So 9*12?
#help-19
faint stump
indigo dirge
yes
indigo dirge
mk, now that u know the area formula, find the area in another way
faint stump
Uhm okay
indigo dirge
use the hypotenuse this time
faint stump
Idk what to do
vivid wyvern
Do u know the other formula for finding area of a triangle using base and height
faint stump
Wdym?
vivid wyvern
Can u identify the height and base of the triangle ABC
faint stump
15 n 12?
vivid wyvern
Which is which
faint stump
15 base 12 height
faint stump
hm
vivid wyvern
But height isnt 12
faint stump
9?
vivid wyvern
Nah that's the area
Sorry it isnt
vivid wyvern
U can consider base and height in two ways here
Anyway
U already used the perpendicular and base for first one
So u should use the hypotenuse as base and the height on hypotenuse as height
vivid wyvern
So which one is ur height
faint stump
Idk that where I’m confused
vivid wyvern
Alr height is the perpendicular on ur base
faint stump
Yes
faint stump
12?
vivid wyvern
Nope
faint stump
Alr
vivid wyvern
Cuz it is perpendicular on base BC
faint stump
YES
vivid wyvern
So can u use the formula (1/2)bh here
faint stump
1/215ad
vivid wyvern
Careful with the bracs
Otherwise yea
faint stump
👍
vivid wyvern
And u found the area of the triangle before right?
faint stump
7.5ad?
faint stump
now do I set equal?
vivid wyvern
now do I set equal?
Exactly
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mystic saffron
looking at ii), and the answer key, my question is, if I have 2 linear transformations T(x) and I(x), will (T+I)x be the same as T(x) + I(x)?
pastel orbit
looking at ii), and the answer key, my question is, if I have 2 linear transform...
yes
mystic saffron
why is this the case
pastel orbit
> why is this the case
addition is defined pointwise for function vector spaces 
that's really all it is
pastel orbit
mystic saffron
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neat scaffold
Hey, how did they get the values for $P(A \cap ( B \cup C))$?
neat scaffold
the .04+.05+.03
vivid wyvern
Do u know how union and intersection works
odd edgeBOT
@neat scaffold Has your question been resolved?
neat scaffold
Do u know how union and intersection works
yes
vivid wyvern
So can u mark BUC
neat scaffold
vivid wyvern
Yes
vivid wyvern
How about A intersection (BUC)
neat scaffold
vivid wyvern
And that part contains the following 0.4,0.3 and 0.5
neat scaffold
is there another way to solve this without the ven diagram?
vivid wyvern
is there another way to solve this without the ven diagram?
Yeah lengthier way
neat scaffold
yeah, I was trying to use properties of set but that didn't get me anywhere
vivid wyvern
Using P(MUN)=P(M)+P(N)-P(M (Intersection) N)
Wait i m missing
Stuffs
Lemme edit
neat scaffold
yeah inclusion-exclusion?
vivid wyvern
yeah inclusion-exclusion?
Oh what does that mean
Sorry cuz i m not well familiarized with terms
Alr now that i think abt it idk which formula to use to crack the intersection
neat scaffold
this is what I was thinking
but yeah idk about intersection
anyway, appreciate the help
.close
odd edgeBOT
vivid wyvern
this is what I was thinking
Oh yea ik that one but well it doesnt seem to be of help
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sage helm
While reading the following passage in book, I tried to practice the rewrite myself.
sage helm
While trying it out, I got to a step where I'm not sure how to proceed. Namely, how to decide on which side of the parenthesis to place the common factor. Since, in this case, this is a matrix-vector multiplication, the order matters. Placing the vector on the right side of the matrix is the ideal position, but what is that choice based on? See the following image.
At the second line, I can either go with the left or the right version; but, what criteria do I base that choice on?
You're saying, if you place it on the left just transpose the vector?
safe vapor
left one.
I mean it's gonna be the same result in a way but convention is the left one
sharp oak
e is on the right of A. That doesn't change even after factoring.
safe vapor
If you try the one on the right, the matrix wont be compatible with the vector (ie. row vs column vector). Or if it is, then you're using the equivilant row vector, which is still fine
Like let M be nxn, then e mist be nx1 (ie. a column vector), in which case you go with the one on the left
sage helm
e is on the right of A. That doesn't change even after factoring.
I like this answer. It makes sense to me.
Ok, I see it, now. In both factors, it was on the right of the matrices, and so it remains there when factored out.
