can you correct this now?
#help-0
sour shore
fast basin
Tension in massless string remains same throughout
fast basin
tension in the slant part of string
and tension in veritcal art of string will be same
*part
T=mg
sour shore
fast basin
Tu=Tl
indigo bramble
ah it makes sense
sour shore
it's the same tension (considering just magnitude)
sour shore
alright try correcting this now
indigo bramble
r^2 = 2 * (mg) ^ 2 * ( 1 + cos 143)
sour shore
alright simplify
sour shore
yeah
indigo bramble
so r^2 = 2/5 * mg^2
sour shore
we can use that I'm unhappy with approximating stuff like that but its what your qn does haha
yeah (mg)^2
indigo bramble
yess
sour shore
ok i hope you clarified some intuition in how pulleys affect forces
indigo bramble
yea
its been a while
so i forgot
and clung to a wrong way of thinking that i saw somewhere a few hours ago
sour shore
for single pulley system they change direction but not magnitude
sour shore
yeppers
indigo bramble
yippee
fast basin
happy new year
indigo bramble
happy new year
altho i dont have the official answer
one friend of mine sent their solution
im concered that their answer is different to what we just figured out
indigo bramble
@sour shore @fast basin is there anything that they did wrong?
fast basin
this is correct
indigo bramble
oh wait sorry
they used sin 53 = 3/5
which is wrong
sin 53 = 4/5
so the correct option comes out to be sqrt(2/5) * mg
sour shore
<@1045188678105976852> <@1191249190756569159> is there anything that they did wr...
by the way in this case while the formula is different, do you understand how your friend got the first equation?
indigo bramble
by the way in this case while the formula is different, do you understand how yo...
I don't think so
How did he get the first equation?
sour shore
so remember we did vector addition with the parallelogram law?
sour shore
How did he get the first equation?
so remember we did vector addition with the parallelogram law?
indigo bramble
so remember we did vector addition with the parallelogram law?
Yes
sour shore
then our convention was that the bottom tension is completely horizontal
indigo bramble
Why the - T i cap
indigo bramble
then our convention was that the bottom tension is completely horizontal
Yea
sour shore
this is a different convention
indigo bramble
Oh cus it's opposite to i cap
sour shore
what direction is icap
indigo bramble
Y axis
sour shore
and what direction is jcap
indigo bramble
and what direction is jcap
X axis
sour shore
yeah
so your friend expressed the force vectors acting on the pulley
in terms of i and j components
probably a better way tbh haha
indigo bramble
Why are we adding the sin and cos components while subtracting the mg
@sour shore
sour shore
oh
i firgot
it's not just y-axis vs. x-axis
i cap is the unit vector pointing up specifically, and not down
j cap is the vector pointing right specifically, and not left
so the terms having sin/cos are the components of the force exerted on the pulley by the upper half of the rope
indigo bramble
j cap is the vector pointing right specifically, and not left
Got it
sour shore
and the negative term (sin 180 deg) is the force exerted on the pulley by the lower half of the rope
solving their sum gives you the force vector they together exert on the pulley in the up component and right component
indigo bramble
Ohh so the rope has two parts, the upper and lower. I thought of it as one. Got it.
sour shore
yeah
indigo bramble
solving their sum gives you the force vector they together exert on the pulley i...
Yes makes sense
sour shore
well any point of the rope divides it to an upper part and a lower part
fast basin
there is only a single rope
sour shore
so you can do the same kind of analysis at e.g. any point below the pulley, but it's quite boring
it's just 0 = mg icap - mg icap
you can do it above the pulley too
indigo bramble
it's just 0 = mg icap - mg icap
Ah
sour shore
it's just 0 = mg sin 53 i + mg cos 53 j - mg sin 53 i - mg cos 53 j
indigo bramble
it's just 0 = mg sin 53 i + mg cos 53 j - mg sin 53 i - mg cos 53 j
This is above the pulley at any point?
sour shore
explain it to me
indigo bramble
So mg sin 53 i is the vertical component and - mg sin 53 would be the opposite direction to it, both of which cancel each other out
Similarly for the horizontal component
sour shore
vertical component of what, and where does the opposite direction come from?
indigo bramble
Vertical component of the rope that's at an angle of 53
Opposite direction because the force there is at an equilibrium
sour shore
ok i have to go now so I'll explain it
choos a point above the pulley, mg sin 53 i + mg cos 53 j is the force the part of the rope above the chosen point exerts at the point (in the up component and right component respectively), and - mg sin 53 i - mg cos 53 j is the force the part of the rope below the chosen point exerts at the point (in the up component and right component respectively)
indigo bramble
choos a point above the pulley, mg sin 53 i + mg cos 53 j is the force the part ...
Okay got it, thanks
fast basin
how old are you
fast basin
i want to know in other countries in which age they study these topics
indigo bramble
11th grade
fast basin
Yes in india also 11th grade
fast basin
no problem
bright scarab
guys can someone explain the difference between parallelogram law and triangle addition in vectors
bright scarab
like some say we have to join tail with tail and some say head with tail
im totally lost
like if i wish to add these free vectors which rule should i follow
bright scarab
is it always the case?
west cipher
yes
bright scarab
or only when both are free vectors
lone heartBOT
grizzled wigeon
@hybrid flint
if you draw out the resultant vector the magnitude of that vector is the same as length “c” of a triangle it’s a parallelogram
bright scarab
thank u this clarified a lot
grizzled wigeon
thank u this clarified a lot
no worries I was stuck on this as well
lone heartBOT
Available help channel!
dusty geyser
.
so i tried to expand the proof of irrationality of 2 to 4 but somehow i get a contradiction too??
fallen verge
At no point is it necessary that q is even
dusty geyser
well if p² is even, then q² has to be even too, correct?
echo socket
dusty geyser
ohhh i see
so like this
so this means that the proof ends in "no result" basically, right?
which at least means no contradiction but also no clear "okay this is definitely rational"
ofc this is purely hypothetical as we know that 4 is a perfect square but i'm just trying to understand that this kind of proof only gives a proper result if it is actually not a perfect square
lone heartBOT
Available help channel!
drifting gust
drifting gust
it says there are a1 ..... a50 number and all are positive numbers
at least how many different ai +aj sums can be obtained ? ( i is not equal to j )
lone heartBOT
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
exotic belfry
you have the answer and the explanation in the screenshot. @drifting gust
if the explanation in the screenshot is not clear enough, tell what is not clear.
drifting gust
if the explanation in the screenshot is not clear enough, tell what is not clea...
