precalculus

[swift shuttle]

And relies more on geometric understanding

[rocky silo]

It’s not a definition what

It’s proved by tailor series

Get ur facts right 😔

[tulip fern]

it is tho

[rocky silo]

No

[tulip fern]

buh

yes by definition

[hardy venture]

OK

I’ve had this before

[rocky silo]

It’s proved

[tulip fern]

we have to either define e by its power series or force the definition

[rocky silo]

Wdym by definition

[hardy venture]

You have different definitions

[tulip fern]

it's a "proven" result but we take it as a definition

[devout fog]

Trigonometric forme of a complex number

[rocky silo]

I think we have different definitions of definition

[tulip fern]

we either define e by its power series

or take it as definition

[rocky silo]

But definition means it’s not proven

Power of a point is a definition

[hardy venture]

Or by $e=\lim _{x \to \infty}(1+\frac{1}{x})^x$

[obsidian monolith]

YeetusDeletus5

[devout fog]

What is this

Lim quand x tend vers +infi

[hardy venture]

Or by $1=\int_1^e\frac{1}{x}dx$

[obsidian monolith]

YeetusDeletus5

[hardy venture]

My point is the “definition” of e is subjective.

[rocky silo]

Yeah

[tulip fern]

no it isnt

it's proven

but e^x is a definition

[rocky silo]

What

[hardy venture]

What’s the definition of e^x then?

[rocky silo]

The power of a point is defined to be a value

[tulip fern]

its power series

[hardy venture]

Taylor series, ok.

[tulip fern]

wait i thought you were talking about power of a point 💀

[hardy venture]

@rocky silo What is your definition of e^x?

[tulip fern]

assuming it converges on the complex plane (which it does)

[rocky silo]

It’s a number to the power of a variable

[valid topaz]

oh wowsies

[rocky silo]

Yup

[hardy venture]

“Number”? “Power of”?

[devout fog]

What is the solution

[hardy venture]

So e^x=5^x?

[rocky silo]

No I’m saying e is the number

So it’s like 2.71… to the power of x

[hardy venture]

(2.714)^x=e^x?

[rocky silo]

No

I mean

Not like Ron ding

Without rounding

[hardy venture]

I didn’t round

[rocky silo]

E is not 2.714

[hardy venture]

Then what is it?

What makes e so important it needs a symbol?

@rocky silo

[devout fog]

.

.

...

[rocky silo]

Ok so do u want me to say e^x is this

Sure that works too

[hardy venture]

Ok, e is this limit

[rocky silo]

But that still means Eulers is proven from definitions

[hardy venture]

Wait

What’s your definition of exponents?

[tulip fern]

the fact that $e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$ over all reals $x$ (assuming exponentation is a continuous extension of its rational counterpart) allows us to extend its definition to the complex plane

[obsidian monolith]

literally anything

[rocky silo]

X^n would be multiplying x by itself n times

[hardy venture]

What if n is negative?

How do I multiply x by itself -1 times?

[rocky silo]

1/x^-n

[devout fog]

e^x = exp(x) for each x in |R

Fun

Ction

Notation

[rocky silo]

Divide it

Instead of multiply

Go backwards

[hardy venture]

What if n is rational?

1/2 times?

[rocky silo]

Sort

Sqrt

[hardy venture]

And what’s sqrt?

[rocky silo]

Square root??

[hardy venture]

What is a square root?

[devout fog]

What is the problem guys

[hardy venture]

Hol up lemme cook

[tulip fern]

for a rational a, sqrt(a) = inf{x in Q : x^2 < a}

[rocky silo]

Sqrt(n) = x when x^2 = n

[tulip fern]

for a real, take the limit

[devout fog]

I don't understand stand very well

En isn't my language

[hardy venture]

So √(25)=-5 because (-5)^2=25?

[rocky silo]

It could be plus or minus

[tulip fern]

this definition doesn't make sense, sqrt isn't a function then

it's a relation

[rocky silo]

But ig the square root function is only positive

So abs Val

[hardy venture]

Let’s skip real numbers

Complex numbers

x times itself i times

How?

[rocky silo]

Eulers

[tulip fern]

...

and what does that mean?

[rocky silo]

Oh damn

[hardy venture]

So Euler’s formula is how you define complex exponents.

[rocky silo]

So it is a definition

Oops mb

[tulip fern]

of course, it's a natural extension

[hardy venture]

I have cooked, peeps.

[rocky silo]

Yeetusdeletus is the funniest name I’ve heard

[tulip fern]

because if we want a natural extension of e^x to the complex plane, then that's what we do

define it like that

e^x admits a unique power series over the real line (taylor's theorem and power series equivalence)

so extend the power series naturally

and it naturally gives the identity $e^{ix}=\cos x+i\sin x$

[obsidian monolith]

literally anything

[tulip fern]

thus it is natural to define it that way

[hardy venture]

You can absolutely 100% define e^x as its power series, Euler’s formula is then a consequence of the definition.

