help-27

[cunning valley]

today

work energy and power

[misty crest]

hmm

[cunning valley]

its only a few formulas anyways

I think I can remember them from heart atp

[misty crest]

yea but learning it takes more than passively consuming some videos

[cunning valley]

I did a few problems

[slate kite]

actually i knew how to do calc way b4 gr 10, i just use it as a reference for the easiest things i be doing on the most simple math problems

[cunning valley]

U = mgh

K = 1/2 mv^2

[misty crest]

like i was taking practice tests and doing 40 question long mcq pages per unit

[cunning valley]

Ui + Ki = Uf + Kf if theres no non conservative forces

[misty crest]

after reading a textbook and watching khan academy videos

to learn physics you need to do problems

not just glance at the formulas and watch a few videos

[cunning valley]

Ui + Ki + Wother = Uf + Kf

then obv W = Fd

W = F cos theta d

[cunning valley]

I did do rpoblems

I did the problems at the end of the chapter

princeton review textbook

they were alr I could do more problems tho

[misty crest]

oh those books can be pretty quick actually

[cunning valley]

yeh

[misty crest]

i didn’t use them to learn the material though

[cunning valley]

I watched a organic chemistry tutor video at 1.5x speed

aswell

then did the textbook

exact same concepts

some examples

[misty crest]

you should do more than just that

like download practice tests per unit

[cunning valley]

true yeah

[misty crest]

i forgot where but i had like 30-40 mcq questions per unit for those two physics classes

college board questions

[cunning valley]

so you did like 2 units a week?

during a break though right

[misty crest]

yea

[cunning valley]

or 3 units a week

[misty crest]

i was literally doing it 12 hours a day

[cunning valley]

I find it hard to focus for longer than 10 ngl

after 8-9 hours I start tweaking

[misty crest]

uhh idk what my pace was like

i think i was doing around 1 unit a day

🤔

[cunning valley]

yeah school teaches slow af

[misty crest]

yea i think it was 1 unit a day now that i think about it

[misty crest]

indeed

[cunning valley]

you got 5 on both exams?

[misty crest]

yessir

just don’t ask me what i got on my ap history exams

shh

[cunning valley]

hmm

what if

I pass english

[misty crest]

wasn’t that bad tbh

double 4s

[cunning valley]

but I don't do early admissions

and I actually lock in next year

bro my school only lets grade 12s do calc/physics 2 exams usually

[misty crest]

i mean if it’s just english then it shouldn’t hurt you that bad

[cunning valley]

so if I can self study these 2 and get 5s

next year will be pretty much guaranteed 98% gpa

[misty crest]

it’s more important that you have great letters imo and grades in stem

[cunning valley]

cuz I'll just be doing review of the shit i self studied next year

[cunning valley]

which teacher do I glaze then

physics?

idk

I handed in two of my labs late

2/2 labs late

[misty crest]

i got letters from my ap calc teacher, ap chem teacher, and ap physics c mech teacher

[cunning valley]

do you try getting letters from grade 12 teachers

or grade 11

it might be too late for grade 11 bruh

[misty crest]

well there was some overlap for me

[cunning valley]

I slept in biology already

hmmm

my math teacher

idk

theres like fucking 40 people in my class

[misty crest]

i mean if you’re really this physics enthusiast then shouldn’t your teacher be aware of this

[cunning valley]

lmao idk

I only started learning physcis this year tbf

he said no one has gotten 100% on one of his ap physics tests before

so maybe I could be the one\

[misty crest]

bruh

are people in your school dumb

[cunning valley]

I doubt it though those AP questions are kinda conceptual af

[misty crest]

😭

[cunning valley]

idk

[misty crest]

no one got a 100 once?

🤔🤔🤔

[cunning valley]

yeah that's what he said

HGMMMm

[misty crest]

i mean it’s not like getting a 100 on the ap exam

[cunning valley]

wait so

physics 1

will be

[misty crest]

it’s just a classroom test

[cunning valley]

1 semester only

then the school makes you review yourself

[misty crest]

why?

bro what

[cunning valley]

idk

[misty crest]

that’s dumb as hell

[cunning valley]

so we're tryna learn from cirriculum

the cirruculum and ap physics

overlap like

5 units I believe

[misty crest]

curriculum

[cunning valley]

6 units

then they teach you some ap topics at the end

so I was thinking

I do really good on all his ap tests

then after the semester ends

I go up to him in practice sessions

for literally no reason

and ask a bunch of questions

for literally no reason

and maybe he'll see how hard working I am!

[misty crest]

i mean don’t say stupid shit

[cunning valley]

yeah

hmm

[misty crest]

like pop science questions

💀

[cunning valley]

I already slept in my biology class

but biology is full year

hmmm

I think its still recoverable

i think

[misty crest]

you say you study math/physics all day

what books?