That is not clear at all
exotic belfry
what exactly?
lone heartBOT
@drifting gust Has your question been resolved?
strange thunder
Can someone help me solve this?
lone heartBOT
Can someone help me solve this?
strange thunder
I've got (x+y+z)^2=4(xy+yz+zx)=2(x^2+y^2+z^2)
strange thunder
Don't know what to do next
strange thunder
You can you Am Gm inequality
You mean just x, y, z and 1/2xyz?
Do you mean this?
cloud nacelle
yeah
strange thunder
@cloud nacelle
cloud nacelle
or try various combinations
cloud nacelle
Yeah but how can you say for sure that this satisfies the given conditions??
Am Gm inequality is applicable for positive real numbers
so
it always satisfies
strange thunder
Am Gm inequality is applicable for positive real numbers
Yeah but we can't really just say that the min is going to be 4 x fourth root of 2 right
cloud nacelle
Yeah but we can't really just say that the min is going to be 4 x fourth root of...
Do you have the answer?
strange thunder
Do you have the answer?
No
cloud nacelle
this should be the answer
strange thunder
this should be the answer
Then whats the use of the given condition?
cloud nacelle
Then whats the use of the given condition?
no use lol
strange thunder
Doesn't this feel wrong
cloud nacelle
i mean what's wrong in the inequality
strange thunder
i mean what's wrong in the inequality
There's nothing wrong
The question is to find the min when the real numbers satisfy the equation given
strange thunder
And when we apply the inequality we just get that it will be greater than or equal to a value
I don't think we can say it's the minimum value
cloud nacelle
it is the min value
its greater than or equal to ( )
so ( ) becomes the least value
strange thunder
so ( ) becomes the least value
I get that
cloud nacelle
yes
strange thunder
But at that value it may or may not satify the equation given no
cloud nacelle
But at that value it may or may not satify the equation given no
this equation is always valid
strange thunder
And anyways the answer should be a non negative integer less than 100
means 00 to 99
cloud nacelle
well
strange thunder
<@&286206848099549185>
ocean sealBOT
Wither
cloud nacelle
do you know what is $(x + y + z)^2$
hey can't we directly apply Am Gm
the condition isnt necessary
strange thunder
do you know what is $(x + y + z)^2$
You mean the expansion?
real parrot
You mean the expansion?
yes
real parrot
hey can't we directly apply Am Gm
?
strange thunder
$(x^2+y^2+z^2+2xy+2yz+2zx)$
ocean sealBOT
Ishaan
cloud nacelle
the condition doesnt seem necessary
real parrot
AM GM inequality
yeah it would not be very good here
real parrot
**Ishaan**
alright
strange thunder
yeah it would not be very good here
That's what I was saying too
cloud nacelle
@real parrot
real parrot
<@798225726586880030>
don't you get the direct ans tho?
cloud nacelle
don't you get the direct ans tho?
isnt that what we want?
strange thunder
don't you get the direct ans tho?
Well even if we say that indeed is the answer
real parrot
yea
strange thunder
What to do then?
cloud nacelle
What to do then?
so we can factorize that as (x-y)^2 + (y-z)^2 + (z-x)^2
(x-y)^2 + (y-z)^2 + (z-x)^2 = 0
the only way
is
if x=y=z
strange thunder
(x-y)^2 + (y-z)^2 + (z-x)^2 = 0
How did you get this?
undone ledge
so we can factorize that as (x-y)^2 + (y-z)^2 + (z-x)^2
not sure where you got this?
oh wait nvm i see you
undone ledge
if that is true , x , y z will become 0
^it wont ignore
cloud nacelle
if that is true , x , y z will become 0
why
x=y=z = k lets suppose
so
3k + 1/(6k)
undone ledge
^it wont ignore
^
cloud nacelle
the min value of this
0 is not possible
so
and now we apply AM GM
on 3k + 1/(6K)
undone ledge
sounds good
cloud nacelle
root2
alpine sable
I have a problem just to have fun
undone ledge
I have a problem just to have fun
unless you mean in this context (like if you are reffering to the ongoing question in the server)
undone ledge
never thought of it , why do you think it is not defined?
cloud nacelle
never thought of it , why do you think it is not defined?
it was a part of question
basically i was supposed the max value as the answer
and i got two results which 0 and 2
the answer was 0
somehow
undone ledge
could be outside the range of the question
cloud nacelle
could be outside the range of the question
it wasnt
undone ledge
context of question would be apriciated
cloud nacelle
context of question would be apriciated
forget the question
undone ledge
basically i was supposed the max value as the answer
and i got two results whic...
did you use calculus?
undone ledge
forget the question
do you mean forgot , or forget , as in you implying to me?
undone ledge
it was a calculus question
i mean if you used double derivative the other one should be the minima
cloud nacelle
is (-ve)^irrational defined?
just answer this
cloud nacelle
i mean if you used double derivative the other one should be the minima
no no, i asked my sir my solution was correct
and there was an important observation
which I missed
everything was correct
idk why 0 was the answer
so thats why i was asking
undone ledge
what did your sir say about the topic ?