Thanks all
🙂
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errant plank
is there anyway to simplify this further? I can't really think of any
narrow crypt
!show
odd edgeBOT
Show your work, and if possible, explain where you are stuck.
vernal yacht
is there anyway to simplify this further? I can't really think of any
It looks extremely dubious, check again
narrow crypt
is there anyway to simplify this further? I can't really think of any
U can combine fractions
If I’m right, u can factor 1+cosx out
And things will cancel
errant plank
what about it?
unkempt lichen
how did this turn into 1+cos sin underneath
errant plank
how did this turn into 1+cos sin underneath
I'm not sure what you mean
unkempt lichen
i know you meant to multiply the second fraction by sin(u) to equalize denominators
errant plank
oh
vernal yacht
write it like this ?
It looks much clearer now
narrow crypt
I still don't see how I can simplify it further
How so
Have u combined the fractions
errant plank
balmy nimbus
Hey
balmy nimbus
I’m new on here
vernal yacht
I’m new on here
Let’s move to #discussion
balmy nimbus
Ok
narrow crypt
U combined the fractions wrong
narrow crypt
errant plank
errant plank
so far so good?
narrow crypt
No
errant plank
whats wrong
narrow crypt
What is (1+cosx)(1+cosx)
errant plank
ummm
errant plank
errant plank
is it not this?
narrow crypt
What is (1+cosx)(1+cosx)
narrow crypt
No
errant plank
is it 2(1+cos)
(edited)
clever fjordBOT
k
errant plank
u^2
narrow crypt
Good
So (1+cosx)(1+cosx) = (1+cosx)^2
Right?
The squaring is done to the bracket
Not just cosx
errant plank
ohhhh
wait thats what I wrote the first time though I just coppied it wrong by accident
narrow crypt
Okie
errant plank
narrow crypt
With this information, could u simplify it
narrow crypt
Yup
errant plank
which
=
which is fully simplified
nvm no its not
I still have to distribute
narrow crypt
No
narrow crypt
I still have to distribute
Yes
narrow crypt
no as in this isn't a true statment?
It’s not the correct simplification
narrow crypt
No
errant plank
we can cancel
narrow crypt
bruh
unkempt lichen
whoa whoa i don't think we can do that
aa+bb / ab can't be canceled as if they were factors
errant plank
is this right?
narrow crypt
No
odd edgeBOT
@errant plank Has your question been resolved?
narrow crypt
bruh
frosty coral
How to find the general solution ive shown my work
odd edgeBOT
How to find the general solution ive shown my work
frosty coral
consult the unit circle?
oh
could there be two solutions?
since
cos(x)=-1/2
could be
in
two different quadrants
strange aspen
could there be two solutions?
If it’s general there is infinite
frosty coral
what does general solution mean
oh
wait
I think I know what its talking about
is it talking about k?
strange aspen
is it talking about k?
Yeah the from θ+2πk
strange aspen
But first off try finding when
cosθ= -1/2 in (0,2π]
granite bridge
isn't there a shortcut for general solution? Just use that
Wait I'll fetch it for ya
frosty coral
wdym for (0,2pi)
strange aspen
wdym for (0,2pi)
The answer is in angles or radians ?
granite bridge
The answer is in angles or radians ?
Yes
strange aspen
wdym for (0,2pi)
From 0 to 2pi
strange aspen
Same idea I had in mind
strange aspen
Find where cosθ= -1/2 if theta is in 0 to 2pi then you can generalize the solutions
frosty coral
2pi/3
granite bridge
First the solution is in the fourth quadrant
strange aspen
First the solution is in the fourth quadrant
It’s cosθ= -1/2
granite bridge
fourth quadrant
Cos negative in fourth quadrant
frosty coral
thats
strange aspen
Cos negative in fourth quadrant
U sure ?
frosty coral
so is it 2pi/3
granite bridge
Pi minus pi/3
granite bridge
2pi/3 is solution 1
strange aspen
For another time try not memorizing at least understand why somethings are, use reference angles
frosty coral
i think it was 4pi/3
strange aspen
Ok know for the general form
frosty coral
so we have two solutions
strange aspen
Do you know the periodicity of cosine and sine is ?
frosty coral
what does periodicity mean
strange aspen
How the values reoccur
granite bridge
The amt it takes to repaet
strange aspen
It’s a circle
Where if you walk 2π distance you’ll still be on the same position you were in
frosty coral
oh
strange aspen
so for sine and cosine the periodicity is 2π every 2π you’ll do it’ll give you the same value
frosty coral
yeah
strange aspen
2π/3 +6π/3=8π/3
And for integer multiples of 2π
frosty coral
where did you get those numbers?
strange aspen
In other words
2π/3 +2πk
frosty coral
2π/3 +6π/3=8π/3
And for integer multiples of 2π
uh where did you get this
I understand where you got 2pi/3
but where did you get 6pi/3
strange aspen
but where did you get 6pi/3
That’s 2π
granite bridge
but where did you get 6pi/3
No it's 2pi
granite bridge
2pi is just 6pi/3?
No listen
2pi is the time it takes to become same
Sin0 is 0
What is sin2pi
Cos0 is 1
What is cos2pi?