🤔

[cunning valley]

hmm I also have

chemistry tomororw

NO

not tomorrow

next semester

idk who my teacher is

but maybe that

[cunning valley]

I just do competitive math problems

but these 2 days not rly

[misty crest]

did you take the exam last week

[cunning valley]

AMC?

[misty crest]

yea

[cunning valley]

no

[misty crest]

should’ve

[cunning valley]

BECAUSE

there was a strike from the teachers

from literally eaerly october

all the way to just past the deadline

so there was no timeframe to even sign up

tried emailing but they were locked out of accounts

I took another one though

I think I did horrible

[misty crest]

hmm

[cunning valley]

Yeah I did horrible because the exam rules tell you to do everything in exact values

[misty crest]

unfortunate

[cunning valley]

but

I didn't really know what to do for uh

geometry that required trig functions that weren't exact values

like 37 degrees or some shit

[misty crest]

isn’t that just 3-4-5

🤔

[cunning valley]

yeah

wait

I know I got the right answer

but I just wrote

the entire fucking solution

in a single equation

so think of all the adding/subtracting/similar triangles

all in one fucking equation

and when you enter it into your calculator

you get the exact value

okay sorry

I'm yapping

[misty crest]

lol

do you want to major in math

[cunning valley]

maybe

I'm considering it

but my parents

[misty crest]

engineering

[cunning valley]

want me to major in engineering

yes

[misty crest]

yes

[cunning valley]

all my siblings

[misty crest]

figured

[cunning valley]

majored in engineering too

biomedical engineering for one

electrical engineer for other

[misty crest]

i mean

[cunning valley]

mechanical engineer for other

[misty crest]

if you don’t want to be a professor

then it might not be a bad idea to just do engineering

[cunning valley]

well the thing is

if I were to become a professor

[misty crest]

and take math classes as a second major or minor

[cunning valley]

like a math professor

I'd just be a professor

like I wouldn't accopmlish any generational shit

just a professor

[misty crest]

big if here

[cunning valley]

which is kinda sad

[misty crest]

getting into a good phd program is hard enough and then to land a good position as a professor is even worse no

[cunning valley]

like

[misty crest]

yea probably not

[cunning valley]

yeah

[cunning valley]

yeah I don't feel like its worth it

I'm better off becoming a doctor or sum

similar amount of years

4x higher salary on average

[misty crest]

i mean there are things you can do with a math background other than research at university

[cunning valley]

true

wait did you self study BC at all

[misty crest]

i mean i took it in class but i wouldn’t say i actually learned it in class

[cunning valley]

how long is it gonna take me to learn

[misty crest]

my situation was that i learned it the summer prior and just did homework in class then fucked off to do whatever i wanted

[misty crest]

eh this depends

i think if you’re really dedicated you can learn it quite quickly

[cunning valley]

I'm like

[misty crest]

but you’re also juggling it with other things

[cunning valley]

80% dedicated

[cunning valley]

yeah

mostly just biology

and physics

biology I don't think it's rly that bad

and also the other course

[misty crest]

i mean just put whatever time you have available towards it and don’t worry about how long it takes tbh

it shouldn’t take that long

[cunning valley]

I js dont want it to be like

late april

and realize I can barely get a 4

[misty crest]

why not just start now

[cunning valley]

coz like I said I gotta self study rest of physics 1 and all of physics 2

[misty crest]

if you like math

🤔

[cunning valley]

yeah I did start now

what

I learned limits

and continuity

this weekend

i think I know a few things about derivatives

just from being around math a lot

I know the power rule

might know chain rule

rolles theorem

but eh not that much

in depth

[misty crest]

are you using a book or just watching bprp

[cunning valley]

using princeton review

decent book ngl

I think I can pull this off in 12 weeks

[misty crest]

don’t think that’s sufficient to fully learn something tbh

[cunning valley]

ill practice with past mcqs frqs too

and search practice tests

n stuff

I think by the end of the year

I'll probably

lets see

I think done differentiation and onto integrals

then by march def done everything

along with review and whatnot

.close

.close

.closed

[devout snow]

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[cunning valley]

.solved

[devout snow]

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[crimson nova]

wwhy is understandinf how f f' f'' so hard

[crimson nova]

i need some like ways to make it easier to understand

[graceful cosmos]

Just to kinda ask the stupid question:
Do you know how f' relates to f? Like, what f' is describing?

[crimson nova]

f' is slop

e

is this a good way to memorize for ap test

like the middle part of it

[devout snow]

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[carmine elm]

How does one get to the first step?