cloud nacelle
whats a transcdental number?
undone ledge
not sure if i would be able to explain that tbh , consider searching it up
cloud nacelle
oh
undone ledge
not reccuring
cloud nacelle
that's irrational
cloud nacelle
non terminating non recurring
irrational number?
undone ledge
i mean yeah all trans numbers are irrational im pretty sure
cloud nacelle
okay
cloud nacelle
I've never heard of this term before
undone ledge
do tell me if you get a more valid answer about your question , im intrested
cloud nacelle
do tell me if you get a more valid answer about your question , im intrested
sure
I'll dm you
undone ledge
smty!
lone heartBOT
@strange thunder Has your question been resolved?
frail grove
$B=0.\overline{9} \ 10\times 0.\overline{9}=9+0.\overline{9} \ 10B=9+B \ 9B=9 \ B=1 \ 0.\overline{9}=1$
ocean sealBOT
Adam Chebil
tall topaz
frail grove
how come this is true ??
tall topaz
It's not really a rigorous proof
alpine sable
its like really weak yeah
frail grove
where's the wrong step ?
alpine sable
its messing around with the convergence of the thing
if you want a true proof you need to consider a geometric series
frail grove
what's wrong with this proof ?
alpine sable
you are assuming 0.9999... to already exist
vague coral
what's wrong with this proof ?
convergence problem
tall topaz
One could question if, for example, B can be cancelled here.
alpine sable
which is the conclusion
here is a modification to your "proof"
,align
x &= \overline{9}. \
10x &= \overline 90.\
10x+9 &= x \
9x &= -9\
x &= -1
ocean sealBOT
alpine sable
do u think this is true
frail grove
obv not
tall topaz
Obviously, -1 = ...999 
mortal trellis
before you ask yourself whether 0.99999... equals 1, you should first ask yourself what that notation even means
frail grove
so 0.99999999 doesn't even exist ?
tall topaz
mortal trellis
thats not what we are saying
frail grove
mortal trellis
before you do any sort of manipulations with 0.9999... you should first think about what that thing even is
tall topaz
What qylo is saying is, what does it mean when you refer to 0.999...? That expression/symbol doesn't mean anything until you've defined what it means.
mortal trellis
and then the next question is whether it indeed does exist
and if you prove that it does, then you usually already proved along the way that it equals 1
jaunty mulch
lone heartBOT
pseudo ice
Also take note they say “the value of x” - at the end you should only have one…
bright scarab
logbase11 (2x-1) + logbase11(x+4) = logbase11(11)
2x-1 * x+4 = 11
2x^2 + 8x -x -4 = 11
solve this quadratic baby
jaunty mulch
haha thx
jaunty mulch
2x-1 * x+4 = 11
wait i forgot log laws
is the * to the power of?
or multiply
acc i can ask gpt for that
ah yh its multiply
how tf did i forget that
xd
bright scarab
u can always derive it
theres no log laws
from example if u know exponent laws
a^x * a^y = a^(x+y)
log is simply the reverse, so instead of the bases and powers u work on exponents and powers
logbasea(x) + logbasea(y) = logbasea(x*y)
bright scarab
ur welcome bro
jaunty mulch
Where did I go wrong here? The question is what o find possible values of m given that the y equation touches the other equation
bright scarab
yo i couldnt understand ur handwriting dawg
jaunty mulch
Oof
bright scarab
any way u got the pic of the question
jaunty mulch
The line with equation mx – y – 2 = 0 touches the circle with equation x
2 + 6x +y
2
– 8y = 4.
Find the two possible values of m, giving your answersin exact form.
ah fk
hold on
i didnt bother to complete the square
bright scarab
thats nice
bright scarab
yeah so for tangency the condition is D = 0
for equation x^2 + 6x + y^2 -8y =4
put y = mx - 2
then solve for x
after that put D = 0 for quadratic and thats all
jaunty mulch
ah ok thx again
bright scarab
D = 0 for tangents
D > 0 for non intersecting
D < 0 for secants
idk which one is appropriate try all of them
lol
it should specify touches circle at only one point and stuff
it could be tangent or secant
bright scarab
then yeah go for D = 0
lone heartBOT
@jaunty mulch Has your question been resolved?
tardy folio
how can i get better at math i lost all of it im 60 years old
lament pagoda
The biggest red flag is the fact that their profile and that they joined the server today 💀
gray isle
back to basics, speedrun the khan courses to identify knowledge gaps
lament pagoda
Basically just like a response but without saying anything
That’s what I use it for ig
lone heartBOT
@tardy folio Has your question been resolved?
tribal sluice
i have two piles of pens, that contain two types of pens each (cheap and expensive)
pile number 1: 200 pens in total, the pile contains 400$ worth of expensive pens, and 240$ worth of cheap pens.
pile number 2: i paid a total of 600$, the pile contains 80 expensive pens, and 100 cheap pens.
how much does one expensive pen costs, and how do i solve it?
lone heartBOT
i have two piles of pens, that contain two types of pens each (cheap and expensi...
@tribal sluice Has your question been resolved?
turbid salmon
I don't know if this is the fastest way to solve it, but it's the first one I can think of.
Do you have an idea for how to put this information into a form that you can work with? (Namely ||a system of linear equations||)
tribal sluice
unfortunately no
im so stuck that i cant take even one step twords the answer
solemn juniper
The hint is a good one
There are quite a few equations you can construct with the given info
turbid salmon
As an example, in the first pile we have 200 pens in total, X of the expensive ones, and Y of the cheap ones, that gives us 200=X+Y, there's two more equations that can be constructed from the first pile. Any ideas on what they could be?
tribal sluice
200-y=x
200-x=y
?
inner schooner
I don't know if this is the fastest way to solve it, but it's the first one I ca...
i dont think they will be linear
raw stream
just let price of expensive pen be x and of cheap be y
you will get two equations
@tribal sluice
tribal sluice
i think they are
600=80x+100y
200=400/x+240/y
Yes!
tribal sluice
and now simple two equations
raw stream
Yeah, just solve them now..
raw stream
i dont think they will be linear
Why?
raw stream
@inner schooner
turbid salmon
You can rewrite it as that, by simply defining new variables.
