help-27

[copper spoke]

but i have problems with the algebra

[stone stump]

ok so what is a_4

and what is a_5

do not simplify

[stone stump]

good so thats a_4

[copper spoke]

is this true

[stone stump]

now a_5

not quite

[copper spoke]

i get the idea

[stone stump]

$1\cdot3\cdots(2n+1)=\prod_{k=0}^n (2k+1)$

[woven radish]

Denascite

[copper spoke]

jeah but i get the idea

its just the algebra

I mean i have a_n+1 , thats why i have the term (2n+3)^2 , but in the terminator why there is another (2n+1)^2

thats my problem

[stone stump]

humor me

[stone stump]

you forgot the ()^2 btw

now a_4/a_5

[stone stump]

now we notice that 2*5+1=2*4+3

and we have our (2*4+1)^2 factor before that

aka (2n+1)^2

[copper spoke]

ahhh

lel

you can write

2 * 5 + 1 as 2 * 4 + 3

[stone stump]

2(n+1)+1=2n+3

[copper spoke]

thank

you

you really need to write it down one time to really get it

[stone stump]

yes

[devout snow]

@copper spoke Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

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[lofty verge]

How do you even draw this?

My guess is that two points of the parallelogram lie on the circumference of one circle, the other two points on the other circle. These circles are congruent and touch in exactly one point on their circumference

A third circle is drawn with all points on the parellogram nvm thats wrong

like that?

took me forever to draw but i think this is more correct

after i saw definition of circumscribed

i mean its not perfect but

close

but even then i dont see how to find the area of any of those circles

even better drawing ^

[devout snow]

@lofty verge Has your question been resolved?

[lofty verge]

Figured it out using AI (no hope for humanity 😭)

.close

[devout snow]

Channel closed

Closed by @lofty verge

Use .reopen if this was a mistake.

[rain summit]

!noai

[devout snow]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[rain summit]

READ

[lofty verge]

Tried to avoid it but no human could help...
I know the problems with using AI.
I worked the problem myself, only using the steps given, which are correct.

Use of these AI tools to answer other people's help questions is strictly against server rules

Here are the steps anyway

[rain summit]

What did AI tell you tho?

[lofty verge]

(this is my own work, not AI, but based on AI response)

This is the AI response

(sorry, latex did not work in the PDF)

[rain summit]

Anyways this figure is wrong

[lofty verge]

oh no

but it leads to the correct answer

.reopen

[devout snow]

✅ Original question: #help-27 message

[lofty verge]

how is that wrong?

[rain summit]

This one is more correct

[lofty verge]

I thought so too but then I saw

[lofty verge]

Hmm

I see what you mean, but I think the wording is just funny here

[rain summit]

You can easily draw this in geogebra

[rain summit]

Reopen dude

[lofty verge]

it is

[lofty verge]

.

😭

[rain summit]

Sorry

[soft umbra]

it's been reopened

[lofty verge]

But think about it

a circumcircle

is around the triangle

so the circumparallelogram

is around the circles

[lofty verge]

and if you go look at my work here, it leads to the official correct answer

but also, im not sure

i could be wrong and somehow right

and if thats the case, please explain

[devout snow]

@lofty verge Has your question been resolved?

[lofty verge]

@rain summit Sorry to ping you but do you have any further proof I am incorrect here?

I need to ask another question, so I'll need to close this one...

[tender cobalt]

this has a typo

write down that theta = 60, not 30

[lofty verge]

oh yeah true

thank you

[tender cobalt]

also the fractions have the wrong bar size

do fraction (r) (1/2) instead of two fractions

,,\frac r{1/2}

[woven radish]

mtt

[rain summit]

.

[rain summit]

circumscribed basically means the points of the shape has to be on the circumference

[lofty verge]

there

[lofty verge]

yeah, the points of the circle on the circumference of the paralelogram

[tender cobalt]

$\frac r{1/2}$, not $\frac r{\displaystyle\frac12}$

[woven radish]

mtt

[tender cobalt]

the version on the left is more readable

[lofty verge]

i mean maybe your right but can you prove that?

[tender cobalt]

this should not have a 2

[lofty verge]

i see what you mean now

[lofty verge]

oh yep

i need to proofread this 😭

[tender cobalt]

here, remove the s in the subscript

the answer looks to be correct

it doesnt look like youre aware of how to chain = signs together

[lofty verge]

I know it is correct, i saw the answer key. im just unsure if i did the uh circumscribing right

[tender cobalt]

if you want to show a calculation where one equal thing is being simplified/substituted several times, you can do
A = B = C = D = E
instead of
A = B
A = C
A = D
A = E

[lofty verge]

yeah i know

idk i just

started the document like on difference lines

so i wanna stay consistent

[tender cobalt]

you used circumscribed about correctly, yes

about means around

circumscribed around implies the parallelogram is on the outside

the alternative of putting two circles around one shape seems rather forced

and may not be possible

[lofty verge]

thank you 🙂

[tender cobalt]

np

[lofty verge]

thats super helpful

.close

[devout snow]

Channel closed

Closed by @lofty verge

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[tidal breach]

help w part ii please !!