[hollow jolt]

What happens if you call the new x, (x-1)?

Thats basically whats happening

[carmine elm]

so is it basically letting x = x-1

Oh wait is that the transformation

Ohh

Ty

[fossil locust]

no worries

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[errant harbor]

!status

[devout snow]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[fierce matrix]

1

I know where to begin

I have begun!

[young spade]

If you did some work on the problem already, please show, even through photos

[fierce matrix]

I think the intersection of intervals are important

if piecewise the function is continuous, then it is continuous

for example at x=-1 the first two pieces should connect

and at x=2 the second two

and each fn part be continuous

smooth?

[errant harbor]

Correct so far

[errant harbor]

Using this information we can set up an equality

Same thing for x=2

[devout snow]

@fierce matrix Has your question been resolved?

[devout snow]

@fierce matrix Has your question been resolved?

[fierce matrix]

@errant harbor thx, g2g, I continuoues later, thx

🌿

[devout snow]

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[drowsy vessel]

Hey guys! Can anyone explain to me how to study the convergance of the following

[supple knot]

use squeeze theorem

you also need to explain limit of where x is going

[drowsy vessel]

Oh yeah limit is attempting infinity

How would i use the squezze theorem

For this

[winged tapir]

-1 ≤ sinx ≤ 1

[drowsy vessel]

Huh

[winged tapir]

Multiply all three parts with (x+1)/(2x+1)

[drowsy vessel]

Okay and then i have to find the limit of each part.

?

[winged tapir]

Yup

[drowsy vessel]

aha! Thank you so much!! I get it now

[mystic scarab]

(x is approaching infinity 😉 )

[winged tapir]

Cool

[drowsy vessel]

Sorry, i learn this thing in french

[mystic scarab]

Yeah don't worry, I just corrected so that you can learn the right term 🤗

[devout snow]

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[devout snow]

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[waxen hawk]

I made some changes but now some boxes are empty. Can someone let me know if something’s wrong because I feel like there shouldn’t be 3 empty boxes and duplicates but I couldn’t find anything for root test

[waxen hawk]

Everything for root test seems like it has a better spot or ends up equaling 1 and being inconclusive

[terse orchid]

there are multiple possible spots for most of these

are there any rules for how you should prefer one to the other

[waxen hawk]

No the top part is the only instruction

[terse orchid]

then as long as the box you put it in works they shouldn't mark it wrong

[waxen hawk]

Ok

[terse orchid]

for example P can go to the bottom right but it's a more annoying test

geometric series is literally root test

and ratio

[waxen hawk]

Ok yeah I definitely though there was overlap with the tests

.close

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[thin fern]

I feel like I understand why it's true conceptually (e.g. f(x) = x^3 at x=0) but I'm not sure how to formally write a solid proof for this

[oblique belfry]

Write a limited development around that point and use the argument on x^3

[thin fern]

wdym limited development

[grand warren]

taylor’s formula i guess

[oblique belfry]

Taylor expansion

Taylor young in particular

[grand warren]

seems like a french expression 😭

[oblique belfry]

Yes I always forget

[thin fern]

so this might be a bad question but how do I do that 😔

this is taylor's formula from my lecture notes

[oblique belfry]

$$f(x+h) = (d^3f(x)/6)h^3 +o(h^3)$$

Tell me about the behaviour of d^3f(x) then conclude by comparing it to the residual when close to x

[woven radish]

bourbactive

[thin fern]

Is what you're trying to go at is, depending on what direction h is, the value of f could go up or down?

hence it's not a local extremum

(this is kind of just a guess tbh based on the result we're trying to prove)

[oblique belfry]

Yes that's it. It will either increase or decrease around X because x^3 changes sign around 0

[devout snow]

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[rocky marlin]

Need help either #5

[terse orchid]

To graph (a) you need to know the general shape of curves in the form 1/x

and plug in nice values of x to get points that are easy to graph

for example x=4 gives you f(4)=5

x=2 gives you f(2)=3

[rocky marlin]

So how can you find those nice points, and also what is the general shape of this equation?

[terse orchid]

This is $y=\frac{1}{x}$

[woven radish]

flynger

[terse orchid]

If you want to think about why

[rocky marlin]

Why is there two lines?

[terse orchid]

lets look at numbers x > 0

1 / x is positive if x is positive

does that make sense?

and since we're doing division, the smaller the number we divide by, the bigger the result 1 / x is

1 / 0.1 is bigger than 1 / 1 which is bigger than 1 / 10

so it gets bigger as you go from x = 1 to x = 0.5 to x = 0.1

and smaller when you divide by a bigger number

[rocky marlin]

Ah ok so when its negative it keeps shrinking and from positive it keeps shrinking but im wondering why its not just one line

[terse orchid]

let me ask you this

what is 1/0.1 equal to?