Though I admittedly kinda didn't think of just taking the reciprocal of the price to ignore the amount
raw stream
you got the soln?
tribal sluice
one moment
raw stream
one moment
it should not take that long
tribal sluice
im doing it with explanations on the way, so i cant get to it later and read it again
tribal sluice
iv got two possible answers
x=5 y=2
x=3 y=18/5
yeah
nice
@tribal sluice
type .close
if you are done
and have no doubts
tribal sluice
.close
clear tulip
topic: Antidifferentiation and U-Substitution
problem: why do I need to substitute -u with v, can i not operate on e raised to -u?
clear tulip
.close
cinder breach
.close
lone heartBOT
Available help channel!
slim chasm
Please can someone explain the first question to me because I joined physics late and am unsure how to do it, thanks!
lone heartBOT
@slim chasm Has your question been resolved?
neon shuttle
Please can someone explain the first question to me because I joined physics lat...
what do you not understand?
infact what is to be found in the question?
slim chasm
infact what is to be found in the question?
Oh so sorry
Didn’t realise the question was cut out
This equation doesn’t make sense to me @neon shuttle
I don’t understand how dividing by the distance can somehow give the support force of the other pillar
lone heartBOT
@slim chasm Has your question been resolved?
outer swallow
supposed to prove this
lone heartBOT
supposed to prove this
outer swallow
unsure why this solution works because the equality they arrive at also holds for y>x, b>a
remote heron
hint: consider distributing (a-b)(x-y)
oh, you mean
you see where it comes from
but not why it holds
outer swallow
im unsure why (a-b)(x-y)>=0 implies a>=b and x>=y
remote heron
im unsure why (a-b)(x-y)>=0 implies a>=b and x>=y
the second part is given
its part of your assumption
we assume that $a \geq b$ and $x \geq y$
ocean sealBOT
jan Niku
remote heron
and want to show that $ax+by \geq ay + bx$
ocean sealBOT
jan Niku
outer swallow
oh so the solution starts from the right
remote heron
yea
its up to you to convince yourself or the reader that the manipulation doesnt mess anything up
remote heron
**jan Niku**
then you can just cite the assumption in the last step to show its all positive
or at least nonnegative
misty salmon
Chips are placed one after another on the initially empty 8 × 8 board . The chip
can only be placed in a free cell that borders on the side of at least two free cells. What is the largest number of
chips we can put on the board according to these rules?
misty salmon
Can anyone help
I found 53
But i'm not sure of my answer and I didn't prove it yet
Is anyone available here !?
rustic jungle
does diagonal cells count?
rustic jungle
this smells pigeon principle
misty salmon
I don't feel so cs we can have common free adjacent cells for the same cell
I mean I tried and it didn't work
If u have a good idea using php I would be glad if u can help
lone heartBOT
@misty salmon Has your question been resolved?
lone heartBOT
@misty salmon Has your question been resolved?
lone heartBOT
@misty salmon Has your question been resolved?
misty salmon
.close
old pendant
!help
lone heartBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
tacit arch
.close
lone heartBOT
lone heartBOT
Available help channel!
solar pebble
lone heartBOT
solar pebble
why is this true?
getting from the block inequality to the inline inequality
(more context)
lone heartBOT
@solar pebble Has your question been resolved?
deep kiln
Hello
lone heartBOT
@deep kiln Has your question been resolved?
dense hornet
Is this true?
Which part are you asking about being true
deep kiln
like everything xD especially the part where i remove the | |
deep kiln
i dont know thats the problem
dense hornet
Look at the function
deep kiln
the function itself can be negativ
dense hornet
Notice they replace 1/x with 1/2
deep kiln
i replaced 1/x with 1/2
dense hornet
Since x <= -2
But see, x<=-2, so -1/2 <= 1/x
I think your inequality there might be the wrong direction
deep kiln
since x€(-oo,-2] |1/x|<= 1/2 for all x in the intervall
dense hornet
since x€(-oo,-2] |1/x|<= 1/2 for all x in the intervall
a < b means 1/b < 1/a
deep kiln
whatever x you pick 1/2 will be larger
|1/x| <= 1/2 for every x u can pick in the intervall or not?
if not name one
after that since u got x^2 everything u do is positive so u can remove the ||
also since 0 isnt in the intervall i can do x=x^2/x
but is this true tho in the internet they say its only true that its uniformly converging in [a,oo), a>0
dense hornet
|1/x| <= 1/2 for every x u can pick in the intervall or not?
Yes
dense hornet
but is this true tho in the internet they say its only true that its uniformly c...
Yeah because you’ve made a small little error
You’re taking the supremum yeah
what’s e^x as x->infinity
dense hornet
Nope since ||f_n|| is already infinite
deep kiln
i never take the sup
dense hornet
good point but i remove the sup completly
also, the e^nx^2 you have in the L’Hopital is wrong
You changed the negative to a positive
If you did L’Hopital you’d get 1/0 basically
deep kiln
no no lim n and sup x
dense hornet
It’s inside a supremum
dense hornet
no no lim n and sup x
Whatever variables, you can’t do it
deep kiln
it matters
dense hornet
Having that sup in there makes it so it doesn’t work
deep kiln
u cant take lim if inf isnt even inside the intervall
dense hornet
You can’t apply L’Hopital “through” it
deep kiln
You can’t apply L’Hopital “through” it
but why?
dense hornet
As you’ve seen
The sup is always infinity
But you magically get a finite thing at the end
It doesn’t make sense
You can’t do it
(The actual function you’d have to apply it to is [sup_x f_n], which isn’t a quotient anyway)
Also, you’re doing L’Hopital by differentiating wrt x
But the limit is in n?
