Oh wait
#general discussion
frigid hollow
What? @wild pendant he is right
wild pendant
Oops
frigid hollow
Yeah its in the name itself jado
wild pendant
I read that as irrational
frigid hollow
I see
wild pendant
Sorry XD
frigid hollow
Jado, if a number is rational it means it can be expressed as a ratio of two integers
frigid hollow
Yess
dusty swallow
U guys still on this shit
frigid hollow
So 1/2 aka 0.5 is rational cuz 0.5 can be written as 1/2 or 2/4 or 3/6
burnt nest
yeah
wild pendant
Apparently so
burnt nest
10/5
frigid hollow
U guys still on this shit
We’re helping a middle schooler understand roots of a function vs square roots and rational/irrational numbers
burnt nest
5/10*
frigid hollow
We’re helping a middle schooler understand roots of a function vs square roots a...
So if you’re not gonna help don’t be a jerk to those trynna learn ❤️
No matter how miserable your life may be 🙂
burnt nest
so what is f(x)
frigid hollow
A function that takes in an input x and returns a unique value
burnt nest
f(x) = 2
wild pendant
When you have a f(x)
frigid hollow
Is not a function
frigid hollow
f(x) = 2
Because you are not returning any unique value
Jado let me show u an example
burnt nest
2 is unique
burnt nest
oh
frigid hollow
F(3)=2
wild pendant
Yee
frigid hollow
F(1) =2*1 =2
burnt nest
f(x) = 2x
but what if f(3)=3×2
f(3) = 9
frigid hollow
F(2) = 2*2=4
wild pendant
F(3)= 2*3=6
frigid hollow
Nooooo
Jado
You’re confusing squaring a number and multiplying it by 2
F(3) =2*3 =6
@burnt nest
burnt nest
yeah exactly
frigid hollow
Is f(x)=2x
burnt nest
f(10) = 20
frigid hollow
If*
wild pendant
yeah
We have f(x) 2*x
burnt nest
yes
wild pendant
2*10
frigid hollow
So do you see how for every unique x, we are getting unique f(x) values?
burnt nest
Yeah
frigid hollow
Like f(0) is a different number from f(1)
burnt nest
the a x axsis
wild pendant
XD
burnt nest
f(2) is different from f(4)
frigid hollow
And is f(5) the same as f(15)?
burnt nest
no
burnt nest
that is not a function
wild pendant
Because there's no x anywhere
burnt nest
The answer must have an x
wild pendant
No matter what values x will have, f will always be 2
frigid hollow
The answer must have an x
The function must have an x*
wild pendant
Because there's no x anywhere
burnt nest
yes
frigid hollow
Thank you jado
burnt nest
so x asis is always a root ?
frigid hollow
Im an insomniac but our interaction has fried my brain enough for me to sleep
burnt nest
axsis
frigid hollow
so x asis is always a root ?
Nope
wild pendant
Depends on the function
frigid hollow
A root is simply the value of x for a function where f(x) =0
For instance
For f(x) =2x
burnt nest
I think i will learn that next year
frigid hollow
The toot of the function is where 2x=0
burnt nest
the axsis
frigid hollow
So x=0
burnt nest
X is always 0
frigid hollow
The root is a point where x=0 and y =0
burnt nest
because f(x) = 0
wild pendant
Ye
frigid hollow
Not for all functions
For f(x) =x+1
The root of the function is not at 0
Because f(0) = 0+1=1
But
F(-1) = -1+1 =0
wild pendant
For x = -1
f(x) is 0
frigid hollow
Hence the root of the equation f(x) =x+1 is when x=-1
burnt nest
oh
frigid hollow
So minus 1 is the root of that function
burnt nest
-1 is root of f(x)=0
frigid hollow
So the roots change depending on the function
wild pendant
Some functions don't have roots at all
burnt nest
one last question
frigid hollow
^
wild pendant
They all differ from each other
frigid hollow
one last question
Gonkn
burnt nest
what if f(x) = 1 what is root of dat
burnt nest
Oh yeah
frigid hollow
We just went over this
burnt nest
shit
frigid hollow
💀
wild pendant
Poor ram XD
frigid hollow
Try again tho
Give me another function thats actually a function
Ik u can do this jado
burnt nest
f(x) = x
wild pendant
Yes
frigid hollow
Y=x
wild pendant
It's a really important function too
frigid hollow
Wow jado im so proud of you
burnt nest
y = 0?
wild pendant
No
frigid hollow
Its called the reflection function
dusty swallow
U guys still talking
wild pendant
That's the x axis
frigid hollow
Stfu dapz
frigid hollow
Shut the fuck up
wild pendant
y = 0 is the x axis
burnt nest
hello dappy
dusty swallow
Hello Jado
burnt nest
why dont you join us
dusty swallow
Are u having fun pretending to be stupid
frigid hollow
Like i said if you’re gonna be of no help stop being a jerk to those trynna learn ❤️😌
burnt nest
Are u having fun pretending to be stupid
i actually do not know what is a function
pronto
dusty swallow
Ye ofc u don’t 
frigid hollow
Dw about dappy jado she’s a miserable and frustrated lil chap
dusty swallow
Ram we were having a moment here
dusty swallow
A romantic one if I might add
dusty swallow
U should
frigid hollow
Still don’t
dusty swallow
It’s prob the only time u will see it in ur life
dusty swallow
It means u will slowly suffer loneliness
frigid hollow
you mean you and your non existent romantic life 💀
burnt nest
hemu gets no bitches
dusty swallow
you mean you and your non existent romantic life 💀
I got Jado
frigid hollow
Dappy gets no dick/pussy
burnt nest
he got dic
frigid hollow
I got Jado
Im so happy for you
frigid hollow
he got dic
Wtf
dusty swallow
Who do u have
Ping a single person who cares about u
Exaxctly
Leave while u still can
dusty swallow
A more direct “ur not wanted” prob doesn’t exist
frigid hollow
And her female owner 💀
burnt nest
,time @frigid hollow
vivid gulchBOT
This user hasn't set their timezone! Ask them to set it using ,ti --set.