[rocky marlin]

10?

[terse orchid]

yes

and what is 1/0.01

[rocky marlin]

100

[terse orchid]

1/0.001

[rocky marlin]

1000

[terse orchid]

so as you get close to x=0 from right, 1/x goes to infinity

[rocky marlin]

So will continuously go hugher as it gets closer to 0 but it will never reach zero?

[terse orchid]

Exactly!

and we cant divide by zero so theres no point at x=0

[rocky marlin]

Ok so how would i go about graphing this with these other numbers in the equation?

[terse orchid]

first ill answer this

for negatives its the opposite

1 / -0.1 = -10

1 / -0.01 = -100

1 / -0.001 = -1000

and on and on

so it goes down forever to -infinity

when you approach 0 but from the negative side

thats why theres two curves as they are not connected

[rocky marlin]

Ah ok and so i was thinking to read this as graphing form but im not sure if it is

[terse orchid]

do you know what happens for a function $f(x)$ if you graph $f(x-a)$ instead?

[woven radish]

flynger

[terse orchid]

you are subtracting every x by a

[rocky marlin]

Uh im a little lost

[terse orchid]

lets say we have a line

f(x)=x

if you look at f(x)-1 that is equal to x - 1 right?

so the y value is subtracted by 1 at every point

we brought the graph down by 1

[rocky marlin]

Ah ok

So your subtravting 1 from the y

[terse orchid]

Yea sorry i meant $f(x)-1$

[woven radish]

flynger

[terse orchid]

have you learnt quadratics yet?

[rocky marlin]

Ye

[terse orchid]

this is $f(x)=x^2$

[woven radish]

flynger

[terse orchid]

What will happen if I graph $y=f(x)+3$ instead?

[woven radish]

flynger

[rocky marlin]

It will move up three

[terse orchid]

yes exactly!

So how do we move it horizontally?

the secret is instead of changing $f$ which returns the y-value from the outside we have to change the input $x$

[woven radish]

flynger

[terse orchid]

For example if I want to move the graph to the right by 1

[rocky marlin]

So would it be f(x-1?

[terse orchid]

yes

that will shift the graph to the right

$f(x-1)=(x-1)^2$

[woven radish]

flynger

[terse orchid]

because 1-1 = 0

is the new vertex

[rocky marlin]

In the equation that didnt happen

[terse orchid]

This same logic applies for our graph $f(x)=\frac{1}{x}$

[woven radish]

flynger

[terse orchid]

What will $f(x)-1$ look like if the below is $f$

[woven radish]

flynger

[rocky marlin]

Shift left 1

[terse orchid]

are you sure

im talking about $y=f(x)-1$

[woven radish]

flynger

[terse orchid]

we're changing the output value $y$ not the input

[woven radish]

flynger

[rocky marlin]

Plus 1?

Wait wby is it y=f(x)-1?

[terse orchid]

it will go down by 1 because we are subracting by 1

I'm saying if $y$ was equal to that INSTEAD of $f(x)$

[woven radish]

flynger

[terse orchid]

So if we subtract the right side of $y=f(x)$ to get a different graph

[woven radish]

flynger

[terse orchid]

Are you more comfortable thinking in terms of functions?

$g(x)=f(x)-1$

[woven radish]

flynger

[terse orchid]

If we copied f but moved everything down by 1

does that make sense?

[rocky marlin]

Yes but im just wondering how to do the the equation f(x)=1/x-3+4

[terse orchid]

ok then answer this real quick

$g(x)=f(x)+b\$
$h(x)=f(x-a)$

[woven radish]

flynger

[terse orchid]

How are $g,h$ related to $f$?

[woven radish]

flynger

[rocky marlin]

H(x) is minus whatever compared to f and on the x axis g(x) is plus whatever on the y axis compared to f

[terse orchid]

We would say

$g$ is the graph of $f$ moved up by $b$

$h$ is the graph of $f$ moved right by $a$

[woven radish]

flynger

[terse orchid]

if the $b$ was negative then we are moving down by -b

[woven radish]

flynger

[rocky marlin]

Ok so how would we do my problem

[terse orchid]

if the $a$ was negative then we are moving left by -a

[woven radish]

flynger

[terse orchid]

How is $\frac{1}{x-3}$ related to $\frac{1}{x}$

[woven radish]

flynger

[rocky marlin]

So its shifted left three up 4?