That’s wrong
lone heartBOT
barren bolt
last ether
r
stable minnow
s
lone heartBOT
@alpine sable Has your question been resolved?
sand ridge
lone heartBOT
sand ridge
unsure how to start
noble notch
the left hand side looks similar to the rhs
i.e same operator etc
so consider what exact value for sin
would give you root 3
and try get to the rhs
serene oracle
i need some help pls
Edgar, an NFL running back, rushed for an average of 154.4 yards per game this season, which is 30% higher than his average was last season. What was his average then?
Round your answer to the nearest tenth.
noble notch
Edgar, an NFL running back, rushed for an average of 154.4 yards per game this s...
open. new help channel
sand ridge
and try get to the rhs
unable to get it that way tho
noble notch
what did you get stuck on
sand ridge
we have not been thaught any value of sin which is root 3
noble notch
yeah ok
so which one of those values
when plugged into sin or cos
are going to give you a root 3
or result in some answer with a root 3
sand ridge
60 and 30 give root 3/2
noble notch
yes exactly
so if you are looking for your lhs to be the same as your rhs
which value do you want to plug in
i.e do you want the root3 to be from the sin or cos
if that makes sense
dull wraith
For games played until a winner, is the chance of a person winning always 1/players if chances are equal? (Bingo, for example)
placid zinc
dull wraith
I know
lament forge
the probabilities have to add up to 1 (assuming that exactly one player will win), and there are players of them, and they're all equal, so they must all be 1/players
dull wraith
Spent a while trying to calculate the chances of winning at bingo
lament forge
the probabilities have to add up to 1 (assuming that exactly one player will win...
but yeah once there's any asymmetry this argument doesn't work
so the probability that you win given that you have some nonzero advantage is going to be complicated
wanton folio
hello
alpine sable
do you know linear algebra
It seems you're not used to this server. You should just post the question without probing whether there's someone who might be qualified enough to answer it
I'll know whether I can answer it once I see it
wanton folio
Okay
We got vector spaces V. Let's look at the simple case V=Rn over the field R
And even simpler only work with n=2 or n=3
We can represent these "spatial vectors" as arrow in Rn with a magnitude direction and no fixed position right?
But take for example $t\cdot\vec{(1,2)}$ with $t\in\mathbb R$.
lament forge
(that is not an equation)
ocean sealBOT
epic_morphism
wanton folio
Now everyone would say ah it's simple that's a line through the origin and the point (1,2). But since vectors are not fixed in space can't we also interpret the above as another line not going through (1,2)? Obviously 0 must lie on it since t=0 is a possibility but I imagine two seperate lines and then a discontiuinity at zero.
lament forge
Now everyone would say ah it's simple that's a line through the origin and the point (1,2)
...i wouldn't say that
that's a vector, and which vector depends on what t is
if t = 0, then it's the trivial arrow of length 0 that starts and ends at the same point, there's no particular reason why that would have to be the point (0,0) if we're saying that vectors don't have a fixed position
if t = 1, then it's just the vector (1,2), the operation of scaling the vector by 1 has not changed anything
if t = 5, then it's the vector (5,10), a vector with the same direction but five times the magnitude, the fact that it's scaled ...does not affect the fact that we can declare its start position to be wherever we want
if t = -1, then it's the vector (-1,-2), scaled by a factor of -1 and therefore now pointing the other direction
etc.
wanton folio
I should have made it clearer, $t$ runs through all real values. So I'm talking about the set ${t\cdot\vec{(1,2)} \vert t\in\mathbb R}$
ocean sealBOT
epic_morphism
lament forge
ah ok, so that's a set of vectors
...well you haven't given any particular spatial interpretation for a set of vectors
it would make a lot of sense to declare that all of them have the same starting point, although you could make the starting point (1823,7) if you want to and then it doesn't go through (1,2), there's no particular reason to start it at (0,0)
or you could have them all start at completely random locations
wanton folio
Exactly that's my point - why does this set in all of linear algebra represent a simple line through the origin - because it could also represent any set of vectors completely displaced from one another with all a different starting point? And if you fixed one for all of them - it doesn't even have to be the starting point (0,0).
lament forge
...well the fact that you can do this doesn't mean that you have to if it's... not useful
in practice it's more useful to consider the set {t * (1,2) | t \in R} as a line through the origin instead of a completely random mess
wanton folio
That I agree with HOWEVER
lament forge
you can think of a vector as not having any particular starting point, but you can also think of it as a point, or as a line but starting at (0,0)
wanton folio
The motivation for my question stems from the definition of the span. One says the span of (for example) the vector (1,2) IS EXACTLY EQUAL to a line through the origin and through the point (1,2). It explicitly doesn't give other possibilities.
sour verge
One other thing to note is that linear algebra is mostly interested in vector spaces, their properties, and their subspaces. The set where the line does not go through the origin is by definition not a subspace of Rn, so in a way it's less interesting in relation to Rn as well..
wanton folio
I agree that the idea is always more useful for a simple line through the origin however this has never been stated explicityl anywhere and that's my problem
lament forge
The motivation for my question stems from the definition of the span. One says t...
ok well i think that's... reasonable actually
like, the type of thing that the span of a vector is isn't "a set of vectors and also an interpretation of what that means geometrically", it's just a set of vectors
defining the span of (1,2) as "the line through the origin and the point (1,2) (we are interpreting vectors as points)" is the same thing as defining the span of (1,2) as "{t * (1,2) | t \in R}, and by the way we interpret this set as being a chaotic mess where all of the vectors start in random places", because those are the same set of vectors
wanton folio
One other thing to note is that linear algebra is mostly interested in vector sp...