dusty swallow
Anyway
burnt nest
good bye guys im about to go pray
burnt nest
cya
urban dust
So this is Euler’s number expressed as a summation and maybe I’m misunderstanding something about summations but why does n start at 0 instead of 1?
Actually nvm lol
It’s a factorial
normal arrow
@frigid hollow f(x) = 1 is a function tho
frigid hollow
Shouldda made it clear by a “function” we were speaking of strictly injectice functions
Injective*
so no f(x)=1 is not an injective function
normal arrow
you managed to spend so much time specifically saying this is not a function and you never made yourself clear that you were talking about injective functions specifically
insane
frigid hollow
i did actually
I mentioned one to one
Which is injective
if your definitions are up to date 😂
frigid hollow
You are bi
dusty swallow
“Time parameter” 
frigid hollow
Get your life up to date
normal arrow
ikr
dusty swallow
Mf
dusty swallow
Hm
frigid hollow
basically the codomain and domain is equal for bijective
dusty swallow
Actually this time ur right
frigid hollow
Not for injective
dusty swallow
They call it just one to one
frigid hollow
Wtf is surjective though
Isn’t that not a function by definition
Cuz it’s many to one
Oh nvm
x^x is still a function
dusty swallow
X^2
dusty swallow
-1 and 1 are mapped to 1
normal arrow
what
frigid hollow
Bruh
normal arrow
surjectivity isnt about that???
frigid hollow
Meh, guess my high school teachers were just cappin
vivid gulchBOT
vivid gulchBOT
burnt nest
,w sqrt(8)
vivid gulchBOT
frigid hollow
Rational or irrational @burnt nest
frigid hollow
Yeah u’re right surjectivity has nothing to do with many to one
normal arrow
for any b in B, there exists a in A for which b = f(a), how does that relate to (-1)^2 = 1^2 = 1
burnt nest
,w vector
vivid gulchBOT
normal arrow
if i was a redditor, i'd say this is a certified r/confidentlyincorrect moment
but i only use reddit for nsfw material and memes
burnt nest
,w Function
normal arrow
so whatever
frigid hollow
wasn’t confident about anything😂
vivid gulchBOT
frigid hollow
surjective is injective and bijective
So basically, the range always has to be greater than or equal to the domain?
burnt nest
,w f(x) = 0x
vivid gulchBOT
frigid hollow
Reviewing definitions of functions with john, you’re right jado i was wrong
dusty swallow
Ah wait I confused non injective with surjective
frigid hollow
f(x) = 2
burnt nest
That is not a function.
frigid hollow
Ah wait I confused non injective with surjective
Surjective*
frigid hollow
That is not a function.
It is i was wrong, but id rather you call it a constant than a function
frigid hollow
Facts
burnt nest
,w constant
vivid gulchBOT
vivid gulchBOT
burnt nest
,w f(x) = √2x
vivid gulchBOT
burnt nest
@frigid hollow i found a root
normal arrow
lmao
normal arrow
f(x) = sqrt(2x) does indeed have a root at x = 0
frigid hollow
This man was
Actively confusing roots/zeroes of a function to square roots
So glad knowing the progress you’ve made
Actually impressive for a 6-7th grader
(💀)
normal arrow
well get off your high horse then and explain better
frigid hollow
why don’t you lol
normal arrow
i have exams in two days
burnt nest
functions at 9th grade right
frigid hollow
instead of choosing to nitpick randomly
dusty swallow
i have exams in two days
gl
normal arrow
thank you
dusty swallow
If u don’t get A’s
frigid hollow
Then go study, your little rants aren’t gonna help you pass
dusty swallow
I will Hunt u
frigid hollow
Is the exam on maths
wild pendant
Good luck John
normal arrow
Then go study, your little rants aren’t gonna help you pass
says your ass laughing at a kid
normal arrow
gluck indeed
иди нахуй
frigid hollow
What grade are you ian
wild pendant
11th
frigid hollow
There ends my knowledge of russian
burnt nest
i got a 100/100 in my last math exam, thank you @frigid hollow
loud lakeBOT
@burnt nest has given 1 rep to @frigid hollow
wild pendant
The exam will have
Systems, monotony, max and min of functions, trigonometry, polyonyms and exponentials and logarithms
Ez stuff
burnt nest
You did not help, the exam was about vectors and inequalities, but you gave me a boost to study harder.
burnt nest
ok
dusty swallow
Yes
wild pendant
It works because you move y^2 on the other side and you do the difference of squares and then you'll have:
y=x or y=-x
Right?