[tulip fern]

the other approach is defining it that way

there is no rule saying that we can't define things

if we couldn't, the complex numbers wouldn't exist

we define i to be a number such that i * i = -1

so what's the harm in defining e^(ix) = cos x + i sin x explicitly

i mean, there's a natural extension described above that has it as a corollary

[rocky silo]

Damn yg are smart

Thanks for the help tho

[hardy venture]

np

≠p

[rocky silo]

Wait what grade are yg in

[tulip fern]

7th (13 yrs old)

[hardy venture]

11th

[rocky silo]

Woah orz

Why is one of ur role undergrad math

[tulip fern]

because i like doing it

[rocky silo]

Oh

[valid topaz]

cuz it basically means they're learning a topic of a typical mathematics major

[valid topaz]

hmmm

[hardy venture]

These are my mom’s math notes from when she was learning trig

[rocky silo]

W mom ❤️‍🩹

[hardy venture]

They’re jumbled tho

[rocky silo]

Why do we need this?

[hardy venture]

I thought someone might appreciate them

[rocky silo]

I very much apreciate

But u should put it in the trig section

[devout fog]

Guys are you in highschool

??

[rocky silo]

Amost

Almost

8th

[devout fog]

The final year I mean

[rocky silo]

12th?

[devout fog]

3rd year in highschool

Yes

[rocky silo]

No I’m not

[devout fog]

Ok

Where are you from

?

[devout fog]

I m sorry for this question

But I m curious

[rocky silo]

Texas

[devout fog]

Ok

I m Moroccan

We are learning maths in french

[rocky silo]

Oh

Ok

[hardy venture]

$i^2=-1$

[obsidian monolith]

YeetusDeletus5

[rocky silo]

Wait is ln(-2) =ipiln(2)

Ln(-2)

[devout fog]

Nombre imaginare pur iR

[rocky silo]

@hardy venture

[hardy venture]

Euler’s formula

[rocky silo]

Ohhhh

Cus e^(ln(-2)) = e^(ipiln(2))

-2 = -1*2

[swift shuttle]

Btw e^x comes from exp(x) which is a function defined as

exp(x) = 1 + x + x²/2 + ... + x^n/n!

So that's where it comes from, aka how it was defined

[hardy venture]

That is a definition of e^x

[rocky silo]

Taylor series

[rocky silo]

Why does it stop at n it goes on forever

[hardy venture]

,rotate

[hardy venture]

Because it’s not a polynomial otherwise

[swift shuttle]

Ok ur right yeah

[swift shuttle]

You expect me to continue?

[rocky silo]

Just right … at the end

[hardy venture]

*write

[rocky silo]

Oops

AMC 10 ahh silly mistake

[swift shuttle]

Idk I'm used to series sums ending at n

[hardy venture]

He’s asking why the series isn’t infinite

Actually wait

He’s right

[rocky silo]

WOAH

[swift shuttle]

It is infinite

[rocky silo]

ACTUALLY

[tulip fern]

ook

[hardy venture]

You’d use approximately equals for the approximation

[tulip fern]

so basically

in set theory infinite things exist

[swift shuttle]

Is expa(x) a thing?

[tulip fern]

thus you can make infinite sequences that converge to something

[swift shuttle]

Yeah

[devout fog]

.

[hardy venture]

But he’s not using an infinite series, the series clearly terminates at n

[tulip fern]

thing is, this definition is almost completely wrong

you need the series to "go on to infinity"

[valid topaz]

hmm

[swift shuttle]

exp(x) = Σn=0 -> inf x^n/n!

[tulip fern]

formally, you need to take the partial sums and see what value they converge to (assuming they converge, which they do)

formally, you do that with a limit (calculus)

[rocky silo]

This starts and ends with the same thing

[tulip fern]

thus

$\exp(x)=\lim_{n\to\infty}\sum_{k=0}^n\frac{x^k}{k!}$

[obsidian monolith]

literally anything

[hardy venture]

$\sum _{n=0}^\infty \frac{x^n}{n!}=e^x$

[obsidian monolith]

YeetusDeletus5

[tulip fern]

the limit is discrete btw 😁

[hardy venture]

I love how we’re talking about calc in the Precalc channel

[tulip fern]

lmao

[nimble wolf]

ikr!

[swift shuttle]

Is that even calc

[valid topaz]

it's fine

trust

[tulip fern]

yes, this is calculus

[rocky silo]

What is a discrete limit

[hardy venture]

It’s limits and Taylor series

[hardy venture]

U know what limit is?

[tulip fern]

a way to say something converges

[rocky silo]

Yeah

[swift shuttle]

Since when are limits calc wth

[rocky silo]

Ok I don’t understand this converges thing

I’m in precal

[swift shuttle]

Don't you study that before derivatives

[nimble wolf]

I mean, limits are the fundamentals of Calc

[tulip fern]

$\lim_{n\to\infty}a_n=a$ iff for all $\varepsilon>0$ exists $N$ such that if $n\ge N$ then $|a_n-a|\le\varepsilon$

[hardy venture]

(0.9, 0.99, 0.999, 0.9999…) what is the biggest thing here?

[obsidian monolith]

literally anything

[tulip fern]

that's the formal definition

[rocky silo]

0.9 repeating forever

1?

[tulip fern]

the informal definition is that it's the value the sequence gets close to

[hardy venture]

Except

1 is not here

[valid topaz]

i like this one

[rocky silo]

Ohhhh

Wait I understand

[valid topaz]

delta epsilon is weird

[hardy venture]

So we say the sequence CONVERGES to 1

[tulip fern]

you can also have sequences like 1, 1/2, 1/3, 1/4, ...

that converges to 0

[hardy venture]

Or the limit as the sequence goes to ∞ is 1

[rocky silo]

Wait

Exp(x) is e^x?