(I think you haven't read my message: I'm thinking of constructing the "possible span" as two disjoint lines and a disjoint origin.)
sour verge
It still wouldn't be a subspace though.
wanton folio
It still wouldn't be a subspace though.
Why is that? It includes the origin and should also be closed under addition and sclaling
lament forge
defining the span of (1,2) as "the line through the origin and the point (1,2) (...
vectors don't actually have a "start point"
there aren't multiple (1,2)s that aren't equal to each other because they start in different places
so it doesn't matter
wanton folio
defining the span of (1,2) as "the line through the origin and the point (1,2) (...
But WE ARE defining the span of one vector as a set of vectors giving a line through the origin. (Or of two linear non dependent vectors even as R2$
lament forge
no we aren't
the definition of span does not include a line that says that if you interpret this set as a chaotic mess instead of a line through the origin then a set of vectors will arrive at your house and kidnap you
if it does then that's the wrong definition and you should find a better source to learn linear algebra from
wanton folio
the definition of span does not include a line that says that if you interpret t...
I've seen the statement "the span of a single vector in Rn over R is a line through the origin" MULTIPLE times in DIFFERENT sources
lament forge
"vectors are points" is a perspective, it's the same kind of thing as what language the text is written in
if i define a bee as
a stinging winged insect which collects nectar and pollen, produces wax and honey, and lives in large communities.
then i haven't "defined" a bee as being an entity that inherently "exists in English", i've just defined it in English, and it would be possible to give an equivalent description describing an identical object in French
lament forge
I've seen the statement "the span of a single vector in Rn over R is a line thro...
did any of those sources then say that if you interpret it as not being a line through the origin you will be kidnapped by vectors
did they all literally explicitly say that
wanton folio
I understand what you're trying to say but let me give you an analogy where I'm coming from: Textbook says the solution to x^2=1 is x=1 (and doesn't expand at all). That's the same thing now because in a sense it's true as 1^2=1 but in a sense also false as the solution to that equation is x in {-1,1} and you can't just omit the -1 in my opinion and not explain it at least a little bit.
lament forge
I've seen the statement "the span of a single vector in Rn over R is a line thro...
the span is a line through the origin. it is also a chaotic mess of vectors that all start in different places. it's the same object whichever angle you look at it from, and describing it from one angle is not an assertion that the other angles do not exist
lament forge
I understand what you're trying to say but let me give you an analogy where I'm ...
...for the analogy to really work here you would also need 1 and -1 to actually be just two different ways of looking at the same number
wanton folio
So basically the problem is the english word "is" becuase in English (and all other languages too) we do not differentiate between "=" and "subset of". For example a dog is a dog (=). But we also say a dog is an animal (subset and not "=*).
lament forge
the span *is* a line through the origin. it is *also* a chaotic mess of vectors ...
this happens everywhere in maths. the group of actions on the rubik's cube is a collection of states a rubik's cube can be in. it is also a set of sequences of moves you can make, modulo a certain equivalence relation. it is also the set of natural numbers below 43,252,003,274,489,856,000 with just an enormous table defining composition. these are all the same object, there is no one correct way to view it
lament forge
So basically the problem is the english word "is" becuase in English (and all ot...
...i don't think that's it?
it's more that english doesn't distinguish... i guess, abstractions
there's a difference between an abstract object, like a set of vectors, and any one particular interpretation, like "some kind of geometric thing" or "it's a rule that takes a pair of real numbers and produces either 0 or 1" or whatever
or... hm
...or tbh i think it mostly isn't a linguistic issue? i think it's just that you're confused about abstractions
wanton folio
No, don't worry that's not the problem. I'm thinking very rigorously about it right now.
I want to ask you a question
Would you consider the following statement true or false
${t\cdot\vec{(1,2)} \vert t\in\mathbb R}={(x,2x) \vert x\in \mathbb R}$
lament forge
...i don't know what the set on the right means
wanton folio
stop wait, sorry
ocean sealBOT
epic_morphism
lament forge
ah ok
well
the set on the left is a set of vectors, the set on the right is a set of pairs of points
so... it depends
is $(1,2) = \vec{(1,2)}$?
ocean sealBOT
bee [it/its]
wanton folio
mmh i see
lament forge
if i saw this expression without context i would probably guess that, if it's intended to be meaningful, the thing on the left and the thing on the right would be sets of the same type, so the "=" would be implicitly an identification between "2D vector over R" and "pair of real numbers", and there's one obvious way to do that with which the statement is true
but like, you could declare that when you interpret a vector as a pair of real numbers, $\vec{(1,2)} = (5,6)$
ocean sealBOT
bee [it/its]
wanton folio
sorry i have an apple in my hand cant type too quickly atm
I see. my original confusion was that I very often read that the span or in general linear combinations are lines (in the sense that they are that) and not that they can be represented as lines.