dusty swallow
That would be one way to argue, yes
ancient helm
Had you only had one of the lines, it would be a linear function
ancient helm
Two occasions?
wild pendant
The fact that there are 2 equations for the same thing confuses me
frigid hollow
The inverse operator of ^n where n is an even number is +-^(1/n)
ancient helm
Why when n is even
frigid hollow
The fact that there are 2 equations for the same thing confuses me
If you tried doing y^3=x^3 this would not occur
The two lines
frigid hollow
Yeah but not +-
vivid gulchBOT
vivid gulchBOT
ancient helm
Wolfram ain’t working
ancient helm
Yea the other lines are in the complex planes
ancient helm
Thats what 98% of graphs are
frigid hollow
Exactly
So why mention the other complex lines
When it isn’t in the scope of this discussion XD
ancient helm
Complex >>>>
frigid hollow
Lol that’s one way of saying you hate RA
ancient helm
Never done RA before
But considering my name is the polar form of complex numbers…
Or the conversion from polar to cartesian
ancient helm
Im in 10th and just finished Pre-Calc
ancient helm
Don’t think my high school has RA
frigid hollow
ap courses?
frigid hollow
Real analysis is a part of precalc
Like functions and their domains is also real analysis
ancient helm
Oh so it prob wasn’t called that
frigid hollow
but again, fairly introductory
ancient helm
Like functions and their domains is also real analysis
Yea we’ve done that
wild pendant
Next year I'll only be taught limits, derivatives and how to solve integrals (only through the u substitution method)
ancient helm
Next year I'll only be taught limits, derivatives and how to solve integrals (on...
Thats it?
wild pendant
Yeah
frigid hollow
real analysis is literally simply analyzing the properties of real functions, but yeah i see what u mean
wild pendant
Next year they used to also teach us complex numbers and matrices
But the government changed it
frigid hollow
CA the same but for complex numbers
wild pendant
Removed them
frigid hollow
that’s dumb
ancient helm
Yea the main diff between Calc AB and BC is polar/parametrixs
wild pendant
Matrices scare me
frigid hollow
Polar form in calc has a rather different meaning than in usual complex analysis
Polar form could mean polar coordinates
Which have nothing to do with imaginary numbers
But instead is a way of converting the coordinates from rectangular (cartesian) to cylindrical( you guessed it, polar)
ancient helm
Cartesian fans doing a degree 10 binomial expansion
(They could easily do ^10 times 10)
wild pendant
Our algebra teacher told us that we can use derivates and limits if we want on the exams
(we haven't been taught them yet)
ancient helm
Our algebra teacher told us that we can use derivates and limits if we want on t...
What the exam over?
frigid hollow
Limits are really intuitive
ancient helm
Yea
frigid hollow
Here’s a fun one
wild pendant
Algebra
Systems, functions, trigonometry, polyonyms, exponentials and logarithms
ancient helm
Hmmst
wild pendant
He said that if yk how derivatives work, you can use em
ancient helm
Wonder when derivatives would help
wild pendant
(for example if we want to check the monotony of a function)
That's the only occasion they can help
I think
For the exam
But I still find it funny how he allowed us to use em
ancient helm
For Alg 2 for “turning points” of a polynomial our teacher just said to do degree-1
ancient helm
For Alg 2 for “turning points” of a polynomial our teacher just said to do degre...
For the amount of turning points
When really it’s the maximum
wild pendant
Our algebra teacher doesn't care
He just wants to be retired XD
It's his last year
ancient helm
My Alg teacher somewhat cares
wild pendant
Lucky
ancient helm
The class is easy af for me so he could literally just not be here and I’d still be chilling
He does take a lot of days off tho
wild pendant
I'm the only one who understands what the teacher teaches, especially in geometry
(the algebra teacher also teaches us geometry)
So in the classes, I'm literally the only one who talks
ancient helm
My former geo teacher was tough but he actually taught students
wild pendant
And answers
ancient helm
Like he’d do basic deriving and make questions harder than the test
wild pendant
Damn
ancient helm
I’d assume you’d have to prove lim x->inf x^n/n^x = 0 for natural n larger than some value
wild pendant
in f?
ancient helm
Or = inf
frigid hollow
Algebra
Systems, functions, trigonometry, polyonyms, exponentials and logarithms
I can see a limit question relating to trig
wild pendant
Or lnf
wild pendant
Ooh
frigid hollow
Algebra
Systems, functions, trigonometry, polyonyms, exponentials and logarithms
“Prove using limits that for any small x, sin(x) ≈x”
And u need differentiation too
ancient helm
Engineers be like
wild pendant
I highly doubt he'll do that
Because limits and stuff are next year
Also
I'm the only student in my class who knows at least a bit math
frigid hollow
Because you have to use L’hopitals rule
ancient helm
American students are something else
frigid hollow
“Prove using limits that for any small x, sin(x) ≈x”
To show this
wild pendant
Literally all my classmates make the mistake:
(a+b)^2 =a^2 +b^2
ancient helm
My chem teacher had to teach times tables to some of the kids cuz they couldn’t do 1d by 1d multiplication without calculator
frigid hollow
if for a small x, sin(x) is about x
ancient helm
Yea
frigid hollow
Then it should satisfy the limit
frigid hollow
Then it should satisfy the limit
Lim(x goes to 0) = sin(x)/x goes to 1
ancient helm
f'(x)/g'(x) = f(x)/g(x)
If f(x) and g(x) = 0,inf, or -inf
frigid hollow
How do we prove that?