Do u get it?
strange aspen
I’ll leave it to you @granite bridge a little busy atm
granite bridge
I’ll leave it to you <@732464419954622525> a little busy atm
No big deal, thank you
granite bridge
I mean in degrees
granite bridge
You can convert it to radians later
frosty coral
yeah I understand that
granite bridge
yeah I understand that
The thing is trig functions repeat values as you turn multiple times around that circle
frosty coral
I was confused why you guys changed 2pi to 6pi/3
is it to make the addition
easier
granite bridge
If I had my notebook id show u the theory but I don't
So I'll explain
Basically costgeta was -1/2. So you had to consider both general and particular solutions
First pi minus pi by 6 to get 2pi/6 cuz that's the first one
(I'll explain from the start, hope you don't mind)
frosty coral
yeah i dont mind
granite bridge
Then after that though, we have to account for the general solution
The general solution for cos theta
Is 2n times pi plus or minus your angle
Here it was pi/6
But this was if theta was positive, here its more like
pi+- pi/6
So you get it???
frosty coral
yeah
granite bridge
This shortcut I did works only if cos is positive, so u modify it to make it negative
You can try putting it in the formula 2n pi plus or minus pi/6
What do you see?
cos is........ 1/2
But from our original Q, it was -1/2, so it became pi plus or minus so as for cos to enter the second or three third quadrant where cos is neg
I hope I'm clear enough
Any questions, dear?
frosty coral
uhh i dont think so
granite bridge
uhh i dont think so
Glad to know🙏
frosty coral
In other words
2π/3 +2πk
this?
granite bridge
The domain of cos is zero to pi, and two by three times pi is in the domain, so it works
granite bridge
alr and dont we need to add something like
It means k is an integer
frosty coral
which is true i think
granite bridge
Think what the expression means
He means after every rotation, were adding 2pi/3
Which is fine. What I meant with my shortcut was that
After every half rotation, I'm adding or subtracting pi/3
Both are the same
It's just worded differently
frosty coral
so it has to be an integer
frosty coral
also u called me dear but im a dude and im in highschool too 😭
granite bridge
You're doing well for someone who's just been introduced to trig
frosty coral
yeah im doing alg 2 trig over the summer with no teachers
granite bridge
also u called me dear but im a dude and im in highschool too 😭
My teacher used to do it to me
granite bridge
You should try khan academy or the org chem tutor for this
frosty coral
so would the answer be 2pi/3+2pik , k E Z
granite bridge
so would the answer be 2pi/3+2pik , k E Z
Yes
granite bridge
That would be it. If the option has many angles in each, then put in done values of k and see which one satisfies
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mystic saffron
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mystic saffron
,rccw
clever fjordBOT
mystic saffron
How do i determine when i shud use cross or dot product
elder vault
In this case, since I am asked to find the angle, I will use dot product
mystic saffron
Well sin also gives angle isnt it?
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crystal charm
Well sin also gives angle isnt it?
that's for finding the angle between a line and a plane
odd edgeBOT
copper quarry
that's for finding the angle between a line and a plane
what is your doubt
do you need help sir
unborn dragon
,help
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manic grail
Is this first case of the proof correct ?
odd edgeBOT
Is this first case of the proof correct ?
low locust
what other case is there
amber veldt
what other case is there
infinite limit?
low locust
ok I guess
odd edgeBOT
@manic grail Has your question been resolved?
manic grail
what other case is there
Thats another question i had
If we say lim f(x) >0
Does it going to infinity count ?
amber veldt
Does it going to infinity count ?
id say depends on context
in calculus no, in analysis yes
amber veldt
Is the theorem true even when it goes to infinity
you tell me
amber veldt
Is writting oo> 0 a thing
exactly my point
manic grail
I dont understand what the course has to do with it
Isnt math supposed to be true in any course
amber veldt
I dont understand what the course has to do with it
convention, thats all
manic grail
Oh
amber veldt
yes i would
manic grail
Alr
amber veldt
know the definition?
i put them up on proofwiki years back so i know theyre there
infinite limits
manic grail
know the definition?
