ad hoc patch up your mistakes
you need to tell me how i can produce each term in the sequence ad infinitum
cedar stone
||in other words, if n is even, product of all eventh primes till the 2n-4th prime, the 2n-1st and 2nth prime, if n is odd, product of all oddth primes till 2n-5th prime, the 2n-1st prime, and the 2nth prume||
pallid bramble
Ok I’ll have a think about the construction
hushed egret
||let f(i) = product of first i primes, a1 = 6, a_i = f(2i)/(a\_(i-1)*p\_(2i-4))...
putting it like this is rather uninformative id say
when you can quite literally just write down the pattern as you did the first time round
still temple
Can someone teach me how to do this bc my teacher assigned it to us but she never taught it to us
marsh citrusBOT
Can someone teach me how to do this bc my teacher assigned it to us but she neve...
regal coral
,rccw
elfin berryBOT
marsh citrusBOT
@still temple Has your question been resolved?
marsh citrusBOT
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wet hinge
in the answers, how would i know to write 1 as combintation of 2 and n. I don't understand that jump.
marsh citrusBOT
in the answers, how would i know to write 1 as combintation of 2 and n. I don't ...
@wet hinge Has your question been resolved?
main idol
in the answers, how would i know to write 1 as combintation of 2 and n. I don't ...
Probably should read theorem 0.2 and its corollary
marsh citrusBOT
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woeful shadow
how do i solve 13b?
marsh citrusBOT
how do i solve 13b?
woeful shadow
i wanna expand the definite integral i wrote and evaluate it but idk how to
i got 0 as my answer
bear with me a little my wifi is slow
not sure if i should use my calculator for this it doesnt say
smoky siren
The signed area beneath the x axis is negative. Without verifying anything else there should not be an absolute value here.
smoky siren
Use part (a) and split it up into two integrals where one curve is always on top for the entire sun interval per piece.
woeful shadow
what if in one interval the area is both below and above the x axis
from pi/8 to pi/4 it is above
smoky siren
Then you didn’t do what I just wrote
Oh I misread
That’s irrelevant.
Any area below the x axis will cancel with an area above but that is already captured in the usual integration process as is.
woeful shadow
hm
silk sable
Any area below the x axis will cancel with an area above but that is already cap...
You are confusing calculating a single curve with the x-axis versus calculating the distance between two curves.
smoky siren
Actually I misread it completely. You do need the absolute value here
smoky siren
You are confusing calculating a single curve with the x-axis versus calculating ...
Yeah I wrote signed area at the start but I didn’t really read the problem.
silk sable
Using an absolute value sign is technically correct but makes things a bit more complicated. It will be easier if you split the integral into two parts.
woeful shadow
Using an absolute value sign is technically correct but makes things a bit more ...
the splitting part confuses me because there are sections where the area is below and above the x axis
like i can easily do from pi/8 to pi/2 and then comes the area below from pi/4 to pi/2
silk sable
You want the length of the line segment being integrated to be positive, so on one interval you will use f(x) - g(x) and on the other interval, you will use g(x) - f(x).
woeful shadow
ohhhh
i can see that
you’re right
after point B 2cos2x is above 2sin2x
let me try
no im still stumped 😭
my answer still came out as 0
somehow
silk sable
Can you show your work?
woeful shadow
i dont know how to do it on paper
woeful shadow
Can you show your work?
hold up
it didnt come out as 0 actually
but
oh wait
i can see what’s wrong
in the fourth line
i think those are all meant to be positive
unless im wrong
silk sable
You should have gotten 4sqrt(2).
woeful shadow
yes
the first definite integral
that i expanded
all those root 2 / 2 ‘s are meant to be positive
silk sable
LPT : Don't skip writing out each step.
silk sable
You will be more prone to making math mistakes.
woeful shadow
i evaluated it in my head
silk sable
Yeah, don't do that until you are really good at it.
silk sable
yw
marsh citrusBOT
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paper pike
hi guys i have a tiny question.. can someone please help me on how to do this operation? i'm trying to understand but i can't, i tried several times in my calculator but it gives me 0
marsh citrusBOT
hi guys i have a tiny question.. can someone please help me on how to do this op...
paper pike
i don't know how to do this operation
i don't understand.. should i multiply the numerator by denominator, or numerator by numerator..?