ancient helm
,w la’hospital
vivid gulchBOT
Requested by DoubleG#3422
frigid hollow
Differentiate the top and bottom
ancient helm
Boi
frigid hollow
Because then it becomes
The limit
Cos(x)/1
Aka cos(x)
And we know cos(0) is 1
Hence
wild pendant
lim sinx/x = 1
X->0
frigid hollow
Sin(x)/x =1 for an infinitesimal x
Hence
Sin(x) =x for an infinitesimal x
Infinitesimal = almost 0 but not 0
wild pendant
The Bolzano theory
What exactly does it say?
Because I've seen it so many damn times and every time I hear a different definition of it
wild pendant
frigid hollow
I am incorrect
Bolzano was actually a very sound mathematician
I read his “theory of sensing information” and concluded it was pseudoscience
frigid hollow
My working is based on circular reasoning so disregard what i said
About the sin(x)/x as x approaches 0
Problem is you cannot use l’hopital’s rule
Because to use l’hopital’s rule you have to take the derivative of sin(x)
Taking the derivative of sin(x) requires us to plug in the definition of a derivative (lim(h goes to 0) (f(x+h)-f(x))/h and so in our question it is lim(h goes to 0) (sin(x+h)-sin(x))/h, but to resolve this limit, we get a term that is the limit (sin(h)/h) as h goes to 0, which is the original limit we’re trying to resolve
So we have to assume this goes to 1 to begin with to prove the derivative of sin(x) is cos(x) and to use l’hopital’s rule to prove sin(x)/x is 1 as x goes to infinity
Aka circular reasoning
Yeah you prove this limit using squeeze theorem not L’hopital’s rule
burnt nest
$\sqrt{0}$
vivid gulchBOT
Jado 🇲🇦
frigid hollow
rational or irrational? @burnt nest
buoyant basin
Whats the square root of 10
buoyant basin
Thanks
wild pendant
Np
buoyant basin
Ok
agile roost
Yeah you prove this limit using squeeze theorem not L’hopital’s rule
small angle approximation, derived using geometry
burnt nest
rational or irrational? <@721087564144050187>
Error
agile roost
maclaurin series are based on derivatives, however you can derive the first term of the sinx series using geometry
(x)
or even more, but the first term is incredibly trivial
although squeeze theorem is nowhere close to circular reasoning so meh
hollow ice
Because to use l’hopital’s rule you have to take the derivative of sin(x)
You can use L’Hôpitals. There is a clever way to avoid taking this limit again when differentiating: using the series form. The series form is the formal analytic definition of $\sin(x)$. So, if we term-by-term differentiate and result with $\cos(x)$, then L’Hôpital still applies.
vivid gulchBOT
an evil woman in a suit
hollow ice
There is a third approach that can be taken.
Consider a local approximation of $\sin(x) \sim x$.
vivid gulchBOT
an evil woman in a suit
hollow ice
We have that for all non-negative $x$, $x \ge \sin(x)$.
vivid gulchBOT
an evil woman in a suit
agile roost
hollow ice
Consider then that $\lim_{x \to 0} x - \sin(x) = 0$.
vivid gulchBOT
an evil woman in a suit
hollow ice
As both functions are continuous, we have that $\lim_{x \to 0} x = \lim_{x \to 0} \sin(x)$.
vivid gulchBOT
an evil woman in a suit
agile roost
feeling devious, might delete your latex 😏
agile roost
besides the fact you literally said the exact shit I did 
just in significantly more words
hollow ice
I said it more formally.
agile roost
"more formally" == "significantly more words"
hollow ice
I dislike the Squeeze Theorem for most cases.
agile roost
it's because you're extremely skinny, never have to squeeze theorem a pair of jeans
hollow ice
it's because you're extremely skinny, never have to squeeze theorem a pair of je...
There are so many possible functions you can use to squeeze theorem your way into this one.
agile roost
ok mini pekka
hollow ice
Did you know that $\lim_{x \to 0} W(x) - W(2x)$ is not defined for the Weierstrass function?
vivid gulchBOT
an evil woman in a suit
hollow ice
There are also points which are locally Lipschitz on the Weierstrass.
frigid hollow
You can use L’Hôpitals. There is a clever way to avoid taking this limit again w...
Yes but using L’hopital’s without proving the limit of sin(x) is cos(x) is circular reasoning😂
Which is all i said
ancient helm
Feel like you could use sin sum angles formula
frigid hollow
Feel like you could use sin sum angles formula
Lol not you could you need the sum of angles formula to expand sin(x+h)
ancient helm
sin(x)cos(h) + sin(h)cos(x)
Subtract sin(x) cuz def of derivative gets sinx(cosh-1] + sin(h)cos(x)
frigid hollow
see the issue?