I know it but there is a problem
I am just using this book to prove all theorems on limits and continiouty with ε,δ
But i havent got to limits going to infinity in proofs
amber veldt
right that's why I'm aksing if you need a definition
and looks like someone renamed the proofwiki page
manic grail
So I don't have an idea of how i would say for every Μ there is δ>0 so that if 0< |x-xo| <δ then f(x)>Μ
manic grail
Ik the definition
Idk how proofs work for these type of limits
Like how would i prove |1/x| has limit infinity at 0
Thats the only issue but ig for this theorem might be easier wait lemme do it
manic grail
Wanted to think about it when i got to infinite limits
Anyways lemme do this for the theorem
And ill send my work to se if done correctly
amber veldt
do it for 1/x^2 then
there's also a theorem you can use
proof is finicky but not conceptually difficult
amber veldt
yup, thats all you need
amber veldt
you know the sequential characterization of limits? it's often easier
but they dont teach it until calc ii
but its not difficult
manic grail
you know the sequential characterization of limits? it's often easier
Probably
Can you show it idk names in English most of the time sorry
manic grail
you know the sequential characterization of limits? it's often easier
You mean the definition?
odd edgeBOT
amber veldt
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thick dove
hi
amber veldt
lim x->c f(x)=L if and only if f(x_n)->L for all sequences x_n->c
thick dove
i'm from brazil and i trying to study math in english
amber veldt
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manic grail
What does f(x_n)-> L mean
amber veldt
the limit of the sequence f(x_n) goes to l as n->infinity
thick dove
i'm a proggramer
manic grail
Real
tacit wasp
i'm a proggramer
!occupied
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Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
manic grail
I know it the other way around
warped badger
i'm a proggramer
Go to #help-13
manic grail
I think
manic grail
the limit of the sequence f(x_n) goes to l as n->infinity
Why would this be the case for all sequances
What if a sequence converges to 0
f(0) doesnt have to be L
tacit wasp
Huh?
amber veldt
Why would this be the case for all sequances
for all sequences xn to c
all sequences that go to c
it replaces "arbitrarily close" with "for sufficiently high terms in a sequence"
conceptually
manic grail
Oh
Isnt this just a substitution
You just sub in x_n to x
And find the new limit
amber veldt
what
manic grail
Its just a substitution to the starting limit
Let x= x_n
Then x_no = lim x-> c x = c
So its lim x_n -> c f(x_n)
@amber veldt
I have a theorem that says if f,g have a limit at xo and f(x) <= g(x) near xo then lim f(x) <= lim g(x) at xo
Does this include if the limits are infinite
Can one limit be bigger than another even if they both go to infinity
amber veldt
Can one limit be bigger than another even if they both go to infinity
no
there is such a thing as "some infinities are bigger than others" with you deal with cardinalities of sets
but not with limits like this
good question though
wait but
your question written and your written in words are different @manic grail
you can say +infinity <= +infty in analysis
manic grail
you can say +infinity <= +infty in analysis
What's the difference
Oh =
So theorem stands for infinite limits aswell
?
So that means that all limits going to infinity equal
sturdy cape
So... yes*
*usually; but you often have to verify this
sturdy cape
We have defined what lim f(x) = oo means
Well because we define an infinite limit slightly differently to a finite limit
manic grail
So if lim f= oo and lim g = oo then oo = oo => lim f = lim g
warped badger
So that means that all limits going to infinity equal
A limit going to infinity doesn't really represent an "infinity" per se
All it says is that the bigger the input gets, the bigger the function gets with no upper bound
amber veldt
even in analysis, where we're more comfortable with the infinity symbol, it's more of a legal fiction
than an actual number
warped badger
It characterizes the behaviour of a function, not really anything related to what we usually mean when we talk about "infinity"
warped badger
I dont understand
You can simply think of "this infinity" as sup(R), if a limit tends to sup(R) there exists no real number that bounds it above
warped badger
Is lim x-> +oo x = lim x-> +oo x^2?
Yeah
manic grail
You can simply think of "this infinity" as sup(R), if a limit tends to sup(R) th...
How can you write sup(R)
There isnt a bound
Infinity is a concept not a number
sturdy cape
so you can think of R as any seq...
warped badger
There isnt a bound
Of course, we simply define an element like that and represent it as "infinity"
sturdy cape
How can you write sup(R)
suprema aren't necessarily bounds
we don't define them quite the same
manic grail
Suprema is liturately the upper bound no?
warped badger
It's more of a convenient thing than anything else
tawny violet
Infinity is a new element that we have to adjoin to the real numbers, which is definitionally greater than every real number.
warped badger
Suprema is liturately the upper bound no?
The least such element
amber veldt
as I said, it's more of a legal fiction
tawny violet
forget supremum that's confusing and irrelevant here
amber veldt
like joining blockbuster just becasue you want a 10% off the vhs rental even if you never go to blockuster again after that one time
youre still a member of blockbuster
manic grail
The least such element
Yes
sturdy cape
we invent that shit
amber veldt
like joining blockbuster just becasue you want a 10% off the vhs rental even if ...
just realized that the student might not be old enough to even understand the reference ...
hidden roost
just realized that the student might not be old enough to even understand the re...