[tulip fern]

oh yeah btw a sequence should be denoted with ()

[hardy venture]

lol my bad

[tulip fern]

yes, they're identical over all of R

so they're equal

[rocky silo]

So it can be anything to the power x

[hardy venture]

Why is “over ℝ” necessary?

[tulip fern]

thus it's natural to extend e^x to the complex plane by exp

[devout fog]

Is just a notation prolonged to R

[hushed sphinx]

No, exp(x) when x is a real or complex number always means e^x specifically.

[tulip fern]

idk just specifying

[valid topaz]

he's being fancy

[hushed sphinx]

Some people like the "exp" notation better, partly because it makes sense to define that function before you define what raising arbitary numbers to arbitrary powers even means.

[tulip fern]

if we interperet e^x by "extension of a continuous function over Q" then asking about its value over the complex plane doesn't make sense

[devout fog]

But e^x is better to manipulate

[hushed sphinx]

And partly because the exp function (defined via power series) can be generalized to some settings where speaking of "raising to powers" in general doesn't make sense at all.

[rocky silo]

Woah I got like 5 ppl tutoring me now

[hushed sphinx]

Ask an interesting question, and you'll (perhaps) get interesting answers.

[hardy venture]

like ℝ because multiplying √(2) a total of √(2) times makes no sense

[hushed sphinx]

More like for example, there's an extension of exp where it works on matrices and gives you matrices back, and it satisfies several of the nice properties of e^x too. Then it can feel more honest to write it as exp(A) rather than e^A, because it doesn't make sense to raise any number other than e to the power of a matrix.

[hardy venture]

n^A= e^(ln(n)A) makes perfect sense to me.

[hushed sphinx]

Well, yes, but people don't actually do that.

[tulip fern]

true, but i'm pretty sure exp is more natural (matrix diff eqs)

in particular, e^Ax is a solution to dy/dx = Ay

[hardy venture]

Calculus in the Precalc channel again

[tulip fern]

what do you think we've been doing this entire time

[hardy venture]

Explaining why some people like exp(x) over e^x

Not really calc

[rocky silo]

Why doesn’t the harmonic series converge

[tulip fern]

hint: bound the terms below

1 <= 1

1/2 <= 1/2

[rocky silo]

No ik the answer I just wanted to see what u would say

[tulip fern]

1/4 <= 1/3

oh ok

[rocky silo]

Wait what

[tulip fern]

yeah it's because the series is bounded below by n for 2^n

[valid topaz]

i remember watching sal khan prove it

[hardy venture]

P series p=1

[tulip fern]

thus by the comparison test it diverges

[tulip fern]

integral test also works 🙂

[rocky silo]

Wait I saw this thing online

S/2 = 1/2+1/4+1/6…

[hardy venture]

OH NO

[tulip fern]

this diverges too

wait a minute

[hardy venture]

NOT -1/12 AGAIN

[tulip fern]

me when someone mentions "-1/12" (they don't even know what the riemann zeta function is)

pop math strikes again

[rocky silo]

So S - S/2 = S/2 =1+1/3+1/5+…

Which is greater than 1/2+1/4+1/6…

[tulip fern]

well the thing is

[hardy venture]

$\sum _{n=1}^\infty \frac{1}{x^n}$

[obsidian monolith]

YeetusDeletus5

[tulip fern]

the series are not well defined so you can't do that

[hushed sphinx]

Both series diverge, so one is not really greater than the other.

[rocky silo]

So S/2 > S/2

So diverge

[tulip fern]

this isn't a contradiction; S was never a real number

[hardy venture]

Yeah.

[valid topaz]

nadat prive riemann zeta function's non trivial roots

[tulip fern]

no

[hushed sphinx]

Oh, perhaps that was the point of the argument.

[valid topaz]

but 1 million dolla

[tulip fern]

pop math strikes again

[hardy venture]

You can’t do S/2-S/2=0 because ∞-∞ is undefined

[rocky silo]

Yeah

But since it’s S/2 > S/2 it diverges

[tulip fern]

arguably it's not even infinity (extended reals)

[rocky silo]

Cus they’re both infonites

[tulip fern]

actually, this argument doesn't work generally

i'm pretty sure

[rocky silo]

Oh

[tulip fern]

because there's a notion called "conditionally convergent series"

and it could create a similar contradiction and still converge

[hardy venture]

Rearranging terms cause different outputs

[tulip fern]

aka Riemann Rearrangement Theorem

hey we're all 🤓s here

[valid topaz]

nadat stop wowing people

with ur knowledge

[tulip fern]

this is calculus bc knowledge at most

[valid topaz]

it sounds fancy

[hardy venture]

Bro I didn’t know it was called that, I knew what it was

[tulip fern]

a real series is conditionally convergent if and only if every rearrangement of it produces every extended real number or diverges

otherwise it's absolutely convergent

[hardy venture]

Time to stir up controversy

$\sqrt{-1}=i$

[obsidian monolith]

YeetusDeletus5

[tulip fern]

pop math is a beautiful thing

[hardy venture]

Pop math? Popular math?

[tulip fern]

yeah

[hardy venture]

What’s wrong with this again?

[tulip fern]

sqrt isn't defined for negative numbers

thus such a definition is nonsensical

furthermore, if sqrt(-1) = i, then sqrt(-1) = -i, doesn't it? what classifies i as positive?