fervent rampart
to multiply two fractions you multiply the numerators together, and separately multiply the denominators together
the algebra of complex numbers isn't really any different in that respect
paper pike
so -6 +6 x 5 + -5?
fervent rampart
you need to include parentheses and the i
paper pike
so
something like
(-6) + 6 x 5 + (-5) x i x i?
oh wait
give me a second
-6 x 5 + 6i x (-5i)
fervent rampart
just place parentheses around the entirety of each numerator
the parenthesis is implied by them being on separate fractions
paper pike
(-6)x5+(6x-5)i?
fervent rampart
just place parentheses around the entirety of each numerator
what is the numerator of the first fraction?
fervent rampart
what is the |?
paper pike
its just a pipe i writed to difference the first numerator from the second numerator
its just a separator
fervent rampart
instead of the pipe a more standard notation would be to put each numerator in parentheses.
paper pike
(-6 + 6 x i) (5 -5 xi)
fervent rampart
yes
paper pike
alright
so
to multiply two fractions
its numerator to numerator right??
also i have another question
when doing 6 x 1 and -5 x i
its something like
(-6 + 6) x (5 + -5)i right?
so i guess that the final result
it gives me 19
fervent rampart
(-6 + 6) x (5 + -5)i right?
where is this coming from?
fervent rampart
(-6 + 6 x i) (5 -5 xi)
this is correct
paper pike
i get 19
fervent rampart
where did the i go?
fervent rampart
no it isn't
fervent rampart
only some of the terms had i
fervent rampart
(-6 + 6) x (5 + -5)i right?
so moving the i around like here doesn't work
paper pike
hmmm
you suggest something like
(-6 x 5) + (6 x -5)?
or
(-6 + 6) x (5 + -5) * i square?
paper pike
-6 x 6 x i + 5 x -5 xi?
fervent rampart
what?
paper pike
(-6 * 6 * i) + (5 * -5 * i)
fervent rampart
you seem to be swapping addition and multiplication symbols
paper pike
a ver wait
first of all
i want to get this result
to do this i multiplied 5 + 5 x 5 + -5
this is 50
the i i didn't multiplied but i guess that when having the i, you multiply the result of the multiplication by i, so it changes the symbol
right?
fervent rampart
i suspect that you got the correct result on the denominator with the wrong method
fervent rampart
treat i like a variable
distribute out the multiplication like you would for an algebra expression
then remember that i² = -1 at the end
for example:
(2+3i)(5-6i) = 2*5 + 2*(-6i) + 3i*5 + 3i*(-6i) = 10 - 12i + 15i - 18(-1) = 28 + 3i
fervent rampart
yes
paper pike
hmmm
but i did the same as you
look
(-6 + 6i) x (5 + -5i) = (-6 x 5) + (6i x -5i)
fervent rampart
that's not distributivity though
you just multiplied the first term with the first term, and the second term with the second term. FOIL stands for "First, Outside, Inside, Last", and you only got the F and L there
paper pike
ooh
you mean that i need to distribute for example the -6 with the two 5
right?
like
(-6 x 5) +(-6 x -5i) + (5 x -6) + (5 x 6i)
paper pike
like this?
fervent rampart
but now you still have the 5 and -6 multiplied twice and are missing a different combination of terms
paper pike
(-6 x -5i) + (6i x 5) + (-5i x 6i)
fervent rampart
ok now you have 3/4 terms but are missing the -6 and 5 term
fervent rampart
yes
paper pike
yayy
now another question
when doing 6i x -5i
i guess that the result, will be multiplied by -1 right? (because i x i = -1)
fervent rampart
yes
fervent rampart
does your calculator support complex numbers?
fervent rampart
could you show the input you are giving?
fervent rampart
including the i?
paper pike
i guess that the result, will be multiplied by -1 right? (because i x i = -1)
.
you said its correct right?
i mean, when multiplying two numbers that have i, it means result x i right? (in this case, i x i = -1) so result x -1
fervent rampart
if it has i * i, then it will result in -1. but if there is only one i, then it will just have i
paper pike
I THINK I DID CORRECTLY THE MULTIPLICATION
AAAAAAAAAAAAAAA
LOOK
LOOK
in order from left to right
-30 + 30 (because 6i x -5i = -30, but since we have two i, then -30 x -1 = 30) + -30i + 30i
= 60
is this correct?