yes
amber veldt
What does that mean
it means we create a definition in order for the law to be a certain way
for example
we might want an organization to be tax exempt
so we turn it into a religion by some technicality
warped badger
Look, think of it like this, consider the set R of real numbers
warped badger
Say we want to "create" a new element that satisfies the property that any chosen y in R is lesser than it, say this element is x
amber veldt
it's a convenience, not a fundamental truth
warped badger
Then, define a new set by RU{x}, that is, the real numbers with this element we literally made up
amber veldt
it's easier to say "all monotic sequences have a limit" than to say "all monotic sequences that are bounded have a limit and if they're unbounded they have behavior similar to a limit but kind of not"
sturdy cape
so we turn it into a religion by some technicality
-# you know that's how televangelists get away with shit
warped badger
Your limit as x goes to infinity would be the input of a function f on the x we defined over that set
hidden roost
these are crazy
tawny violet
Maybe this also helps clarify something:
The cardinality of the naturals/integers/rationals and the cardinality of the reals are different "sizes of infinity", yes.
But these are a completely different concept with a different purpose relative to the symbol ∞ that we use in analysis, which is (as it's been said) a useful fiction that allows us to talk about "forever" or "approaching" behavior
sturdy cape
what did we invent
We invented infinity, so to speak
sturdy cape
Maybe this also helps clarify something:
The cardinality of the naturals/integer...
We lie and describe this like a number for the sake of convenience, but the underlying implication is this
tawny violet
It's a number in the sense that it's an element we adjoin to the reals and endow with behavior under addition and multiplication, but it also breaks a lot of algebraic structure that we want to see if we're going to call something a number
warped badger
Maybe this also helps clarify something:
The cardinality of the naturals/integer...
On a more pragmatic note as well, unless you're studying measure theory, topology or set theory, you won't encounter explicit treatment of "infinities of different sizes", so whenever something like this shows up, think of it like a convention to denote a nonexistent upper bound
coarse wyvern
i mean saying we "invent" something is a bit philosophical and invites questions like "is mathematics invented or discovered" which clearly belongs in philosophy server not math
sturdy cape
Unlike a counting number this "number" has certain properties we need to take care of if we want to treat this like a number
well shit we all philosophers in this mine
tawny violet
Without infinity, the real numbers form a field. We like fields, they "behave well" and we can do linear algebra and stuff. With infinity, you have like a semigroup at most or something lol
The best you can get I think is like a loop if you identify +infinity with -infinity
warped badger
i mean saying we "invent" something is a bit philosophical and invites questions...
This is all for the purpose of helping the guy get intuition
tawny violet
iirc
warped badger
I'm not trying to abduct him into the logicist sect
coarse wyvern
Is lim x-> +oo x = lim x-> +oo x^2?
but the asker perhaps doesn't need all of this, for example he asks this
tawny violet
Sure, just explaining why we generally don't consider infinity a number
coarse wyvern
maybe it's enough to stop at we define the limit approaching a number and the limit approaching infinity differently
odd edgeBOT
@manic grail Has your question been resolved?
tawny violet
thats semantics
manic grail
maths is basically just based on axioms and definitions definition is just saying what the thing we write means/represents
tawny violet
was your question answered
tawny violet
yeah
But that's not a very interesting observation tbh, it's a bit like saying that any real number subtracted from itself is 0, therefore a-a always equals b-b
but yes there is only one infinity in the context of analysis
odd edgeBOT
knotty kindle
Pretty sure that isn't allowed
low locust
we dont do problems for you
wooden python
offering to exchange money for completion of work
or vice versa
<@&268886789983436800>
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tropic remnant
We cannot allow the offering of money for work on this server.
blazing shuttle
b&
tropic remnant
Thanks rocket
amber veldt
all cheating must be done for free
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idle fulcrum
Exercise 1.2.7(c) In Abbott’s Analysis. There was already a counterexample showing why g(A)intersectg(B) is not necessarily a subset of g(AintersectB). However, I was considering the train of thought used in the solutions manual (first pdf online) for 1.2.9 b and saw that you can write
xEg(A)Ug(B), and there are preimages for each, so we write g“inverse”, not really inverseE(A)intersection(B), then take g again to obtain xEg(AintersectB). Clearly there is some flaw in the reasoning, but its not clear where. What is the flaw?
odd edgeBOT
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craggy wyvern
is there a reason why there isn't a "oneths" place behind a decimal point in math?
lavish jackal
what is oneths?
craggy wyvern
units but for unitths idk english isn't my main
but why isn't their a one units for the other side
runic swift
but why isn't their a one units for the other side
because its the same as the units place
runic swift
0.6168 = 0 * 10^0 + 6 * 10^-1 + 1 * 10^-2 + others
the 6 is called the "tenth" because its divided by 10
hundredth: divided by 100
if you want a "oneth", you divide it by 1
but thats the units place
craggy wyvern
does it affect it at all?