[terse orchid]

What do you do to the graph

shifted right three up 4

[rocky marlin]

Cus i wasnt sure if its that or just up 4 for a y intercept

[terse orchid]

its shifted right because everything copies the value 3 from its left in the graph $\frac{1}{x}+4$

[woven radish]

flynger

[rocky marlin]

Ok and because the a value isnt negative its not flipped?

[terse orchid]

yes since a is positive its to the right

if it was $\frac{1}{x+3}$ that would be 3 to the left

[woven radish]

flynger

[terse orchid]

But thats the explanation of how the 3 and 4 translate the graph

to draw it

just plot points 1 or 2 apart

until it starts to look like the graph of 1/x

[rocky marlin]

Ok and now for the second one do i just plug in points that can be fully rooted?

[terse orchid]

yes

by the way

what values of $x$ can you put in?

[woven radish]

flynger

[terse orchid]

into the square root

what's the domain?

what numbers can you take the square root of

Can you take the square root of $-3$, $0$, $3$? what makes sense

[woven radish]

flynger

[terse orchid]

you should say stretches vertically by 3

because it could also be stretched horizontally, you should specify

And in your homework below, $\infty$ is not a number, so you should not do $=$

[woven radish]

flynger

[terse orchid]

Never do something like $x \le \infty$

[woven radish]

flynger

[terse orchid]

because $x$ cannot equal $\infty$

[woven radish]

flynger

[terse orchid]

you should do $-\infty < x < \infty$

[woven radish]

flynger

[rocky marlin]

Ahhhh ok

Tysm for the help

[devout snow]

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[restive river]

Explain why an m × n matrix with more rows than columns
has full rank if and only if its columns are linearly independent.

[tall knoll]

what's your definition of "full rank"?

[restive river]

in this question full rank means rank = n

[tall knoll]

alright

and how do you determine the rank of a matrix?

[restive river]

by dim col and dim row i guess?

[tall knoll]

if you want, but now the question becomes how do you compute this "dim col"

i assume you use that to mean the dimension of the column space

[spring oasis]

col is the span of the columns

row is the span of the columns

if the dimension of the span of the columns is greater than the dimension of the span of the rows

then according to your definition it won't be full rank, no?

[terse orchid]

$\dim(\text{span}{v_1,...,v_n}) =|B|$ for any basis $B \subseteq {v_1,...,v_n}$

[woven radish]

flynger

[terse orchid]

Thats the basic definition of $\dim$ of a vector space

[woven radish]

flynger

[devout snow]

@restive river Has your question been resolved?

[devout snow]

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[potent compass]

can i get help w this

[fossil locust]

this is just the unit circle definition

y = sin theta and x = cos theta

[terse orchid]

yup

[potent compass]

but how do i solve this

[crystal dawn]

when is sine negative? when is cos negative?

[potent compass]

sin is negative in the 3rd ad 4th quadrant

cos is negative in the 2nd and 3rd quadrant

[pseudo basin]

ok and which one are they BOTH negative in

[potent compass]

3rd quadrant

[potent compass]

does it become negative bc you minus both sides by cos?

[fossil locust]

wdym by 'minus both sides by cos'

[potent compass]

actually imn not sure

i wanna know how they become negative

or why they are negative

to chose the quadrant

[fossil locust]

do you understand this step?

[potent compass]

yes

but hwy are they negative

[pseudo basin]

are you familiar with the unit circle as a concept

[potent compass]

no

i dont think so

[pseudo basin]

ok then you have to go learn that

[fossil locust]

this is already the definition

[pseudo basin]

bc its not really gonna work with some random lady on discord yapping at you about it

but ok let's step back from that

are you familiar with the coordinate plane @potent compass

or the cartesian plane or the xy plane or however you call it

[potent compass]

yes

[fossil locust]

cos(theta) = adj/hyp = x/1 = x

[potent compass]

oh yes ik tht

[pseudo basin]

ok right

do you know the quadrants on the plane and can you correlate signs of x & y with quadrants

[potent compass]

yes

[pseudo basin]

the idea's the EXACT SAME with trig shit, only now your "x" is called "cos(θ)" and your "y" is called "sin(θ)".

[potent compass]

ic

but what makes them negative in the wauestion

to look for the quadrant tht they are negative in

[fossil locust]

the question says that sin(theta) < 0 and cos(theta) < 0

[fossil locust]

now using this, you have y < 0 and x < 0

[pseudo basin]

"negative" means the same thing as "less than 0" @potent compass

[potent compass]

ohh

tysm

Miss

what about this pls

for number 5.2.2

would i use the angle (2)

to find st

but it doesn’t have a value

[devout snow]

@potent compass Has your question been resolved?

[crystal dawn]

assuming you meant the top (2), you can find it by first finding the (1) next to it.