To resolve the derivative we need to show the lim as h tends to 0 of (cos(h)-1)/h is 0 and sin(h)/h is 1 to conclude the derivative of sin(x) is cos(x)
ancient helm
Maybe we could prove la’hospital
frigid hollow
So if you were differentiating sin(x)/x top and bottom to resolve the limit we need to conclude the limit sin(h)/h is 1 to begin with😂
Aka circular reasoning
ancient helm
Is there some way to use squeeze?
frigid hollow
Maybe we could prove la’hospital
L’hopital’s is proven to not work for some discontinuous functions
ancient helm
Sad
frigid hollow
Is there some way to use squeeze?
Yeah iirc its establishing for all x, sin(x)>tan(x) ( by definition sin is opp/hypo and tan is opp/adj
ancient helm
Wouldn’t sin < tan
frigid hollow
Yeah the other way
ancient helm
As hypo > adj
frigid hollow
Then do 1/ to both of them
ancient helm
tan(x)/x ?
frigid hollow
To get 1/sin(x)>1/tan(x)
Actually
You could start with
Sin(x)<x<tan(x)
Where x is in radians
This would apply
Then flip it so 1/sin(x)>1/x>1/tan(x)
Now
Multiply by sin(x)
1/tan(x) is cos(x)/sin(x)
And 1/sin(x) times sin(x) is 1
So we get
1>sin(x)/x>cos(x) as the limit x tends to 0
1>sin(x)/x>1
By squeeze theorem
Sin(x)/x =1 as x tends to 0
Cheers!
agile roost
By squeeze theorem
great, now you can use L'hopital's
frigid hollow
Redundant
agile roost
nooooo, you're kidding!
frigid hollow
Nope
agile roost
you've proven the result that sin(h)/h = 1
frigid hollow
So why would i use L’hopital’s 😂
agile roost
therefore now you can use L'hopital's to show that sinx/x=1
frigid hollow
Already proved the limit, hence using L’hopital’s would be redundant
agile roost
why wouldn't you use L'hopital's?????
frigid hollow
Because…. It’s redundant 💀
agile roost
of course not
frigid hollow
If the limit we were trying to prove is already proven💀
In this case it is redundant
agile roost
no
frigid hollow
Yes
agile roost
prove it again
frigid hollow
No
agile roost
use L'hopital's
agile roost
prove via epsilon delta
agile roost
then sketch the graph of sinx/x and integrate over ℝ
frigid hollow
❤️
frigid hollow
then sketch the graph of sinx/x and integrate over ℝ
I combined both steps and graphed the integral of sin(x)/x
Easy
vague dagger
0/0
frigid hollow
=e^pi
agile roost
Easy give me a real challenge pls ❤️
yeah I'm putting that one down as a fail, you have to do it by hand
frigid hollow
yeah I'm putting that one down as a fail, you have to do it by hand
Still ez
agile roost
and graph it
agile roost
yup, cheating again
frigid hollow
You asked me to graph it
agile roost
you gotta graph it by hand, that's an overall fail on the assignment
frigid hollow
Not graph it by hand
agile roost
better luck next time
agile roost
yeah I'm putting that one down as a fail, you have to do it by hand
I said you have to do it by hand here
you failed, cope.
dusty swallow
frigid hollow
Will do
Learnt something new tho didn’t know you could express trig integrals in closed form expressions
agile roost
👴👍
if you mean Si(x) then that's incorrect
closed form has a finite number of operations, Si has an infinite
same with Li etc
frigid hollow
closed form has a finite number of operations, Si has an infinite
learnt something else too thank you
loud lakeBOT
@frigid hollow has given 1 rep to @agile roost
frigid hollow
if you mean Si(x) then that's incorrect
could you still assert that Si(x) can be expressed using elementary algebraic/arithmetic operators and functions?
agile roost
could you still assert that Si(x) can be expressed using elementary algebraic/ar...
the operators themselves are elementary, but there are infinitely many, so it's nonelementary
honest summit
If i have math problem i need help with outside of school and uni can i post it here
frigid hollow
Define outside of school and uni
honest summit
General math problem about probability, dw though i already got help in the main general chat
frigid hollow
Yeah, trust me i wasn’t worried💀
honest summit
Fair enough i wouldn't be either but idk incase of anxiety or something
frigid hollow
😟
@jagged elk sorry to ping you outta nowhere but i wanted to discuss some math with you since you're a phd student
your insight would be invaluable
frigid hollow
@pulsar beacon
@normal arrow
well aight the formula is pretty useless cuz u end up having to solve another equation featuring tetration BUT it’s of 1 degree lower (n-1)
😂😂
frigid hollow
Basically
Take that decimal mumber
Number
And raise it to itself 3 times
Its pretty cool what u end up getting no?
frigid hollow
but thanks for showing it to me
1.47668433735787^(1.47668433735787^1.47668433735787)
paste that anywhere
normal arrow
,w 1.47668433735787^(1.47668433735787^1.47668433735787)
vivid gulchBOT
frigid hollow
🙂
dusty swallow
John get eaten alive
frigid hollow
John get eaten alive
pulsar beacon
only mathematical shit i have ever focused on
Nah I think that's quite the method
agile roost
well aight the formula is pretty useless cuz u end up having to solve another eq...
so you have a recursive formula, solve for the general formula
normal arrow
took me a while to rederive it but basically uhhh
this is literally just substitution in disguise
and your method is really useless because you now have to solve the fucking uhhhh
agile roost
shh!!!
normal arrow
xe^xe^x = a
agile roost
solve the recursion
dusty swallow
Alex lmao
agile roost
what 👼
dusty swallow
Ram when he finds out his life efforts are useless
And Alex is playing with him all this time
normal arrow
agile roost
I'm innocent on all counts
agile roost
idk what you're talking about 😇
agile roost
dusty swallow
frigid hollow
Ram when he finds out his life efforts are useless
“Life efforts” why project your tendency to waste your life on me😂
Im sure someone will validate you for all the riddles you post!
dusty swallow
I can luckily assure u I have a little bit higher life effort than posting riddles on this server, thank u
agile roost
I can luckily assure u I have a little bit higher life effort than posting riddl...