runic swift
affect what
craggy wyvern
Understanding Decimals
runic swift
why would it
craggy wyvern
I was just asking because I didn't see any of them while trying to study for it
and I'm kind of new to this
runic swift
the "oneths" place you asked?
craggy wyvern
yep
lavish jackal
this should help ya
odd edgeBOT
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fallow idol
-(x+2)(x-1)(x-6)<=0
odd edgeBOT
-(x+2)(x-1)(x-6)<=0
indigo dirge
whats the question?
fallow idol
-x-2<=0
-x<=2
x<=-2
fallow idol
-(x+2)(x-1)(x-6)<=0
glass vault
-(x+2)(x-1)(x-6)<=0
usually with these inequalities i make sure that the highest power term is positive in the expansion
fallow idol
well they gave me a graph
but yeah
after mutliplying with -1
i got
x>=-2 x>=1 or x>=6
glass vault
it is the same as $(x+2)(x-1)(x-6)>=0$
clever fjordBOT
ImOakley
fallow idol
x>=-2 x>=1 or x>=6
but apparantly the correct answer is
glass vault
in any polynomial the final leg is going up if the highest power is positive
fallow idol
-2<=x<=1 and x>=6
weary pelican
but apparantly the correct answer is
yeah no that's not how you gotta do it
remember how you can make a product of 4 things negative
(I'm counting the -1 coeff as a term)
negative * negative makes a positive
so to get a product of 4 things negative, you need an odd amount of the coefficients negative
so
if you have three negative terms for example
it will be (positive) * (negative) * negative * negative = negative
best thing to do in those cases
is to do a sign table
you make a table that registers the exact moments each term is negative, positive or 0
and then to get the sign of the product
fallow idol
idk how to make a table
fallow idol
ive not been taught how to make a table
weary pelican
p(x) = 3(x-2)^2 * (x+3)
weary pelican
ive not been taught how to make a table
well it's a good thing you can learn now
it's not that complicated
weary pelican
starting with the leftmost column
it just sums up all terms that appear in the product we want the sign of
weary pelican
-(x+2)(x-1)(x-6)<=0
so in our case, we'll put in that column:
-(x+2)
x-1
x-6
and finally we'll place the entire expression
-(x+2)(x-1)(x-6)
fallow idol
-(x+2)(x-1)(x-6)
brother
fallow idol
-(x+2)(x-1)(x-6)<=0
weary pelican
?
weary pelican
I don't understand what's your point
Right now I'm teaching you how to make a table to figure out the sign of -(x+2)(x-1)(x-6)
fallow idol
oh
weary pelican
specifically when it's <= 0 is what interests us
weary pelican
-(x+2)(x-1)(x-6)
so let's call this p(x) so I have enough space to put it in my table
fallow idol
alr
weary pelican
so to get the sign of p(x) at a particular value of x
all we need is to multiply the signs of each term
so all we need for now, is to get the signs of each term
The top row will serve as a record of the x-axis, and so it'll keep track of every important x value
by that I mean specifically the x values where any term switches signs
let's start with -(x+2), when is it positive, when is it negative, when is it 0
weary pelican
alr
weary pelican
.close
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karmic stirrup
Whats the difference between a constant and a variable. Like if you write a, as the constant and x as the variable i feel like they both represent the same thing. Becuase you could just as easily display a graph of all the a values as you could with x values
meager juniper
Context is the difference
indigo dirge
the"constant" part just means that a stays the same within the problem while the x varies
meager juniper
@proven wigeon I don't think you understand the purpose of the question...
karmic stirrup
Kinda the question i want to know that becuase a constant still can be almost every number usually
indigo dirge
Kinda the question i want to know that becuase a constant still can be almost ev...
a constant is fixed within a problem
novel hinge
Kinda the question i want to know that becuase a constant still can be almost ev...
Yeah constant is any number
but variable can vary
indigo dirge
if you are given ax+b and a pops up later in the problem, you can be sure that the first a is equal to the second a
novel hinge
but variable can vary
but variables can just be unknowns you need to solve
in essence constant doesnt have different values and isnt an unknown
meager juniper
We can actually treat constants as variables and they function the exact same way as variables.
For instance, if we have a problem in physics where we drop an object off of a building, we have y(t) = -g/2 t^2 + y0
novel hinge
We can actually treat constants as variables and they function the exact same wa...
stupid projectiles
meager juniper
So in this problem, y0 and g are constant, they are taken from information in the problem, and while we have abstracted away their values here, they are considered constant
karmic stirrup
We can actually treat constants as variables and they function the exact same wa...