[hardy venture]

Principal branch of square root

[tulip fern]

but what is the "square root"?

the "principal branch" relies on i itself, so it's a circular definition

more formally, we extend the field R by appending an element i

in particular, we let the elements of C be ordered pairs of R, and define arithmetic as if i was a thing

[hardy venture]

$\sqrt{ae^{i\theta}}=\sqrt{a}e^{i\frac{\theta}{2}}$

[obsidian monolith]

YeetusDeletus5

[tulip fern]

circular definition

[hardy venture]

This is how to define sqrt over C

[tulip fern]

i am aware, but what is C if we're defining i?

[hardy venture]

Ok, I see how i^2=-1 is the correct definition

This does not disprove √(-1)=i

[tulip fern]

if you define sqrt using the principal branch, this would make sense

and you've already predefined i

[viscid thistle]

$e^ipi=/-1$

[obsidian monolith]

SUBLAVA

[hardy venture]

Yes, I am referring to principal branch of square root

[viscid thistle]

Oops lol

Complex numbers are cool

[hardy venture]

Fix this @viscid thistle

[viscid thistle]

I have zero idea how to use latex

$hello world!$

[obsidian monolith]

SUBLAVA

[hardy venture]

#latex-help

[tulip fern]

ok so more on the definition of i

it's like the "Euler's identity" situation

we say it's a theorem with modest assumptions

but we define it that way

[viscid thistle]

$i=imaginary$

[obsidian monolith]

SUBLAVA

[viscid thistle]

here

easy

[tulip fern]

i'm talking about the formal definition

[hushed sphinx]

That's where the name comes from; it tells you essentially nothing useful about the mathematical idea the name is a name for.

[tulip fern]

addition and multiplication have to be defined for complex numbers, we don't really assume the properties (at most, we assume the properties of reals)

funnily enough, "imaginary" was supposed to be derogatory towards complex numbers

but they ended up being a cornerstone of math lol

[hardy venture]

$(a+bi)+(c+di)=(a+c)+(b+d)i$

[obsidian monolith]

YeetusDeletus5

[tulip fern]

we use the notation a+bi, but what we really mean is (a,b), interpereting C as R * R with extra structure

thus i = (0,1)

[hardy venture]

$(a+bi)(c+di)=(ac-bd)+(ad+bc)i$

[obsidian monolith]

YeetusDeletus5

[hardy venture]

I know

[hushed sphinx]

(There are several equivalent ways to slice it formally, but the "R×R with extra structure" one ought to serve nicely for curing philosophical hiccups).

[tulip fern]

i think that the R^2 with extra structure definition is nice

but some other ways to define it implicitly prove the properties of the complex numbers

for example, they form a field

[hardy venture]

So does ℚ

[rocky silo]

I’m back

[hardy venture]

Hi

[steady sentinel]

$phi

$phi$

bruh i wanted the phi symbol

[hardy venture]

$\phi$

[obsidian monolith]

YeetusDeletus5

[steady sentinel]

ooh ok

[astral spade]

calculus ez asf 😭

[undone pumice]

frfr

[valid topaz]

uhhh

ig maybe

[hardy venture]

#calculus

[hallow spire]

yeah bro. the derivative of e^x is literally e^x

how hard can it be?

[undone pumice]

lol

[urban sage]

This is the only channel i understand cuz i suck at math

But i've been studying

i wanna get better cuz i wanna enter into computer engineering (?) (not sure how it's called in English) at college

[undone pumice]

i think maybe yiu mean computer science? Like coding type stuff

[undone pumice]

i see i see… thats okay! im sure your hard work will pay off

[urban sage]

oh yeah yeah that

i've seen ppl call it different ways so i wasn't sure

i feel rlly dumb nowadays but hey i guess everyone started from zero

I don't understand anything in the general chat when they talk about math 😭

[undone pumice]

mmhm yeah, its okay

[undone pumice]

lmao me too its okay

im in precal at school so i like to hang out in this channel too

[frail stag]

yeah its the group theory, general relativity, and topology stuff that i dont get 😭

[undone pumice]

all the weird shapes and terms 😭

[frail stag]

yeah bro

[urban sage]

same even tho im on my last year of school

[frail stag]

nd tensors

[undone pumice]

mm i see

[frail stag]

metric tensor

[undone pumice]

im first year of high school

[urban sage]

the random symbols and definitions bruh what's going on

[undone pumice]

lol

[frail stag]

pro

[urban sage]

that's pretty interesting! i must be delayed cuz my country's teaching really, really sucks. We're on the last place in the maths ranking in my continent

[frail stag]

today;s my bird day btw 🥳

[urban sage]

oh my gosh

[undone pumice]

o-oh wow

[urban sage]

Happy birthday!!!!

[frail stag]

same

[undone pumice]

Happy bday! Yay!

[urban sage]

hope u have or had lots of fun today

[frail stag]

thank you!

[frail stag]

thanks!

[frail stag]

the day just started

[undone pumice]

im not sure, im in high school in the united states, so idk how different it is here versus hwerever you are

i mean, im ahead i thnk

[frail stag]

which country

i ask out of curiosity

[urban sage]

yea im pretty sure

[undone pumice]

the seniors (last year) at my school typically take precalc as a 4th year of math

so thats about the same i think

[urban sage]

Paraguay! It's in South America. Between Argentina and Brazil

[undone pumice]

though we have “advanced” tracks and such

[frail stag]

i went there last year

took alg2

[undone pumice]

ah

nice

[frail stag]

ohhh ok

our thingy doesnt even have that type of system

its just grade 1 math all the way to grade 10 math

😭

[urban sage]

yeah i'm a senior

[frail stag]

im a sophomore

turning senior this april

[urban sage]

That was second year right?