i think i missed something
- x - = + right?
noo
minus x minus = +
right?
fervent rampart
yes
fervent rampart
-30 + 30 (because 6i x -5i = -30, but since we have two i, then -30 x -1 = 30) +...
how did you get -30i here?
paper pike
(-6 x 5) + (6i x 5) + (-6 x -5i) + (6i x -5i)
-30 30i 30i 30i
the -30 is because -6 x -5 = -30 but since - x - = + then it turns out to be 30
fervent rampart
(-6 x 5) + (6i x 5) + (-6 x -5i) + (6i x -5i)
-30 30i 30...
i think you made a typo on the last one here
fervent rampart
yes
paper pike
what typo?
fervent rampart
the result of it below
paper pike
30 i?
fervent rampart
yes
paper pike
6 x -5 = -30 right?
fervent rampart
yes
paper pike
but i x i = -1
fervent rampart
yes
paper pike
so -30 x -1 = 30
fervent rampart
i * i = -1, when you do the multiplication you have "used up" the i
paper pike
ooooh
so the result doesn't have the i
alright
i have another question
We will multiply by the conjugate of the denominator to have a real number in the denominator
thoes that meas 5 - 5 or 5 + -5?
i don't understand
fervent rampart
the conjugate of a complex number is the same number but you swap the sign of the imaginary part
fervent rampart
so the conjugate of 5 + 5i is 5-5i, and the conjugate of 2-3i is 2+3i
paper pike
but i don't understand
you are talking about the addition
5 + 5 = 10
and its conjugate its 5 - 5 = 0
or +5 + +5 and its conjugate its +5 + -5?
do you understand my doubt? if not i can try to explain it better
fervent rampart
the conjugate is only something which makes sense for a complex number
you flip the sign of the number which is multiplied by i (that's what we call the imaginary part)
if i had -3i + 2 then the imaginary part is -3, because that's the number multiplied by i
so the conjugate of -3i + 2 is 3i + 2 = 2+3i
if you have a real number like 10, you can write it as 10 + 0i. so the conjugate of that is just 10 - 0i = 10. so the conjugate of a real number is just itself
fervent rampart
yes
paper pike
yayyyy
thank you
i have another question
i did the distributive property
(5 + 5i) x (5 + -5i) = (5 x (-5)i) + (5 x 5) + ((-5)) x(5)i) + (5 x 5i)
and i get
50i
because 25 + 25i = 50i
right?
hmmm
any idea guys
i'm kinda confused right now
fervent rampart
(5 + 5i) x (5 + -5i) = (5 x (-5)i) + (5 x 5) + ((-5)) x(5)i) + (5 x 5i)
how did you get the third term?
paper pike
i applied the distributive property
5 + 5i x 5 + -5i = (5 x 5) + (5 x -5i) + (5 x 5i) + (5i x -5i)
fervent rampart
5 + 5i x 5 + -5i = (5 x 5) + (5 x -5i) + (5 x 5i) + (5i x -5i)
this is correct
paper pike
the result i get is 25 + 25i
fervent rampart
-25i + 25 + 25 + 25i
this is correct
fervent rampart
that is incorrect
paper pike
then what's the correct result?
fervent rampart
when you multiply by the conjugate you should end up with a real number
fervent rampart
what is -25i + 25i?
paper pike
the result of this
fervent rampart
-25i + 25 + 25 + 25i
how do you get -25i + 25 + 25 = 25 + 25i though?
paper pike
wdym?
fervent rampart
-25i + 25 = 0
25 + 25i = 50i
neither of these are true
you can add terms which don't have an i together and terms which do have an i together
but you can't mix real and imaginary
paper pike
but you said this was correct
paper pike
5 + 5i x 5 + -5i = (5 x 5) + (5 x -5i) + (5 x 5i) + (5i x -5i)
.
unless you mean that i should do
5 + 5i x 5 + -5i = 5 x 5 + 5i x -5i + 5 x 5 + -5i x 5i
fervent rampart
-25i + 25 + 25 + 25i
this is correct so far
it is not correct to add it up to 25 + 25i
paper pike
ooh
you mean then an complex number addition right'
like real + real and imaginary + imaginary
fervent rampart
yes
marsh citrusBOT
@paper pike Has your question been resolved?
marsh citrusBOT
ruby mulch
Inspired by the grammatical alignment meme, I tried to solve the following question "How many possible partitions of a set of 6 items. The number of partitions are free, i.e. you have to count partition into 1 set, 2 sets, 3 sets, 4 sets, 5 sets and 6 sets together. Order doesn't matter". I got 203 with the following calculation:
-
There are 1 way to partition a set of 6 items into 1 set.