Yea but theres a different i think becuase i think d/da (a) either doesnt make sense or is 0 while d/dx (x) is 1 . Dont qoute me on this i was told this im not sure if it is real
meager juniper
While t is considered an independent variable and y is considered a dependent variable.
novel hinge
Yea but theres a different i think becuase i think d/da (a) either doesnt make s...
not real
d/da (a) = 1
karmic stirrup
Ok
meager juniper
However, we can imagine situations where we allow g or allow y0 to vary, such as examining entire families of problems
So we can treat constants as variables if we have motivation
Sometimes this can be valuable
So truly the only difference is context, as I mentioned before
novel hinge
I think we're making this more confusing than it is
karmic stirrup
Ima be fr 5 people talking isnt the best XD
indigo dirge
@meager juniper you should explain, you were here first
i suggest everyone else leave
karmic stirrup
I am english tis just my ign
novel hinge
I think he speaks english just well
meager juniper
@karmic stirrup well, if you have a moment to review what I wrote, I will be happy to answer follow up questions. 
karmic stirrup
I will read what you wrote thoroughly
Then let you know
I understand. If i find constants in calculus like d/da am i able to treat it as a variable?
meager juniper
So we can treat constants as variables if we have motivation
An example of such a motivation would be if we wanted to find (for instance) a building that would result in the object dropped reaching the ground after 4 seconds. We would allow y0 to vary instead of t, and now y0 is the variable and t is the constant.
karmic stirrup
Yea that makes sense.
meager juniper
I understand. If i find constants in calculus like d/da am i able to treat it as...
Yeah, absolutely. The fact that you are taking the derivative with respect to it means you are allowing it to vary
karmic stirrup
Couldent we make the constant a variable if we added another axis to graph it or sm
meager juniper
Exactly.
karmic stirrup
Ok makes perfect sense now
meager juniper
In fact, this sort of abstract space is called a configuration space
They are very commonly used in physics and engineering.
For instance, I'm a nuclear engineer, and we consider how neutrons travel inside of reactors using the neutron transport equation, which is a differential equation in a 7 dimensional configuration space.
karmic stirrup
That sounds fun
meager juniper
Did you have any other questions for me?
meager juniper
That sounds fun
Realistically, we just use numerical methods on a computer to solve them, which is not quite as interesting.
karmic stirrup
Yea i mean thats how your gonna do it can visualize 7d that well
What parts do you do for nuclear or do u oversee or sm
meager juniper
I don't work on reactors, I work on detectors mostly.
meager juniper
My favorite isotope is Ta-180m
karmic stirrup
I love hydrogen +1 charge
meager juniper
So for an isotope, you have a few characteristics that help determine, roughly, how stable an isotope is.
First, is whether or not the number of protons and neutrons are even or odd. Most stable nuclei are even-even, there are fewer even-odd, and odd-even, and fewer still odd-odd.
Second, a nucleus is unlikely to be stable if it is not in its ground state.
There are in fact, only 5 stable odd-odd nuclei: H-2, Li-6, B-10, N-14, and Ta-180m
karmic stirrup
Ta stands for which
meager juniper
Tantalum, element 73
The m means it's in its first metastable state. So not in its ground state
karmic stirrup
How stable is it. Like lead or
Wait 74 is my favorite element kinda wiere how their close
meager juniper
In its ground state, Ta-180 decays with a half-life of 8 hours, like any good odd-odd nuclei should
karmic stirrup
8 hours is plenty of time to move it around
meager juniper
But in its first metastable state, it has never been observed to decay.
karmic stirrup
Dam
meager juniper
The reason for this is because the decay paths are all forbidden. It can't decay directly to Ta-180m because it would need to simultaneously emit 8 photons, but it only have 70 or so keV excess energy to do so.
And due to some insane fluke of nuclear chemistry it also cannot β decay to tungsten or hafnium
karmic stirrup
So it really wants to decay but its just hella inconvenient for it to do so
meager juniper
Is it possible to have hydrogen +2 so it has -1 electrons
No, lol, but it is possible to have a 2/3 bond between atoms.
karmic stirrup
Interesting
karmic stirrup
Are you good with relativity stuff i got a major question ive always needed answered
karmic stirrup
When you approach a black hole time dialation slows down to infinite. So it would take an observer infinite time to observe you fall into the black hole. But hawking radation says the black hole should decay before an infinite amount of time has passed for you fall into the event horizon
meager juniper
As I understand it, locally it doesn't really matter that it takes you an infinite amount of time to cross the event horizon, you just cross it, and then the direction of time for you stops being a place in the future, but instead a specific point in space.
But, this is definitely not something that we know for certain as a species, and in specific it's also not something that I know entirely.