[frail stag]

school starts april 20

[frail stag]

yes

[undone pumice]

you dont get a choice based on scores or something? 😭

thats…wow

[frail stag]

no we dont 😭

theres no optional subjects

[undone pumice]

Nuuu thats silly

[undone pumice]

gulps in wow what the sufferin-

[urban sage]

i was second year until recently too!! Turned senior on february

[frail stag]

math, physics, chem, bio, social studies, english, fl, sl and tl english

these are all mandatory

[undone pumice]

whats fl and sl

[winter comet]

what?

[urban sage]

Huh? U can choose ur subjects?

[winter comet]

what?

[frail stag]

?

[frail stag]

pro

[undone pumice]

Like

[urban sage]

what happened

[frail stag]

first lang and 2nd lang 😭

[undone pumice]

at our school

[winter comet]

what happened to junior?

[frail stag]

• third lang english

[winter comet]

are you on the quarter system? how do you become a senior in february

[undone pumice]

In like 7th grade u can go into regular or accelerated math whcih lets u be more ahead

And such (i mean we also habe electives)

[urban sage]

My school year starts on february and ends on late november!

[frail stag]

the fl and sl are what we get to choose when we are 1st grade for fl and 4th grade for sl

[undone pumice]

w-wow i would die

[frail stag]

same bro

[undone pumice]

im taking chinese this year

[frail stag]

i hate them so much

[winter comet]

so the school year is spring-fall instead of fall-spring?

[urban sage]

I need to take chinese too (and English) cuz i wanna study abroad

[undone pumice]

then a second yearand im stopping foreign langauges

sigh

[frail stag]

so im trilingual

next year i have to learn a 4th lang 😭

[undone pumice]

I heard people saying they like chinese… but thats because the chinese1 teacher onoy hates the heritage kids x.x

[frail stag]

but they remove the fl and sl langs

[undone pumice]

ooppp

ummm

🙏

[frail stag]

ye

apparently we wont go too deep into it tho

like

writing 1 sentence answers at best

ive heard

[undone pumice]

Oh

interesting

[frail stag]

yeah bro

[undone pumice]

in my chinese class we habe to write full paragraphs

[frail stag]

my 9th grade in the us was peakest school year

[undone pumice]

speak for several minutes of recordings

[frail stag]

took up mechatronics course

[undone pumice]

And we have to do full presentations with “complex” sentences (and write only complex sentences) and this is just the first year of chinese..

Heritage track is supposed to build fluency I guess. Not taking chinese3 tho. (I can understand fine, like hearing, but writing a bit of a struggle)

[urban sage]

It's more like summer-spring cuz on february it's still summer

[frail stag]

for the fl and sls we have to write whole essays and discourses (letters, convos, notices, advertisements, etc)

[undone pumice]

oh i see o.o

[frail stag]

same goes for english but grammar is slightly more complex

[undone pumice]

i see i see

[frail stag]

the grammar is relatively simple for fl and sl

the difficult part is the vocab

[winter comet]

ohh wait do you live in australia or something

[urban sage]

Bro that's too hard

[undone pumice]

They live in like southern america iirc

[frail stag]

hw many letters are in the chinese alphabet

i ask

[undone pumice]

*south america or something agh i frogt the name

[undone pumice]

uhhh

its not english lettrs

[winter comet]

strange

[urban sage]

south america :) southern hemisphere so it's the opposite from the US. When it's winter for u it's summer for us. We have christmas and new years on summer

[frail stag]

yes

[undone pumice]

theres like a hundred thousand “characters”

[frail stag]

ik

damn

[undone pumice]

with diff meaning

[frail stag]

how do tey work

[winter comet]

ah ok gotcha

[frail stag]

fl and sl letters also work differently from english type ones

(hindi is my fl)

[undone pumice]

eg: 你好，我叫Evelyn. 我喜歡數學

[frail stag]

i dont get it but ok 😭

[undone pumice]

也喜歡吃！ and each of those words have a meaning and grammar type structure

[frail stag]

ok

[undone pumice]

each “picture” is basically a word in english

[frail stag]

oh damn

[undone pumice]

And you have to learn to write each one

[frail stag]

how does one remember allat

writing it seems complex too

wow

[undone pumice]

… well, there’s a reason writing is my worst portion

[frail stag]

how do they bear with mandarin 😭

[undone pumice]

my speaking and reading is slightly better

[frail stag]

ok

i can understand

[undone pumice]

Sooo thats how first-year chinese is

[frail stag]

also

whatd u learn in precalc

[undone pumice]

cant wait for the second year!