[devout snow]

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[analog sonnet]

so like if i have an ion its basically just an atom but with or without one or more electrons right?

[fossil locust]

an ion can also have more electrons than the original atom

it just can't have equal numbers of electrons and protons

so it always has a charge

[fossil locust]

the proper names for this are cations and anions

ca+ions, if that makes sense

[analog sonnet]

thanks man

[fossil locust]

so that would have fewer electrons, cause that's less negatively charged

also I think they'll ask you on the test:
atoms and ions have the same number of neutrons and protons
different numbers of electrons

no worries!

[analog sonnet]

.close

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[gleaming hill]

im stuck here

what should i do

[grand edge]

you're meant to arrive at your "k + 1" statement from the inductive hypothesis, not conclude it is true without any justification

[errant harbor]

Hint : $\sum_{i=1}^{k+1} = \sum_{i=1}^k + k + 1$

[woven radish]

Matcha

[grand edge]

yeah, the key is to always use the inductive hypothesis to show that the inductive step is true

[errant harbor]

So, we know that $\sum_{i=1}^k = \frac{k(k+1)}{2}$

[woven radish]

Matcha

[errant harbor]

Then we need to use this to achieve $\sum_{i=1}^{k+1} = \frac{(k+1)(k+2)}{2}$

[woven radish]

Matcha

[errant harbor]

Got it @gleaming hill ?

[gleaming hill]

mm

[errant harbor]

You can ping me if you get stuck

[gleaming hill]

@errant harbor

i dont understand this part

[errant harbor]

Try to write out what $\sum_{i=1}^{k} i$ is

[woven radish]

Matcha

[errant harbor]

@gleaming hill ?

[gleaming hill]

this rihgt

[errant harbor]

No- I meant write it out, like 1+2+ blah blah

[errant harbor]

You'd like to understand why this is true right?

Or am I misunderstanding your question

[gleaming hill]

i didn't know it was like that

but now ik :)

i think i get it

i'll keep the chat open until i solve it, i hope that's fine :)

can i ping you later if i have more problems? @errant harbor

[errant harbor]

Sure

[errant harbor]

Yep

[gleaming hill]

i think i solved it

[errant harbor]

Nice work

[gleaming hill]

this is it right?

[errant harbor]

Your notation is wrong

[devout snow]

@gleaming hill Has your question been resolved?

[gleaming hill]

Wait what would be correct then

@errant harbor

[devout snow]

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[drowsy vessel]

I had to find the transitional matrix from B canonique to the given B (first line), can somebody tell me if i did it right

[devout snow]

@drowsy vessel Has your question been resolved?

[devout snow]

@drowsy vessel Has your question been resolved?

[devout snow]

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[woven radish]

Renato

[little totem]

what's G?

[faint gorge]

i think was cyclic groups and then of order 2n+1

[spring oasis]

it's the group of nth roots of unity the Gn

can i get a helping hand with 3

[little totem]

so there are two parts to break down here

first the "such as" part with G i asked about

can we figure out what are valid values of n here?

once youve figured that out, how does the summation further restrict the values of n?

any step here you're stuck on?

[spring oasis]

how

[little totem]

well, we know that whatever group w^6 is an element of, its order must be odd, yeah?

so lets pick a random odd number, say like 5

is w^6 an element of G_5?

[spring oasis]

yes

i mean no

6 = 1 (mod 5)

maybe?

[little totem]

first of all, can you simplify w^6?

they told you what w was, yeah?

[spring oasis]

yes

[little totem]

what do you get when you simplify it

[spring oasis]

w^6 = ^{12pi.i/33}

I'm not sure wdym by simplify

w^3^2

w^3 = e^{2.i.pi/11}

w^6 = e^{4.i.pi/11}

[little totem]

yes good

[spring oasis]

now what

[little totem]

now lets call w^6 = v for convenience

lets take the roots of unity of order 11

can you visualize what the elements look like?

[spring oasis]

v^11 = 1

@little totem

[spring oasis]

just wk = e^{2k.i.pi/11}

with k in {0,..,11}

[spring oasis]

you here?

[little totem]

yeah

yeah thats good

so we can see that v is in G_11, yeah?

[spring oasis]

G66 contains G11 is your point?

[little totem]

the other way

G66 contains G11

but yes

[spring oasis]

mb

then what?

[little totem]

so we know v is in G11

what are all the Gk such that v is in Gk?

where k is odd

[spring oasis]

ahh who the fuck knows at this point

v is w^6

v is in G6 implies G66

[little totem]

uhh w^66 is a number

did you mean G66

yes v is in G66

is v in G33?

i just remembered that G66 doesn't help us because we need an odd order

[spring oasis]

v is in G66 implies G22 is in G11, G33, G44, G55

[little totem]

yeah see the pattern?