-
There are $\frac{(2^6-2)}{2}=31$ ways to partition a set of 6 items into 2 set.
-
There are $\frac{3^6-3\cross(2^6-2)-3\cross 1}{6}=90$ ways to partition a set of 6 items into 3 set.
-
There are $\frac{6!}{8\cross2!}+frac{6!}{3!\cross3!}=45+20=65$ ways to partition a set of 6 items into 4 set.
-
There are $\frac{6!}{2!4!}=15$ ways to partition a set of 6 items into 5 set.
-
There are $1$ ways to partition a set of 6 items into 6 set.
Is that correct?
marsh citrusBOT
Inspired by the grammatical alignment meme, I tried to solve the following quest...
elfin berryBOT
Xwtek
stoic saddle
btw you can use the enumerate environment to get numbered lists in TeX
anyway your counts are correct
marsh citrusBOT
marsh citrusBOT
Available help channel!
low scaffold
shoudnt the answer of derivative of 0 and 1 be undefined as well since the general derivative is undefined
but in the answer solution, its saying 0
marsh citrusBOT
shoudnt the answer of derivative of 0 and 1 be undefined as well since the gene...
elfin berryBOT
k
stoic saddle
tfw constant function turns out non-differentiable 
low scaffold
**k**
ohh
i swear cuz i always see when theres nothing on the numerator
it just becomes 1
@stoic saddle @rancid geode do u know why this is the case?
rancid geode
its
only true
for multiplication
like
when u cancel stuff
with the denominator
lets say
i want to simplify
For ${a +b \neq 0 }$,
[ \frac{a^2 - b^2}{a+b} = \frac{(a-b)(a+b)}{(a+b)} = (a-b) \cdot \frac{a+b}{a+b} = (a-b) \cdot 1 = a-b]
elfin berryBOT
k
low scaffold
i see now
cuz if the numerator was to get cancelled out and becomes times 0, it would just result to zero
which is not correct
marsh citrusBOT
Available help channel!
chrome swift
Hello!
Just want to make sure : The big V means union?
marsh citrusBOT
Hello!
Just want to make sure : The big V means union?
chrome swift
Or is it different from the big U?
According to my notes, the union should be a big U. Not a big V, so I'm confused.
devout mauve
its an or
chrome swift
Okay I am confused then because it's not what I expected.
Basically, I am trying to follow the demonstration of this proposition :
Then, we determine the probability of ONE of the elements.
Then somehow we use the or by using the upper bound of the addition/union.
static quarry
the key is in the first sentence: complementary event
shy monolith
Okay I am confused then because it's not what I expected.
Union makes sense when you have sets
those in the brackets are not sets, but statements
when you negate a collection of statements true for all h, it means at least one of them is false
chrome swift
Mind if we back off a little bit?
We have the following equation
We have found what is the upper bound of one h:
I understand up to that point.
After that, I'm lost.
Apparently, from the equation P(AUB) <= P(A) + P(B),
We can get the following equation:
I understand the summation.
I don't understand the other part.
I'd understand if it's supposed to be the Union. But if it's for "or". Then I'm lost.
devout mauve
but or is the same thing as union
just for statements
{x: x satisfies P} union {x: x satisfies Q} = {x: x satisfies P or Q}
chrome swift
So, baiscally, it's the same as union in this case.
Since the entire equation is not a set, we use the operator or. But it's operate exactly like an union.
devout mauve
yes
chrome swift
Gotcha. And if all h are complementary, we can then consider the complementary event
P(AUB) = 1 - P(A and B)
chrome swift
But each h are complementary(?), then the intersection should be 0?
each predictor h is complementary (they aren't overlapping)
If I understand well the scenario.
devout mauve
my point is that you cant put and between sets A and B
you have to put the intersection
chrome swift
P(AUB) = 1 - P(A intersection B)
Wait it does make sense kind abit. Each h are different and complementary. But the actual equation can overlap.