I couldn't tell
normal arrow
Lol john really took all this time to derive the substitution 💀
nah i was drying my dog
y'know
you wash it and then you dry it
dusty swallow
I couldn't tell
Same for u and being discord mod
agile roost
you wash it and then you dry it
you dry it then wash it
normal arrow
you are a madman
dusty swallow
you wash it then u wash it
normal arrow
dusty swallow
then u shot it
agile roost
you wash it then u wash it
you wash it then put it in a particle accelerator
or vacuum chamber
no air pressure = water boils off
normal arrow
wow you guys are weird haha
agile roost
wow you guys are weird haha
you clearly have not worked with a particle accelerator
one of the highlights of my time in physics
honest summit
what the FUCK is a logarithm even, why can i just do lg(100000) on a calculator and get the exponent and what does the calculator do what I can't
dusty swallow
log is not a normal formula like a polynomial or e^… It kinda behaves like sin in the sense that if I ask u what sin(2) is, u can’t give me a good answer but atmost a simple heuristik. So what does the calculator do when we plug things in like sin(..) or log(..) - Taylor expansions. I don’t know how familiar u r with Taylor expansions but the idea is to “approximate” hard functions like sin, log and simply describe them using a polynomial. This way only computing the 100th terms of this polynomial already gives an answer so accurate, that for applied math there really is no point in continuing(those terms get smaller over time, therefore adding the first ones does the trick).
@honest summit
pulsar beacon
log is not a normal formula like a polynomial or e^… It kinda behaves like sin i...
Congrats for explaining middle school math
honest summit
It was kinda the exact thing i wondered
Because i have just been frustrated what the fuck the calculator does compared to why you can't just expand functions like cos() sin() log() etc
Into something more sensible than a word that randomly appears into equations
honest summit
log is not a normal formula like a polynomial or e^… It kinda behaves like sin i...
Also thank you
loud lakeBOT
@honest summit has given 1 rep to @dusty swallow
pulsar beacon
Because i have just been frustrated what the fuck the calculator does compared t...
Good question, and the answer is that we do have expansion as dappy explained above but would only be merely an approximation made by hand however a computer can generate a greater accuracy and a great to show the series are the cos and sines.
Btw all this are just operators and their series explain what happens to the input
honest summit
Taylor series and expansion are those different or the same thing
dusty swallow
Same
honest summit
Alright alright i just like to know stuff from the complete ground up and from what I've search i haven't really gotten a good answer as well
I watched a video about which just said that there was a mathematician in 1700's or 1800's who just estimated a stupid amount of logs which was then essentially what was used until calculators
claculated estimations so to speak
Also found an old math book which just had full pages of log estimations
But very well
Taylor series it is at least
pulsar beacon
I watched a video about which just said that there was a mathematician in 1700's...
Oh interesting
honest summit
Most disbelief I've had in math because it was just estimations all the way
I feel there should be another to calculate it because it exists in the natural world
pulsar beacon
Calculate what
pulsar beacon
Well they are operators derived from the unit circle, atlesast for trig ratios
I don't think there exist any rigorous formula that gives the value of a sinx to such an accuracy
ancient helm
We should instead find the exact value of sin for 1000 non co-terminal angles and create a 999 degree polynomial estimation for sin(x) from 0 to 2pi
agile roost
We should instead find the exact value of sin for 1000 non co-terminal angles an...
erm!!!
only gotta do it fo 0 to pi/2
ancient helm
only gotta do it fo 0 to pi/2
Oh yea true
Once you have sin you also can do the other trig functions from there
agile roost
and then that covers cos and tan and sec and cot and csc and hacovercosine etc
ancient helm
Yea
agile roost
i love hacovercosine
ancient helm
Quartercosine
agile roost
no, hacovercosine is real trig!!
dusty swallow
i love hacovercosine
normal type in Ohio
agile roost
,w hacovercosine
dusty swallow
Is ur gf by any chance a function
vivid gulchBOT
agile roost
nah
inland horizon
Sorry but please explain how this is a 3 mark question...
"proceeds to find k" bruh the process of finding k is unbelievably long there's no way that's 1 mark...
To answer this 3 mark question I needed to expand at least 4 quadratics...
and technically speaking since I put "k = 0 or 10/3" instead of just "k = 10/3" I wouldn't have gotten the final accuracy mark, so technically that screenshot is only 2 marks worth of working out...