It's like saying vectors ARE arrows but thats not true they can be represented by arrows.
lament forge
...yeah i guess that's kind of just
saying that something "is" something else in maths is never really a statement about what something, like, "actually is", because that's not a thing
it's not like the physical world where you can look at what something is made of and it turns out to be a big pile of atoms
in maths, what something is made of is a big pile of logic, and any two things that are equivalent piles of logic, even if they're seemingly defined differently, end up being entirely indistinguishable
so saying that something "is a line" isn't incompatible with it also being some other representation, because there isn't really a distinction between a representation of an object and "the actual object"
...well there is in the sense that, even if there are ways to represent a real number as a set of natural numbers, you can't just walk up to a random mathematician and ask them if $5 \in \pi$ and expect an answer without further clarification
ocean sealBOT
bee [it/its]
lament forge
but it is true that you can define + and * operations on sets of natural numbers, and end up with something that you can actually do algebra on the same way as real numbers, and at that point calling that "the real numbers" wouldn't really be incorrect in any meaningful sense
wanton folio
Bro thank you very much, at the end I kind of lost it because I didn't understand quite all of the yapping, but thanks a lot for your patience. It's 4am for me in Switzerland now so I wish you a very good night and beginning of the year and I'll go to sleep. Thanks again very much!
lament forge
it's 3am here, i woke up 7 hours ago, my sleep schedule is a bit of a mess
goodnight
...and yeah not understanding all of that is fair
lone heartBOT
@wanton folio Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
whole otter
Am I blind
lone heartBOT
Am I blind
whole otter
The second one
I’m not sure if inverse tangent is arcsin
Def not but I mean
Collegeboard is probably not gonna make such an error
Have I been
hushed locust
typo
whole otter
lied to my whole life?
hushed locust
should be arctan(x)
whole otter
Yes that’s what I thought
somebody has been using arcsin a bit too much
😂😂😂
Alright ty
.close
lone heartBOT
lone heartBOT
Available help channel!
uncut violet
i
uncut violet
A certain binomial distribution for the random variable X consists of 20 independent trials, each with probability of success p = 0.3. Calculate the normal approximation by determining P (8.5 ≤ X ≤ 11.5), treating X as approximately N(μ,σ2), and using the normal tables.
Please help desperately
or i shall commit suicide
to solve d or not to solve d that is the question
barren portal
standardize the variable
uncut violet
i di d that and got
1.21<z<2.68
i calculated the mean to be 6
and the SD = sqrt 4.2
barren portal
mean= np, and variance= sqrt(npq) right?
barren portal
yes, mb
barren portal
what are the new bounds for x?or z
uncut violet
Thank you
barren portal
I got the standard deviation= 2.04
uncut violet
yeah
barren portal
The calculations are same
barren portal
I mean, stuff before checking table
the table given is cumulative, so you have subtract -infy to 1.21
barren portal
-infinity
barren portal
Required probability is P(-infy<x<2.68)-P(-infy<x<1.21)
this about this graphically, you can see why
uncut violet
yeah that is the area between the 1.21 and 2.68 = z
uncut violet
how do you remove a tail that goes to infinity though
barren portal
the probabilities gives includes the tail
barren portal
https://imgv2-1-f.scribdassets.com/img/document/372870550/original/1b894e2e27/15...
check the picture here
barren portal
Nahh wait
Required probability is P(-infy<x<2.68)-P(-infy<x<1.21)
P(-infy<x<2.68)
This is just for this bit
uncut violet
mhm
uncut violet
yeah
barren portal
so table value of 2.68- table value of 1.21
uncut violet
isnt p( negative infinity) = 0 tho?
barren portal
0.9963-0.8665
barren portal
isnt p( negative infinity) = 0 tho?
we are talking about probablities in an interval
Required probability is $P(-\infty<z<2.68)-P(-\infty<z<1.21)$
ocean sealBOT
.doc
uncut violet
yeah
barren portal
it’s taking the difference between, area from -infy to 2.68 and area from -infy to 1.21
uncut violet
0.9963-0.8665
gives 0.1298
uncut violet
the answer sasy 0.1075
uncut violet
yeah
uncut violet
0.02 error lol
ok we will accept that
barren portal
what we did so far looks correct
barren portal
Good luck
pulsar lynx
hello
coarse marlin
How to continue this
wut is the objective?
like, wut do you want to do with this expression?
feral hemlock
Multiply left side by (m+12n) and right side by (m-12n) isn't it
coarse marlin
Multiply left side by (m+12n) and right side by (m-12n) isn't it
quite the opposite actually 😅
feral hemlock
quite the opposite actually 😅
No way
coarse marlin
yep. thats da wae
pulsar lynx
(m-12)^2 multiply right side
feral hemlock
(m-12)^2 multiply right side
No
We already have (m+12)...
So we need (m+12n) one time
pulsar lynx
you don't multiply same denominator it won't make one denominator
coarse marlin
you multiply m-12n to num and den for the 1st term adn
multiply m+12n to num and den for the 2nd term
you'll end up with the same den afterwards
coarse marlin
No way that I'm wrong
oh wait. yeah. i misread the den of 1st term as m+12n lol.
youre right. absolutely 🫡
coarse marlin
brother. where did you get this from?
pulsar lynx
I typed this, In youtube video they say , to add expression with different denominator we have multiply both side with opposite denominator
coarse marlin
I typed this, In youtube video they say , to add expression with different denom...
this is the step you do after you've written the two fractions as one, with the denominator as the product of the former 2 denominators
you cant do this directly to the prior 2 fractions
pulsar lynx
you cant do this directly to the prior 2 fractions
yes the multiplication is only for one fraction
bright scarab
bruh easy
pulsar lynx
no easy for me
bright scarab
keep (m+12n)(m-12n) as A, 5m/(m-12)^2 as B, 2n/(m+12n)(m-12n) as C, (m-12)^2 as D
dont worry u will get it
hold up
u dont even have to do that
omg i misread the question
wait its still okayish
pulsar lynx
Is this correct I mean can I multiply like this and make same denominator
pulsar lynx
bright scarab
i see
omg
someone complicated a simple question
wait u dont need to always cross multiply
in experience it sucks
instead be more creative
multiply and divide the left one with (m+12)
multiply and divide the right one with (m-12)
and there u go same denom
pulsar lynx
someone complicated a simple question
you mean me?
bright scarab
5m^2 + 60m + 2nm -24n/ (m+12)(m-12)^2
bright scarab
you mean me?
idk?
my statement is never meant to target a particular person
its written cleverly ambiguos
bright scarab
5m^2 + 60m + 2nm -24n/ (m+12)(m-12)^2
try to simplify this
bright scarab
do I have to multiply and divide?
yeah basically wat u are doing is multiplying by 1, so as u can tell by identity property multiplying or dividing by 1 doesnt change value
instead of 1 u are multiplying and dividing by the number of ur choice
as it yields 1
pulsar lynx
I didn't get it
How would I know ,by what I have to multiply left and right in different question?