Since it's a loss function over a random variable.
Looks like we are doing like this
(V being the intersection)
Ahhh I got it. I totally missed something in my notes.
.close
jovial lotus
I'm trying to compute the following integral:
$$\int_0^\infty {x}{a x}x^{-2}dx,$$
where $a \in (0, +\infty) \cap \mbb{Q}$ and ${x}$ is the fractional part of $x$.
I tried a direct computation, but for an arbitrary $a$ it turned out to be hard to complete. When $a=1$ I got $\ln(2\pi) - \gamma$ after some tricks with series. The problem is that it's hard to imagine the subdivision of the real line into segments where ${x}{a x}x^{-2}$ looks like $(x - n)(ax - m)x^{-2}$.
I tried applying Fourier transform to ${ax}x^{-1}$ to use Plancherel's theorem (namely, $\langle f, g \rangle = \langle \hat{f}, \hat{g} \rangle$) but to no luck. I'd really appreciate any help or ideas as to how to compute this integral.
marsh citrusBOT
I'm trying to compute the following integral:
$$\int_0^\infty \{x\}\{a x\}x^{-2}...
elfin berryBOT
tight bound
elfin berryBOT
jewels!
tight meteor
Now ax = u and same thing 
jovial lotus
The problem is that {ax} = ax - m iif m/a <= x < (m+1)/a, and I have to work out all intersections of [k, k+1] and [m/a, (m+1)/a]
tight meteor
Ah I get the problem
nimble prairie
**jewels!**
Idk if I’m cooking but switch order of sum and integral (with verification if need be) and we’ve got a discrete convolution going on no?
nimble prairie
Oh bruh
tight meteor
Unless you wanna sub that out first
odd crest
not op but could you explain how this and the original equation are the same?
jovial lotus
jewels probably meant x + k, not x - k
elfin berryBOT
jewels!
tight meteor
@jovial lotus whats the restriction on a?
is it also an integer
oh nvm
positive rational
jovial lotus
yep
tight meteor
its easier for the case of a being an integer because ak would also be an integer and you could get rid of it from the fractional part
And then double sum maybe
odd crest
is this correct?
jovial lotus
Yeah when a is an integer the segments [m/a, (m+1)/a] and [k, k+1] intersect nicely
odd crest
if so then that's a funky function
jovial lotus
Yeah it looks roughly like that and is rather funky
odd crest
the integral is undefined for many a's
odd crest
then it might not be the same function
jovial lotus
The function is equal to a on (0, min(1, 1/a)) and behaves like 1/x^2 on the rest of the line, so it's integrable
odd crest
when i decrease the step size
it gives more values of a an integral
it might very well be desmos not being able to correctly integrate
jovial lotus
It's certainly desmos, yeah
Numerical integration usually sucks at computing integrals of functions with many discontinuities
jovial lotus
Mostly because it's designed to be accrurate on locally-analytic functions but poorly optimized for funky functions
jovial lotus
how did you come across it?
It's a long story 😅
I was messing around with a certain hilbert space and my questions boiled down to computing a certain Gramm matrix, where each cell can be obtained via the integral I posted
marsh citrusBOT
@jovial lotus Has your question been resolved?
marsh citrusBOT
@jovial lotus Has your question been resolved?
marsh citrusBOT
@jovial lotus Has your question been resolved?
marsh citrusBOT
@jovial lotus Has your question been resolved?
main idol
Probably should just ask on stack exchange with functional analysis tag
marsh citrusBOT
@jovial lotus Has your question been resolved?
tidal totem
how would i solve this differential
marsh citrusBOT
how would i solve this differential
tidal totem
i just dont know how to start
the 4y is confuising me if it wasnt there i would js do u = y' and then solve for u
pure dagger
limber hearth
You need the caracteristic equation
sullen lynx
how would i solve this differential
Do you know what the operator method is?
tidal totem
Do you know what the operator method is?