[spring oasis]

no

[little totem]

ok your reasoning is written weirdly

ok lets back up a bit

[spring oasis]

w is in G33

that is what is given

[little totem]

here

when k=2, we get our v

and so v is in G11

[spring oasis]

w is in G3, G11, G33

[little totem]

good

er, not G3

write out the elements of G3

is w in there?

oh i went back to read, w isnt in G11 either

[spring oasis]

let x be in G3, then x = e^{2.k.pi.i/3}

[little totem]

yeah, what k would give us w?

[spring oasis]

k is an integer so imposible

[little totem]

exactly

so w is not in G3

nor is it in G11

[spring oasis]

its in G99

[little totem]

yep!

[spring oasis]

G66

your point?

[little totem]

so what is v in then?

we said v is in G11

is v in say G22?

[spring oasis]

w^6 = e^{4.i.pi/11}

v is in G11, G22, G33

[little totem]

bingo!

[spring oasis]

wait

[little totem]

so what are all the orders of G that v can be found in?

[spring oasis]

which k gives it that v is in G22

[little totem]

ok going with this definition

but adjusting for G22 instead of G11

[little totem]

just wk = e^{2k.i.pi/22}

k=4 gives us v, yeah?

[spring oasis]

correct

[spring oasis]

yes

[little totem]

clear now?

[spring oasis]

ye

[little totem]

ok cool

did you figure out in general what G's v is in?

[spring oasis]

33 | 2n+1

[little totem]

uhhh "33 such that 2n+1"

can you rephrase that?

[spring oasis]

divisibility

[little totem]

is v in G44

[spring oasis]

no

[little totem]

did you check?

[spring oasis]

wait

wait

w^6 = e^{4.i.pi/11}

[spring oasis]

w^6 is in G11 yes or no

[little totem]

yes

w^k = e^{2k.i.pi/11}
k=2

[spring oasis]

yes

[little totem]

ok so now is the pattern clear

[spring oasis]

no

[little totem]

lemme explain how to visualize this then

G is a cyclic group, yes?

[spring oasis]

y

[little totem]

so we should be visualizing a circle when doing these problems

imagine you drew 11 points around in a circle, evenly

starting at z=1 and going counterclockwise

these give us the G11 roots of unity

can you visualize this?

have you done this before?

[spring oasis]

w is in G33,
w = {2pi.i/33}
w3 ={2.pi.3.i/33} = {2.pi.i/11}
so G11 is contained in G33

[spring oasis]

ok

[little totem]

ok so, we say that v is an element of G11

because it coincides with one of these 11 points we drew

does this make sense so far?

[spring oasis]

yes

[little totem]

ok good

so now if we draw 22 points instead of 11

[little totem]

can you see that we are basically just taking the 11 points and just drawing points in between

we are not taking away or moving any of the existing points

we are simply adding more points in between

this makes sense?

[spring oasis]

yes

[spring oasis]

yes

[little totem]

so if v coincided with the 11 points, they definitely coincide with the 22

yes?

[spring oasis]

y

[little totem]

can you now see that here with 22, we split the segments between points in 2

but we could have split the segments by any number of points

we could have split each of the 11 regions into 7 parts

making 77 points around the circle

that doesn't change that v coincides with G11

and so adding points without removing points makes it still coincide with G77

following?

[spring oasis]

sure

[little totem]

so any multiple of 11 works

G44, G55, G66, G77, etc

v is part of all of them

because v is in G11

[spring oasis]

can we start from the scratch

[little totem]

sure

[spring oasis]

w is in G33

w^6 after simplification we notice it's in G11

so his conjugate of w^6 is also in G11

[little totem]

oh i didnt even see the conjugate, but fortunately it changes nothing about the problem

[spring oasis]

now we want this N to be in an odd cyclic group that is divisible by 11

what now? 11 | 2n+1

?

[little totem]

we just did the hard part of the first step

lets now look at the second part, the summation

notice that no matter how you evaluate this sum, you have to start at j=0

and just keep going up

yea?

[spring oasis]

sure

reminds me of geometric series

[little totem]

yeah

ok so now if you want, you can simplify the summand inside the summation

lets do that first, because it will make everything a lot simpler to see

[spring oasis]

?

either n = 1 or n is not 1

if n is not 1 we can use the geometric series formula for getting rid of the summation

if n = 1 then we can compute the sum manually

[little totem]

ignore the summation for now

just simplify w^11j

how would we write w^11j in a nicer way

[spring oasis]

(w^11)^j

[little totem]

ok and we have w, right? so we can write out what that is

what w^11 is

[little totem]

nice!

now lets look at the summation now

ignore the n+4 for now

for now lets just put a k there

sum from j=0 to k

lets pretend k is 0 to start

evaluate the sum

this should be very easy

[spring oasis]

no idea

[little totem]

you know how to evaluate summations right?

can you do $\sum_{j=0}^0 4$

[woven radish]

Cozmogrgdfschkipkhrshtensi

[spring oasis]

4

[spring oasis]

trick question?