@bright scarab
<@&286206848099549185>
coarse marlin
How would I know ,by what I have to multiply left and right in different questio...
you multiply anything that gives you the lcm of the 2 denominators
take your question for example
denominator of 1st term is (m-12n)^2
den of 2nd term is (m-12n)(m+12n)
their lcm is (m+12n)(m-12n)^2
so, you multiply 1st term with m+12n
and 2nd with m-12n
so that they both have the same denomiator, which is their lcm
pulsar lynx
But when we multiply (m+12n) (m-12n) by (m-12n) it became (m+12n)(m-12n)^2 how ? It should also be multiplied with (m+12n) isn't it?
coarse marlin
no, why wou;ld you do that?
pulsar lynx
Is n't it like this
coarse marlin
correct
pulsar lynx
So here we are multiplying numerator and denominator by (m-12) it should multiply both (m+12) and (m-12)
in right side
coarse marlin
why so?
coarse marlin
it would be (m+12)(m-12)^2
lone heartBOT
@pulsar lynx Has your question been resolved?
pulsar lynx
it would be (m+12)(m-12)^2
How it should also multiply with m+12 and give (m^2-12m+12m-144) -> (m^2-144)-> (m+12) (m-12) and (m-12)^2
pulsar lynx
.close
broken spruce
Hi, Math Experts! I have a question to ask about general assumptions when reading more abstract math texts.
I just started dipping my toes into abstract math, and came across this statement on Wikipedia: "In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist"
https://en.wikipedia.org/wiki/Ring_(mathematics)
The book 'An Introduction to Mathematical Cryptography (second edition)' also says on Page 95 that 'elements [of a ring] are not required to have multiplicative inverses' (the italicized text is my own).
Since a ring is a set with some well-known properties, and since multiplicative inverses exist for numbers like 5 and 10, I was wondering if it was taken for granted in Mathematics that when the text says 'X does not exist for Set R', it also means that X must be taken from within Set R, and we don't consider cases where it exists in other sets?
🙂
mortal trellis
5 does not have a multiplicative inverse inside Z
golden canyon
The definition of a multiplicative inverse states that the inverse has to be in the set, otherwise multiplication isn't even defined
broken spruce
The definition of a multiplicative inverse states that the inverse has to be in ...
Ohh... I was not aware of that
mortal trellis
it has one inside Q or R but that doesnt matter in this case
broken spruce
I thought it was a broader assumption in math that if we were referring to any law/rule concerning a set, we were referring exclusively to elements of that set.
mortal trellis
its relevant for example that you cant solve 5x=4 in Z, precisely because 5 doesn't have an inverse in Z
broken spruce
The definition of a multiplicative inverse states that the inverse has to be in ...
That's an important distinction, so thanks!
broken spruce
The definition of a multiplicative inverse states that the inverse has to be in ...
BTW, where did you get the definition from? What textbook is good for that?
Just want to know before I close, so I can go do my homework properly rather than going from link to link and compiling various informal definitions
golden canyon
I think the standard one is that for all x in S there should exist a y in S such that xy = 1
1 being the multiplicative identity
mortal trellis
the wikipedia article on fields for example has it
any text on abstract algebra has it
(hopefully, otherwise its a bad text)
golden canyon
Except 0 actually (additive identity)
broken spruce
I think the standard one is that for all x in S there should exist a y in S such...
Yes, that would be what it meant for Set S to have an inverse law
broken spruce
the wikipedia article on fields for example has it
That's the thing, I was reading about rings in a textbook
mortal trellis
yeah well its not a relevant axiom for rings
you can also phrase it that the nonzero elements should form a group
and then its in the group axioms
broken spruce
I think the standard one is that for all x in S there should exist a y in S such...
The same is said of the additive inverse law in 'An Introduction to Mathematical Cryptography', but I wasn't sure whether it was a rule of dealing with sets, or if it was a general restriction applied to all instances of inverse laws (that they are only valid if they only involve members of the set they hold true for).
mortal trellis
for specific rings (integral domains), there always exists a larger set (called its field of fractions) in which every element has a multiplicative inverse. in the case of Z that would be Q. so in that sense the axiom would be pretty useless if it didnt require the inverse to again be in the ring itself
its not some sort of general implicit rule that it should always be within the same set. thats just part of the axiom
broken spruce
for specific rings (integral domains), there always exists a larger set (called ...
Ahh, absolutely makes sense.
broken spruce
its not some sort of general implicit rule that it should always be within the s...
So it's not a general implicit rule, but it's implicitly part of that specific axiom?
broken spruce
for specific rings (integral domains), there always exists a larger set (called ...
What you said absolutely, absolutely makes sense.
mortal trellis
not implicitly
very explicitly
the wikipedia article just didnt want to cite an axiom in the first paragraph
broken spruce
the wikipedia article just didnt want to cite an axiom in the first paragraph
I'd better get a good book on Abstract algebra then. Would you recommend any? What I have are books about cryptography, communication theory, and I have 1 more book on number theory on the way, but I don't think they cover Abstract algebra in that much depth.
broken spruce
there are probably some books in <#873809533615628308>
All right, thanks, it popped up in the first result. 🙂
Have a nice day!
.close
zinc turret