no
sullen lynx
Oof the operator method would help solve this question pretty quickly
tidal totem
so what is it
sullen lynx
Search it up on YT
Also there's a property of exponential shift so make sure to learn that as well before trying the above question
tidal totem
uuuuuuuuuhhhhh i cant find a video with a similar example
amber birch
i just dont know how to start
have you learned how to solve something like y'' - 4y' + 4y = 0?
if you can do that, the next step is to follow the variation of parameters method
here g(t) = e^(3t), so g(t) is just the right hand side of the DE
marsh citrusBOT
@tidal totem Has your question been resolved?
jovial shell
is this wrong?
marsh citrusBOT
is this wrong?
stoic saddle
your handwriting's wonky but no, this is not wrong. why are you asking whether it's wrong?
marsh citrusBOT
peak ginkgo
cuz it's -ln(3)
3^(-1)
marsh citrusBOT
Available help channel!
uncut vale
Hello, I saw a proof online of why the Reuleaux triangle has a constant width but I'm bad at geometry so I don't understand it , can anyone explain in an easy and detailed manner please?
marsh citrusBOT
Hello, I saw a proof online of why the Reuleaux triangle has a constant width bu...
marsh citrusBOT
@uncut vale Has your question been resolved?
sweet sparrow
let's say T is on the arc AB
when we consider the floor as fixed, the line perpendicular to the floor at T has to go through the arc's radius, in this case it's C
because CT is constant, when T moves on the floor, C also moves on a horizontal line
you can also see that the object is bounded between two parallel lines so CT is also the object's width in this case
marsh citrusBOT
@uncut vale Has your question been resolved?
potent ember
can someone explain this i dont get it
marsh citrusBOT
can someone explain this i dont get it
stoic saddle
those notations are broken
potent ember
broken?
stoic saddle
yes, broken
i can tell what they were trying to write but that isnt how you write it
if you didnt see the answer options would you know how to solve the equation?
potent ember
if you didnt see the answer options would you know how to solve the equation?
i was just guessing by considering two cases either sin^2x is 0 and cosx is 1 or sin^2x is 1 and cosx is 0 and checking the answers for it but it didnt work out for me
stoic saddle
no guessing
that is a bad strategy
you should not guess and there is no need to guess.
with equations like these guessing is an absolute no-go.
potent ember
with equations like these guessing is an absolute no-go.
then replace sin^2x with (1-cos^2x) ?
potent ember
and you will get an equation with an obvious factorization
so then i got cosx (cosx -1) = 0
cosx = 0 or cosx = 1
so cosx is 0 at π/2 + πn at cosx = 1 at πn but i dont know how that translates to these options
stoic saddle
try to plot each solution set from the answer options on the unit circle
be mindful of +2nπ vs. +nπ
potent ember
be mindful of +2nπ vs. +nπ
whats the difference
stoic saddle
one steps up by a full period for each next value of n, and the other by only half
potent ember
wont cosx also 1 at 2nπ
stoic saddle
... i just realized i fucking missed an error of yours
cosx = 1 at πn
this should be 2πn not 1πn as you wrote
otherwise you would be claiming cos(pi) = 1 when it isn't
at odd multiples of pi cos is -1
it is 1 only at even multiples of pi
sweet sparrow
also period is pi*n only when cos x = 0
potent ember
oh i understand now
potent ember
so then i got cosx (cosx -1) = 0
cosx = 0 or cosx = 1
so cosx is 0 at π/2 + πn...
did i also get cos being 0 at π/2 + nπ wrong
sweet sparrow
that one is correct
potent ember
can someone explain this i dont get it
so it could be option a bc it includes 2nπ for cos being 1 but i can't think of anything else
potent ember
try to plot each solution set from the answer options on the unit circle
ok so i tried this option a gives me the x values which give cos x as 0 and 1 and b also gives me cos x as 0 and 1 but it includes nπ where cosx could be either 1 or -1 so is the option only a) or d) both a and b?
stoic saddle
i dont see where you plotted any of this on the unit circle.
i dont rly like having to repeat my instructions and also being misunderstood
these are the sorts of diagrams i was expecting
i did one option and part of another
potent ember
i dont rly like having to repeat my instructions and also being misunderstood
sorry i wasn't understanding what u meant by plotting but i get it now forgive me
potent ember
these are the sorts of diagrams i was expecting
okay i got the answer thank u
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
marsh citrusBOT
how can I get the 20th term?
iron ocean
I got the 5th term wait
Imma show u im walking on my way home
is anyone still here?