As far as the black hole evaporating before you enter it, nature doesn't actually deal well with infinities, it is likely that only a finite amount of time will pass before you enter due to some sort of tunneling effect or something.
karmic stirrup
Actually that makes sense because if you just tunnel over the place of infinite distoration you will be inside without have to actually cross over it
meager juniper
And even if an infinite amount of time passes, it is likely that any black hole large enough will grow faster than hawking radiation forever, and thus has properly infinite life.
wanton bison
hi
wanton bison
wanton bison
You are right
nocturne belfry
im not sure if thats what youre asking 
because the conclusion is also false i guess as you note so maybe im missing the point
wanton bison
No that's it
I wrote the terms down properly, if I would apply the real part, I would attain exactly what I would get applying the cosine addition formula
The reason I asked this was because I was stuck on an induction proof
but yeah ty
.solved
odd edgeBOT
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sage helm
The passage in a Linear Algebra book mentions how to construct a change-of-basis matrix between two basis B and B'. It shows 2 ways:
The first one is when both bases are orthonormal. Cool.
But, the second one seems to be saying that if the bases are not orthonormal, to make them both orthonormal; unless I'm miss reading it (the blue equation).
sage helm
That seems to preclude making a change-of-basis matrix between bases that have differently scaled basis vectors. For instance, if the "x" basis of B has 1 unit along the x-axis, but the B' has 2 units in "x", they're both going to be normalized to 1 unit.
In general, the columns of a change-of-basis matrix don't have to normalized. So, I'm not sure what it is that I'm missunderstanding from that definition.
faint knot
In general, the columns of a change-of-basis matrix don't have to normalized. So...
they dont, but if say you want to use the first way and your basis vectors are orthogonal but not orthonormal, you can fix them so that you can use the first way on them
odd edgeBOT
@sage helm Has your question been resolved?
sage helm
Ok, but that's potentially messing things up, unless (for example) you only care about orientation. So, say in vector space R^2 that axis units are 1w X 1h in basis B, but 1w X 2h in basis B', that type of normalizing change-of-basis is going to clobber that scale of B', no?
@faint knot ^
faint knot
Ok, but that's potentially messing things up, unless (for example) you only care...
this is not really how basis vectors or bases are considered
its more of whether the bases can or cant reach a vector in the first place
itd be more of "Im in R^3, and these three basis vectors cant reach all of R^3 because theyre all dependent"
the usual basic linear algebra more cares about the direction of the basis vectors than the size
sage helm
But, if you're using the basis as a coordinate system, which is almost all the time, doesn't the scale matter then?
faint knot
well we're not
we're using basis as a vector space
if the coordinates get screwed up then that sucks for the coordinates
not for us really, its still the same vector space
sage helm
It's still the same vector space, but that doesn't seem to me to be the same vector after the transformation. Because the transform is changing the value of the underlying "abstract vector" to something else, do to the normalization.
Say, something like, the underlying vector is (5, 5) in R^2, but after being normalized it gets changed to (1, 1) in R^2; still pointing in the same direction, but at different scale.
faint knot
using the basis vectors {(2, 0), (0, 2)}, the vector (3, 4) in this basis would be (1.5, 2)
if the units are twice as large, the coordinates must be twice as small to represent the same vector
odd edgeBOT
sage helm
.reopen
odd edgeBOT
✅
sage helm
What I'm saying is that if basis B has some orthogonal but not orthonormal vectors, and then the same for B', and B != B' then, when making the change-of-basis matrix, if you normalize both, you loose the relative scale between both basis.
So, in that case, things still point in the same direction, but their scale relative to each other is lost.
faint knot
vulkanoid you have to recognize that the method theyre giving here cant be the only method that exists
sage helm
Of course, I understand that.
faint knot
you can clearly see the method given does not care, that doesnt mean the entirety of linear algebra does not care
just this specific method and most of basic linear algebra
sage helm
What I'm asking is if my interpretation is correct.
faint knot
of course? this was never in contention
sage helm
Ok, then I'm good.
It seems I didn't make my intentions clear, but I wanted to confirm that there was information lost if the second way was adopted.
faint knot
yes you lose information during this conversion
I was thinking they were going to talk about a different process which can take any n linearly independent vectors and force them to be orthonormal
sage helm
The author doesn't mention this loss of info. Basically, that there is a trade off when choosing this route. As I was looking info for another related topic, it occurred to me that this was the case, so wanted to confirm I wasn't missunderstanding something.
So, thanks for the info.
🙂
.close
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lavish jackal
hi
lavish jackal
I need help with this
vivid wyvern
What have u tried
lavish jackal
I tried making it by parts
using shapes like this
im working on yellow shape
why 9 is wrong?
did i do something not right 
@vivid wyvern
lavish jackal
U dont have the height given here
fair how i can find the height?
vivid wyvern
fair how i can find the height?
I was thinking of taking different shapes
wary fossil
lavish jackal
i know
vivid wyvern
vivid wyvern
I need help with this
Thas 8
So to get the height u need to deduct 3 as well as 2 from 8
vivid wyvern
do i need to 8-3 or -2?
Both
vivid wyvern
Not 6