[urban sage]

I have red note/xiaohongshu and it's entertaining to see comment sections cuz i don't understand anything

[undone pumice]

ummm

a lot fo algebra

working on trig

matrice vector work to come (next unit)

[frail stag]

oh

[undone pumice]

learnt logs and inverses too

[frail stag]

algebra

ok

we never learned mch abt vectors and matrices

we learned nothing in fact

[undone pumice]

oo

oop

[frail stag]

i only learned them on my own just bcuz on the 3b1b series

[undone pumice]

Wow

icic

[frail stag]

😭

like bro

[urban sage]

Same

[undone pumice]

i learnt them a bit in 8th grade but i think precalc should fledh out my knowledge

[frail stag]

they didnt teach us abt graphing functions until 9th grade

i was in the us in 9th grade btw

[undone pumice]

i took like

probably alg1, 2, geo, trig level stuff in 8th grade

[frail stag]

until then it was just histograms, linegraphs, and stuff from statistics

we also learnt abt trig just thisyear

for some reason

[undone pumice]

oh icic

[frail stag]

and these are all of the cool topics imo

[undone pumice]

oh yeah also learnt some stats stuff in those courses

regressions and such

normal distribution

right end left end tails, hypothesis testing, confidence levels, bla bla

[frail stag]

yeah we didnt learn regressions, normal distributions

[undone pumice]

i figured

[frail stag]

the furthest we got in statistics is finding mean, median mode, and ogive curves of grouped data 😭

[undone pumice]

igs u can google the course contents, was a course called like integrated 3

[undone pumice]

icic

Sounds about right from what my classmates told me

[frail stag]

ok

india btw

10th grade is def different in india

bcuz of boards

[undone pumice]

Ah

icic

[frail stag]

i wrote 200 exams this year 😭

[undone pumice]

oh wow 😭 😭

i pray for your sanity

[frail stag]

pro

[frail stag]

its mostly fine actually

[undone pumice]

phew

[frail stag]

theres a kid who genuinely became a stickfigure bcuz of these 😭

some kids have a ton of pressure from parents for good marks

and school too

[undone pumice]

damn

[frail stag]

we have sunday and saturday school 😭

[undone pumice]

i kinda understand cus i have asian parents 😭

[frail stag]

ok

[frail stag]

studying this in 8th grade is very imppressive

[frail stag]

😭

[undone pumice]

igs

also integrated 2 (the course before) took in 8th grade

[frail stag]

my parents also expect me to be a topper

[undone pumice]

basically the same thing but a bit easier

[undone pumice]

oof… im not on track to be valedictoriam so igs my parents dont care

[frail stag]

but i only studied like crazy during the first 2 months cuz i felt dumb af at the new subjects

[frail stag]

whats a valedictorian...

?

[frail stag]

now i barely study ah

[viscid thistle]

Person with the highest class in a class

[frail stag]

oh

pro

[viscid thistle]

grades

[winter comet]

bro with the highest class

[frail stag]

i dont think our school does that 💀

[winter comet]

he's just that fancy

[frail stag]

they just have the top 30 kids' names and grades on a banner

free advertisement

[undone pumice]

yeah just #1 grades

[frail stag]

thats literally their motive

for good grades

[undone pumice]

Im like 5th in my class with weighted gpa

[viscid thistle]

I just realized I only have series, paramaterics, and conics before I can abandon this channel for my own studies 😛

[undone pumice]

like 5/1084 or smn

[frail stag]

thats why we have long ahh school hrs

i asked my teachers abt integration once cuz i didnt understand what it was back then

he said it is for area under curve

[undone pumice]

Yeah thats correct

[frail stag]

and said nothing else 😭

[hardy venture]

I can help

[frail stag]

he didnt explain it well

[undone pumice]

So like

[hardy venture]

I can help

[frail stag]

i understand it better now

dw

[undone pumice]

kk

[hardy venture]

Oh ok never mind

[frail stag]

i watched the 3b1b series essence of calculus

peak series

[undone pumice]

Nice nice me too

[frail stag]

they just tossed us 50 formulas of integration and differentiation at us

the moment the lesson started

[undone pumice]

oop wow

[frail stag]

lit not even a basic explanatin

[hardy venture]

I cAn HeLp

[viscid thistle]

for vectors is dot product the msot advanced I go for precalc

[frail stag]

its integrated syllabu

for iit jee

[viscid thistle]

I LOVE THAT SERIES

[frail stag]

same

i finished it just day before yesterday

[hardy venture]

#calculus

[viscid thistle]

what should I learn first, conics, parametrics, or sequences?

[undone pumice]

mm those are all kinda diff kinda hard to pick

[trail hatch]

do conics first

[frank spire]

What's precalculus

[raw hill]

As per the channel description,

[frank spire]

Hmm

[round geyser]

How would we graph this

[swift shuttle]

I'd start by factoring everything

[round geyser]

Guess what

It’s now simplified into x(x+6)/(x^2+4)

[velvet wyvern]

us monoply 🥀

[swift shuttle]

That's a good start

[round geyser]

D: x≠+-2

But their common factors is (x+2) and (x-2)

So HOLE is (2,2) and (-2,-1)

[round geyser]

There is also no HA

And VA is y=0

That’s what I use

[swift shuttle]

That should be enough then, no?

[swift shuttle]

How did u get that tho

[round geyser]

I’ll show you

I flipped

That should be HA

y=0

[swift shuttle]

The thing is

You compute your VAs using your original function, not the simplified versions

So for instance, y = ±2 are potential VAs

[swift shuttle]

If you plug 0 into your original function, you get -2/16 which is a numerical value just fine

So y=0 is not an asymptote

[round geyser]

I meant it’s the power of the numerator is < the power of the denominator is y=0

[swift shuttle]

I'm not sure how else ur meant to graph, but you usually want to define your domain, calculate your limits, study the derivative and extremums and that should be solid ground for drawing

[round geyser]

Numerator is 1 and denominator is 2

[swift shuttle]

Numerator is x⁴ and denom is x⁴

[round geyser]

Y=A/B

[swift shuttle]

You mean the coefficient or?