[spring oasis]

i am telling you it's better if we use geometric series no?

[spring oasis]

if it's a k instead of a n it's zero still

[little totem]

you mean 1

because anything to the 0 power is 1

you forgot that roots of unity go around the circle, they never go to the origin

[spring oasis]

ah

[little totem]

but sure, you can try a geometric series

lemme know what you get with an arbitrary k as the sum upper limit

[spring oasis]

i don't get it

[little totem]

then lets not worry about the geometric series and just take it step by step

now we evaluated the sum up to 0

which was trivial

evaluate the sum up to k=1

you dont really have to evaluate the whole sum

you just need to roughly visualize this

[spring oasis]

w^11

[little totem]

1 + w^11

[spring oasis]

that

[little totem]

and w^11 we said was e^2pi.i/3

so can you see that this is definitely not 0

lets once again use z = w^11 = e^2pi.i/3 for convenience

[spring oasis]

okay

[little totem]

now lets evaluate the sum for k=2

so we take our 1 + z

and we add one more term

this time, i do want you to evaluate the entire sum

can you figure out what it is?

[spring oasis]

w^6

[little totem]

why

[spring oasis]

w^22 = w^6

no?

[little totem]

i dont think so? why is this true

also it would be much easier to think in terms of z instead of w

we said w^11 as z

no need to keep using w

[spring oasis]

lets once again use z = w^11 = e^2pi.i/3 for convenience

w^6 = e^{4.i.pi/11}

ook my bad

[spring oasis]

w^22 = e^4pi.i/3

N=7?

@little totem

[little totem]

uhhh

you could have just done z^2

lol

[spring oasis]

yes

[little totem]

ok so hold up

before making points about N

what is the sum 1 + z + z^2

we didnt evaluate this yet

[spring oasis]

1 + w11 + w22

[little totem]

again, no need to go back to w

it only makes things harder

[spring oasis]

how

[little totem]

its much easier to work with
e^2pi.i/3
than
e^2pi.i/33

why go to the one with bigger denominators

[spring oasis]

it's a geometric series

[little totem]

right, so evaluate it

[spring oasis]

z^(2+1) -z^0 / 2 - 0

z^3 - 1 / 2

[little totem]

z^3 should be easy to evaluate right?

keep going

[spring oasis]

lets once again use z = w^11 = e^2pi.i/3 for convenience

w^33 = 1 because w is in g33

[little totem]

hey

look at what z equals

[spring oasis]

the sum is zero

[little totem]

directly use that and evaluate z^3

dont bother converting to w

[spring oasis]

z^3=1

[little totem]

yeah, see that?

there is no need to go back to w to evaluate that

there is nothing special about w that you have to base everything off of it

ok so

another way to understand why this sums to 0

z is an element of G_n

whats the minimum n here?

[spring oasis]

n is in N

[little totem]

what is the fewest points around a circle you need to coincide with z?

[spring oasis]

2

z0 and z1

[little totem]

close

the exponent is 2pi/3

it divides the full circle (2pi) into 3 parts

G3

[spring oasis]

this is G3

[little totem]

remember this?

visualize 3 points around the circle in a triangle

z is the top point

[spring oasis]

so minimum Gn is G3

[little totem]

yeah

so we are adding

1 + z + z^2

can you see that that is simply the 3 evenly spaced points around the circle?

[spring oasis]

okay, so

how do i do my problem

[little totem]

did you follow the steps so far? i promise ill get to your problem

but im trying to lay out the intuition for you

so that you are able to understand what is happening better

[spring oasis]

Y

[little totem]

youll be able to better understand the other problems

ok so

since they are the 3 evenly spaced points around the circle

the sum is 0

because they all cancel with each other

you can think of it as saying the center of mass is at 0

we summed from j=0 to 2

if we sum from 0 to 3, what happens?

[spring oasis]

1+z + z^2 + z^3= 0

[little totem]

thats sum up to 2, not 3

ok so your left side is correct now

but this sum is not 0

what is z^3?

[spring oasis]

1

[little totem]

try again

yep

roots of "unity" meaning 1

these are evenly spaced around the unit circle

they can never be 0

[spring oasis]

z is in g3

[little totem]

yeah so now

if

1+z+z^2 = 0

can we see that