I'm pissed (even though that was a practise question)
frigid hollow
Good question, and the answer is that we do have expansion as dappy explained ab...
incorrect, Sin() cos() and log() are not operators but functions. making the distinction is important to further analyze the properties of functional operators fundamental to transforming. a function f(x) to another function g(x)
This is because functions have inverses, but operators, I mean binary operators with two variables (x^y takes x and y) can be made 'functional' and thus invertible either by holding x constant or y constant, if you hold neither operand constant then you have an operator
agile roost
didn't know non-elementary functions could have a taylor series expansion
tf do you think the series expansion you derived of Si(x) was then
frigid hollow
tf do you think the series expansion you derived of Si(x) was then
like isn't the definition of Si(x) using an integral more simplified than the series expansion
dusty swallow
How would u measure “simplicity” 💀
agile roost
yes, doesnt change anything
dusty swallow
Don’t tell me there is an actual way
agile roost
How would u measure “simplicity” 💀
he's somewhat right
agile roost
nah
dusty swallow
Ok thx bud
agile roost
no definition that i know of, just less space
also easier to apply
you recognise from a series that it is the riemann sum of a known integral, not the other way round
dusty swallow
I See
agile roost
you recognise from a series that it is the riemann sum of a known integral, not ...
ah yes this integral appears to be the riemann sum of... this integral, therefore it is the known integral, of this integral
dusty swallow
honest summit
does this work..?
i feel i have missed something crucial
because answer made no sense to me
considering i know next to nothing how taylor series work it's very likely i missed something
the picture is from some uni tho
agile roost
yes
honest summit
so i should be able to continue this sequence and get a result that is as close as possible to cos(x) where x is degrees, as usual?
pulsar beacon
so i should be able to continue this sequence and get a result that is as close ...
I think it was for radians
honest summit
makes sense
limpid cargo
What type of maths is this ?
ancient helm
Summation?
agile roost
Summation?
no! silly, it's curly wurly operational calcoolus
honest summit
I think it was for radians
it was radians i have finally calculated cos(x) without writing cos
thank you
pulsar beacon
no! silly, it's curly wurly operational calcoolus
Guy is keeping count of the analysis
agile roost
Guy is keeping count of the analysis
no worries, I'm THE 4σ male
pulsar beacon
Only 4sigma
Can one become so smart they are literally viewed as someone with mental condition
Like so esoteric that you're seen as a stupid person
honest summit
@chrome basin this
chrome basin
<@723981192801550528> this
saw something similar in a 3b1b vid but i can't remember what it is about or how it works
honest summit
who are you asking
honest summit
aight
dusty swallow
second?
honest summit
fair fair
gonna go into enginnering next semester
i would've earlier but my teachers messed up teaching math in high school
chrome basin
<@723981192801550528> which semester
2nd
agile roost
Can one become so smart they are literally viewed as someone with mental conditi...
ehh, Nikola Tesla... Bobby Fischer
it's not too uncommon
but the insanity is usually a byproduct of high intelligence. Not everyone of a certain intelligence will develop it, they'll usually have a higher chance of it though
pulsar beacon
but the insanity is usually a byproduct of high intelligence. Not everyone of a ...
Fair
enough
ancient helm
Hello general math(s) discussion
agile roost
Hello general math(s) discussion
can't divide by a matrix!!
ancient helm
can't divide by a matrix!!
Shush
Also x/x = 1 for all nums x
Seems like it should apply to matrix
agile roost
vivid gulchBOT
r•eⁱˣ = r•cos(x) + ri•sin(x)
Even better, if we can prove that
For some natural number n, n>1:
$$
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}^n
=
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix} * a
$$
For some number a, then taking the n-1 root of a gives the value of that matrix
agile roost
ancient helm
Why tf did I use a as the scalar
When it’s an element of the matrix
Uses e instead
New challenge:
Find a matrix whos value is i
Alternatively : Find a matrix M such that M^3 = -1•M
agile roost
New challenge:
Find a matrix whos value is i
Alternatively : Find a matrix M suc...
1 0
0 -1 smh
wait no
ancient helm
Not checking that (too lazy)
ancient helm
you want M^2=-I
No cuz then divide by M you get M = -1
ancient helm
Also that matrix squared ain’t even it’s negative
agile roost
i is a rotation by 90 deg
agile roost
Also that matrix squared ain’t even it’s negative
tis
agile roost
agile roost
tis what?
-1 0
0 -1 works
-1 0
a 0 for all a
real
0 0
a -1
a b
-(a^2+a)/b -(a+1) too
just solved the system xoxo
agile roost
tis what?
0 -1
1 0 is the closest matrix you'll get to actually i
because if you liken 1 to I then M^2=-I
agile roost
0 -1
1 0 is the closest matrix you'll get to actually i
which if you cube it then yes, you get -M
happy???