Actually

Yeah you don't have VAs I'm just wrong

You do have an HA

[round geyser]

Yes

Y=0

[round geyser]

Yes it’s the coefficient

[swift shuttle]

y = 1

[round geyser]

ok

[swift shuttle]

Cuz their coefficients are 1

1/1 = 1

[round geyser]

Ok

Corrected versions

[round geyser]

HOLE is abbreviated from Holes of discontinuity

[swift shuttle]

Yeah that sounds about right

Now chevk for your extremums

Do you even study that in precalc

Yeah I don't think so

So you're pretty much done

[rigid pond]

can somebody help me with this?

I know I’m supposed to isolate the square root

this is what I need help on

[velvet wyvern]

isolate it

then simplify LHS

[lilac moat]

Make pair of RHS numerator you can see that one factor is x=1 then it is (x-1) devide x-1 with numerator so you will get other factor and then just ez square both side

By my math ☠️

[round geyser]

Yes

[rocky silo]

just simplify LHS

[round geyser]

What if we try to do long division on LHS

[rocky silo]

RHS?

[rocky silo]

Grab

Yeah

[round geyser]

The x^2-1

[rocky silo]

Wait no

I don’t think it’s divisible by x+1

[lilac moat]

Ye what problem ?

[round geyser]

Or let’s try to factor 2x^3-5x^2+4x-1

[lilac moat]

I am saying that only

[lilac moat]

See this

[rocky silo]

Wait I get it

[lilac moat]

You have to simplify numerator of RHS

[rocky silo]

U simplify once

Than add 2

[lilac moat]

?

[rocky silo]

Divide by x-1

[lilac moat]

Ye do that you will get more factor

[rocky silo]

Wait

[round geyser]

Is there a domain

[lilac moat]

?

[round geyser]

X≠+-1

X cannot be positive or negative 1

[lilac moat]

Umm i dont mean that we just have to make simplyfy numerator of RHS

[round geyser]

Cross multiply

[rocky silo]

Ok so u get 2x^2-3x+1

The subtract 2 from both sides

Wait no add 2

[lilac moat]

Yep get factor of that also

Tell me what u get

[rocky silo]

2x^2-x+3

Ohh

[lilac moat]

Nooo

[rocky silo]

Wait I need paper

[lilac moat]

Factor that

[rocky silo]

X -1)(2x-1

[lilac moat]

Ye now write numerator of RHS (X-1)^2 (2X-1) AND THERE YOU GO

[round geyser]

Wow

[lilac moat]

?? Smthing wrong ?

[round geyser]

No nothing’s wrong

I graphed these I got

[lilac moat]

Why graphed?

[round geyser]

X can also be ≈1.324

[rocky silo]

U mean left hand side

[lilac moat]

Yee ye mb

I didn't see questions I go with flow

[rocky silo]

Wait ok but how do u do the problem

[lilac moat]

??

[round geyser]

I would subtract the 2 over

Make it the same base

[rocky silo]

That’s what I did

[round geyser]

Then square them

And cross multiply

[rocky silo]

It’s a third and 4th degree

I’m not doing allat

[lilac moat]

That why i said factorise

[exotic flame]

Why is arccos(-sqrt2/2) 3pi/4 instead of 5pi/4

[round geyser]

2x^3-5x^2+4x-1 cannot factor

[lilac moat]

Umm we just did ask @rocky silo

[rocky silo]

Bro what is this

Never seen a problem like this

[round geyser]

I seen that before

[rocky silo]

Range

0 to pi

Can’t be above pi

[round geyser]

Bc sqrt(2)/2 can be 1/sqrt(2)

And draw a triangle

[rocky silo]

Bro

Lock in

What are I talking abt

[round geyser]

And also Cos is =x/r

[lilac moat]

Is there any chance this question has no real sol?

[round geyser]

-1 is x so it’s definitely gonna be in Q2

[rocky silo]

@exotic flame it has a range [0, pi]

[exotic flame]

So arccos of anything cant be none of the bottom half of the unit circle

[rocky silo]

Yeah

Only top half

[exotic flame]

does that apply to arcsin

[round geyser]

And Q2 is pi-theta

[rocky silo]

Also arcsin and arctan are only right half

[round geyser]

Which is pi/4

[exotic flame]

Ahh ok that makes sense

tyvm

[round geyser]

pi-pi/4 is 5pi/4

[exotic flame]

I see now

[round geyser]

Also it can be Q3

[rocky silo]

Bro lock in more what are u talking abt

[round geyser]

I’m doing that also and I’m having a quiz Tmr

[lilac moat]

I got ans it x=1 ☠️

[rocky silo]

Woah how

It can’t be

[round geyser]

And ≈1.32…

[rocky silo]

Cus divide 0

[lilac moat]

Are we talking about your question right ?

[rocky silo]

That’s not my question

That’s some random

Question

[lilac moat]

I am talking about this

[round geyser]

Ic

[lilac moat]

I didn't see that

[rocky silo]

It’s not 1 bro

[lilac moat]

Oh

[rocky silo]

Undefined

[round geyser]

So then it’s ≈1.324

[lilac moat]

Hmm

[rocky silo]

Maybe

[lilac moat]

What how ?

[rigid pond]

How do you factor and simplify the rational expression, isolate the square root, square both sides, and then check for extraneous solutions within the domain?