<@&286206848099549185>
earnest bear
what is kept the same and what is added between steps of the sequence
proud basin
just post it, people will see it eventually
onyx pine
there’s a pattern here, check the difference of each term
mild stratus
If you can't tell, its an arithmetic progression/ a sequence where the number of balls is increasing by n
mild stratus
Try to form a pattern using n or use APs
marsh citrusBOT
@iron ocean Has your question been resolved?
stone pelican
brave spire
i dont see how the existence of a supremum helps me prove the existence of an infimum
marsh citrusBOT
i dont see how the existence of a supremum helps me prove the existence of an in...
devout mauve
-x
brave spire
what
okay u mean
multiply by -1 and try to reverse the inequality?
okay hold up let me yap a bit
devout mauve
I always love it when those minimalistic hints actually work
brave spire
so assume the least upper bound axiom is true
therefore there exists some real number K such that K us an upper bound of A and for every upper bound K' of A, K <= K'
devout mauve
what is A
brave spire
our non-empty subset
devout mauve
A is bounded below
brave spire
so for every x belonging to A, x <= K and x<=K' and K<=K'
devout mauve
not necessarily above
brave spire
oh right am so fucking stupid
the least upper bound axiom is true
but A is not necessarily bounded above
it is bounded below definitely
okay lets restart
A is bounded below means
there exists some real number K such that for every belonging to A, K<=A
devout mauve
typos
devout mauve
for every a belong to A, K <= a
brave spire
oh shit right thats what i meant 😭
A is bounded below means
there exists some real number K such that for every x belonging to A, K<=x
this means
-K >= -x for every x belong to A
now let B = {y such that y=-x for every x in A}
clearly B is also a non-empty subset of R
devout mauve
of R
brave spire
now this means B has an upper bound
brave spire
now this means B has an upper bound
by our axiom B has a supremum
let the supremum of B be z
brave spire
okay now i got confused myself 💀
brave spire
give the supremum a new name
yeah alr
so for any upper bound z' of B, z'>=z
so for every real number w belonging to be B, z'>=w, and z>=w and z'>=z
so -z'<=-w, -z<=-w and -z'<=-z
Now by definition of B, -w will be an element of A.
Clearly -z', and -z are both therefore lower bounds for the set A
since -z' is arbitary and is always less than or equal to -z, therefore -z must be the infimum
is this right @devout mauve
devout mauve
right idea
wrong direction
you need to start with an arbitrary lower bound of A
and then show that it is <= -z
brave spire
ugh
😭
hold up let me try again once more
consider some non-empty subset of R called A which is bounded below
therefore A must have some lower bound K such that for every element x in A x >=K
brave spire
and then show that it is <= -z
wait what i dont understand
devout mauve
to show that -z is the infimum of A you need to show that for every lower bound z' of A you have z' <= -z
you instead started with upper bounds of B
you really just need to rewrite what you wrote
all the ideas are there
or otherwise you can also show that indeed all lower bounds of A correspond to upper bounds of B and vice versa. but without that there could in theory be a lower bound of A which you dont get with your approach
glacial hedge
i dont see how the existence of a supremum helps me prove the existence of an in...
it's so weird to me that there's people who are that good at computations, who have studied that much computational calculus and arithmetic, yet still don't actually even know how real numbers work because they never actually covered the fundamentals
Finally started uni classes ?
devout mauve
well real analysis != calc
brave spire
it's so weird to me that there's people who are that good at computations, who h...
no i m trying to do howies
brave spire
you instead started with upper bounds of B
if -z is an upper bound of B then isnt z a lower bound for A by definition
fathom ridge
show it with the inequality
devout mauve
I am not saying that these things are hard to show
but they need to be shown rigorously
and not just skipped over
at this level everything needs to be shown
glacial hedge
well real analysis != calc
as someone who did RA at the start of uni, I'm always surprised by how late some countries actually do things rigorously and groundedly
devout mauve
(same)
marsh citrusBOT
@brave spire Has your question been resolved?
marsh citrusBOT
marsh citrusBOT
Available help channel!
obtuse sky
I find the determinant of these vectors