now you can do your illegal maths
ancient helm
Brb gotta verify your claim
Correct. Good job
$
\begin{bmatrix}
0 & -1 \
1 & 0 \
\end{bmatrix}
i
$
vivid gulchBOT
r•eⁱˣ = r•cos(x) + ri•sin(x)
agile roost
i hate you
ancient helm
Why
agile roost
but its sorta right
both do a rotation of pi/2 rad in their respective planes
it's still painful to know how you derived that equality
agile roost
how you derived it is faulty, but the actual thing is somewhat correct
if you have x+iy and then form pairs in R^2 like (x,y) then multiplying by i is the same as using that matrix
ancient helm
Yea also the matrix only equals i when cubing
agile roost
Yea also the matrix only equals i when cubing
agile roost
because if you liken 1 to I then M^2=-I
I^2=1
ancient helm
<:4am:844828541300113408>
Otherwise you could make the claim that real num • real num = complex num
agile roost
the identity matrix is basically 1
agile roost
MATRICES ARENT REAL NUMBERS
ancient helm
As any matrix times it equals one
ancient helm
MATRICES ARENT REAL NUMBERS
I meant when finding the elements of the product matrix
agile roost
ancient helm
You multiply an element (real) by the matrix equaling i and get a complex
agile roost
You multiply an element (real) by the matrix equaling i and get a complex
in R^2 then reverting, yes
multiplying any pair (a,0) will yield (0,a) which is ia
pulsar beacon
Hello general math(s) discussion
This looks so esoteric
ancient helm
I looked up rotational matrix, makes sense
You can use this to make a 2 by 2 matrix of any real or complex number
vivid gulchBOT
an evil magma in a suit
hollow ice
$v = \langle v_1, v_2, v_3 \rangle$
vivid gulchBOT
an evil magma in a suit
hollow ice
$u \cdot v = u_1v_1 + u_2v_2 + u_3v_3$
vivid gulchBOT
an evil magma in a suit
hollow ice
$u = \langle u_1, u_2, u_3, \dots \rangle$
vivid gulchBOT
an evil magma in a suit
hollow ice
$v = \langle v_1, v_2, v_3, \dots \rangle$
vivid gulchBOT
an evil magma in a suit
hollow ice
$u \cdot v = \sum_{n = 1} u_n v_n$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\langle 2, 3, 4 \rangle \cdot \langle -1, 1, -4 \rangle$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\vec{\mathbf{u}} \cdot \vec{\mathbf{v}} = \vec{\mathbf{v}} \cdot \vec{\mathbf{u}}$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\vec{\mathbf{u}} \cdot \vec{\mathbf{v}} = \sum_{i=1} \vec{\mathbf{u}}_i \vec{\mathbf{v}}_i$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\vec{\mathbf{u}} \cdot \vec{\mathbf{v}} = \sum_{i=1} \vec{\mathbf{v}}_i \vec{\mathbf{u}}_i$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\vec{\mathbf{u}} \cdot \vec{\mathbf{v}} = \sum_{i=1} \vec{\mathbf{v}}_i \vec{\mathbf{u}}_i = \vec{\mathbf{v}} \cdot \vec{\mathbf{u}}$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\vec{\mathbf{u}} \cdot (\vec{\mathbf{v}} + \vec{\mathbf{w}}) = \vec{\mathbf{u}} \cdot \vec{\mathbf{v}} + \vec{\mathbf{u}} \cdot \vec{\mathbf{w}}$
vivid gulchBOT
an evil magma in a suit
hollow ice
Lemma 1: $\vec{\mathbf{z}} = \vec{\mathbf{v}} + \vec{\mathbf{w}}$, then $\vec{\mathbf{z}}_i = \vec{\mathbf{v}}_i + \vec{\mathbf{w}}_i$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\vec{\mathbf{u}} \cdot (\vec{\mathbf{v}} + \vec{\mathbf{w}}) = \vec{\mathbf{u}} \cdot \vec{\mathbf{z}} = \sum_{i = 1} \vec{\mathbf{u}}_i \cdot \vec{\mathbf{z}}_i$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\vec{\mathbf{u}} \cdot (\vec{\mathbf{v}} + \vec{\mathbf{w}}) = \vec{\mathbf{u}} \cdot \vec{\mathbf{z}} = \sum_{i = 1} \vec{\mathbf{u}}_i \cdot (\vec{\mathbf{v}}_i + \vec{\mathbf{w}}_i)$
vivid gulchBOT
an evil magma in a suit
hollow ice
We will say $c$ is a scalar.
vivid gulchBOT
an evil magma in a suit
hollow ice
$c(\vec{\mathbf{u}} \cdot \vec{\mathbf{v}}) = (c\vec{\mathbf{u}}) \cdot \vec{\mathbf{v}} = \vec{\mathbf{u}} \cdot (c\vec{\mathbf{v}})$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\vec{\mathbf{u}} \cdot \vec{\mathbf{u}} = ||\vec{\mathbf{u}}||^2$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\vec{\mathbf{u}} \cdot \vec{\mathbf{v}} = ||\vec{\mathbf{u}}||||\vec{\mathbf{v}}||\cos \theta$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\cos \theta = \frac{\vec{\mathbf{u}} \cdot \vec{\mathbf{v}}}{||\vec{\mathbf{u}}|| : ||\vec{\mathbf{v}}||}$
vivid gulchBOT
an evil magma in a suit
hollow ice
$i + j + k$ and $2i - j - 3k$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\cos \theta = \frac{(i + j + k) \cdot (2i - j - 3k)}{||i + j + k|| : ||2i - j -k||}$
vivid gulchBOT
an evil magma in a suit
hollow ice
$\cos \theta = \frac{1(2) + (1)(-1) + (1)(-3)}{||i + j + k|| : ||2i - j -3k||}$
$\cos \theta = \frac{-2}{||i + j + k|| : ||2i - j -k||}$
vivid gulchBOT
an evil magma in a suit
hollow ice
$||\langle a,b,c\rangle|| = \sqrt{a^2 + b^2 + c^2}$