[lilac moat]

??

[round geyser]

1. inspect their domains

Is you see like 1/x

D: x≠0

[rigid pond]

this is what I was talking about

[lilac moat]

I just Factor numerator and it cut demonic (x-1) so that means x=1 in donain

[round geyser]

I agree let’s square them

Substitute sqrt(3x+1) = y

[rocky silo]

Oh nah please

I’m not squaring that

[round geyser]

I’ll do it

[rocky silo]

Take one for the team

🫡

[lilac moat]

Bro I get it

Ans is undefined due to LHS=RHS

@rocky silo done

[rocky silo]

How

Wait lemme do

[round geyser]

Anyway

[lilac moat]

If u see LHS side got irrational root and RHS got rational root so how ?

[round geyser]

Let’s conclude that it’s x≈1.324

[rocky silo]

Yeah that’s what I got

[lilac moat]

Hence proved LHS not equal RHS

[lilac moat]

Nah

[round geyser]

Look

[lilac moat]

Nah I am not look on that my thoritical work said no mean no no solution that all

I said no real solution but solution can be non real

[round geyser]

Anyway check my work

[lilac moat]

Your ans is non real

[round geyser]

Great

Good to hear

[lilac moat]

By the way what class u all study in ?

[round geyser]

I did these when it was Unit 3 of my Pre-calc

[round geyser]

Pre-calc; radians

[lilac moat]

.... Idk that I am in diff world

Any i will start study after 30 min and my doubt will be continetal lvl stay tuned

[round geyser]

Ok

[lilac moat]

Alr bye

[round geyser]

Btw

[swift shuttle]

Well that's just a pain

I'm not sure there's an easier method to do that

[round geyser]

What is it

[swift shuttle]

I said I'm not sure if there is an easier methid

[round geyser]

I don’t think so only graphing them on a Desmos

[viscid thistle]

stinky precalc

I have to start studying conics and paras

[round geyser]

Cool

[round geyser]

Is it y=a(x-h)^2+k

[bronze geyser]

thats vertex form of quadratic relation

[round geyser]

Yes

What’s this method called

[raw hill]

This is really just substitution tbf

[round geyser]

Yes

[round geyser]

This one is easy to graph correct

[raw hill]

Yes

[round geyser]

The form is y=a|x-h|+k

[round geyser]

If we graph it are we subtracting 27 to the 8

[raw hill]

If you want to solve by finding where a graph intersects the x-axis, then sure.

[round geyser]

What are some other ways

Like separate them into two

[raw hill]

Find where the graph of y=5|2x+1|-8 intersects y=27

[round geyser]

Like y=27 and y=5|2x+1|-8

[raw hill]

Yes you can solve it analytically as well

[round geyser]

Ok

[mellow jetty]

can someone help me do this for my homework, i dont understand how to do this

[round geyser]

Since sec is negative

It will not be in Q1 and Q4

[round geyser]

I’m right

[rough leaf]

this is like idk scc

it’s 3 4 5 triangle

ik thst much

for the tan value

[undone pumice]

If sec(x)<0, then its reciprocal cos must also be less than 0

negative tan can only be q2 or 4

Based on cos value must be q2

[round geyser]

Yep

[undone pumice]

So use that to make a triangle sketch

Make sure to remember only one of the sides is negative label the correct one then solve for hypotenuse and solve for the cos value of a triangle with that tangent ratio

Hope that helps @mellow jetty

[round geyser]

I thought of

set tan^-1 (-4/3) =x

and take the tan

[undone pumice]

You could

[round geyser]

To get tan x = -4/3

[undone pumice]

But I feel like that’s a bit complicated

[round geyser]

3-4-5 triangle

Use that value

cos x = x/r

[undone pumice]

Idk we have problems like the one that person posted on our hw so I solved it like our teacher said

[round geyser]

3/5 and done

[undone pumice]

Remember the sign

As I said earlier

[round geyser]

-3/5

ASTC

Q2 is Sin is positive

[round geyser]

My thought process

Also that’s what I did

But there is no -1

[undone pumice]

Mm icic

[round geyser]

Do you remember the teacher given u a value and u have to figure out the six trig values

[undone pumice]

yeah

[round geyser]

I got that all right to get an 91 on the last unit which is in degrees

[undone pumice]

oop

[round geyser]

I wrote down sin x = y/r, cos x = x/r, and tan x = y/x

On top of my paper

[undone pumice]

mmhm

[round geyser]

Now I’m having another quiz tmr just on radians

And evaluate it

[undone pumice]

icic

glgl

supposedly we have our last quiz of the year on friday

[round geyser]

Yay good for u

[undone pumice]

Yah but after this we gotta prep for the ap exam 🥀

[round geyser]

Okay so

[round geyser]

That’s the example I did

My friend needs help so I said

Draw xy coordinate mark ASTC

[undone pumice]

mmhm

[round geyser]

I told her that 8,10, and 14

Will be have triangle on Q1

And where they’re positive

8 is Q3 and Q4

10 is Q2 and 3

Also with 14

[undone pumice]

Right

[round geyser]

You see negatives value cross out Q1 and where they’re positive immediately

If you see they’re positive, use Q1 and where they’re positive

Like 9

They use Q1 and Q3

[round geyser]

Also my teacher said about next one I’m doing after that quiz is to use calculator for this

[undone pumice]

ah

[round geyser]

I assume it’s in radians