[tidal breach]

i kinda dont get the question, is it like finding the range of the radius of a circle w centre 3+5i that passes through the condition arg(z-2) = 3pi/4 ??

[pseudo basin]

not just passes through the ray

but intersects that ray twice

you're correct to be thinking about circles with center 3+5i

[tidal breach]

rihght so i find like where it tangents the ray and also the radius where the circle would just barely pass through the base of the ray?

and then do the range between that

[pseudo basin]

tangents
touches

but yes sure

[tidal breach]

does this look right

and then AC < 2r < AB

[pseudo basin]

yes

[tidal breach]

tysm !!

.close

[devout snow]

Channel closed

Closed by @tidal breach

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[unreal ivy]

Hello sir I am having maths doubts

[inner sorrel]

?

[unreal ivy]

Sir I want to understand why product of two negative numbers is positive

[inner sorrel]

don’t ask to ask just ask

[unreal ivy]

Sir I want to understand why product of two negative numbers is positive

[native gyro]

!da2a

[devout snow]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question!

[inner sorrel]

Ok so lookee here @unreal ivy

there are multiple ways to do this

one using the box method

and one using pattern

Which one you wanna know

[unreal ivy]

Sir I want mathematical proof

[inner sorrel]

no need to call us sir

[unreal ivy]

Ok

[inner sorrel]

so let us take two numbers, say x and y

We can represent them aa x = (m-p) and y = (n-q) right

[unreal ivy]

Yes

[inner sorrel]

Product of x and y is taken as xy

and xy = (m-p)(n-q) right? (Substituting earlier values)

[unreal ivy]

Yes

[inner sorrel]

right now I recommend that you take values of x y m n p and Q

so that it would be easier for you to understand

take easy values

like x = 9, y = 8 or something like that

[unreal ivy]

Can we do without using values

[inner sorrel]

yeah but it would be easier for understanding

[unreal ivy]

Ok

[inner sorrel]

if you took values

so xy = mn - mq - np +/- pq

understood?

here taking values will be helpful for understanding

so take x as 9, y as 8, m and n as 10, p as 1 and Q as 2

so (9)(8) = (10)(10) + (10)(-1) + (10)(-2) + (-1)(-2)

can you simplify this?

[unreal ivy]

No but you applied distributive property here in product of -p(n-q)

[inner sorrel]

yeah

oh you guys didn’t learnt that yet

[unreal ivy]

How do we know that distributive property holds for negative numbers

[inner sorrel]

oh my bad you guys didn’t learn properties of integers yet?

Wait what grade r u in?

[unreal ivy]

I am school passout but wanted to study again

[inner sorrel]

ohk

so negative numbers are also counted as integers

[uneven coral]

love the enthusiasm mate, I wish you all the best

[unreal ivy]

Thank you

[inner sorrel]

And distributive property holds for integers

[unreal ivy]

Is there a proof?

[inner sorrel]

yeah

you want me to show that?

[unreal ivy]

Please tell that

Yes

[inner sorrel]

so we need to prove that a(b+c) = ab + ac even for negative integers

[unreal ivy]

Yes

[pseudo basin]

it holds for all numbers. in fact it's extremely common that even if you're working with things that aren't numbers you still have some kind of distributive law.

[inner sorrel]

he wants a proof

[unreal ivy]

Yes proof

[pseudo basin]

from what axioms

[unreal ivy]

Yes

[pseudo basin]

...that was not a yes/no question

[inner sorrel]

I just recommended number substitution but he’s doesnt prefer it

[unreal ivy]

Ok sorry

[pseudo basin]

where (i.e. what axioms) are we starting off?

[inner sorrel]

Do you know what is an axiom?

basically what do you want to keep as a given knowledge

[unreal ivy]

I just read that it is a statement assumed to be true

[inner sorrel]

yes exactly

[uneven coral]

He's just starting up, has very less base knowledge

[pseudo basin]

yeah but he's also demanding proof

[inner sorrel]

I think best is to consider the basic properties as axioms

[uneven coral]

Yeah that's true, mate you gotta work up basic terms first

[inner sorrel]

for ease of explanation

[pseudo basin]

so like, what are we really looking for? are we looking for something that's a real formal kind of proof, or are we looking for some explanation of the distributive law on Z?

or like, on what number system. i understood that we're talking about integers thus far.

[inner sorrel]

his original question was actually why the product of two negative numbers is positive

[uneven coral]

His initial question was to prove how the product of two negative numbers is positive

[pseudo basin]

see here is the thing

[inner sorrel]

I recommended him to substitute some basic values

but then he branched off into the discussion about the properties

[pseudo basin]

if we are looking for a formal proof,
and if we do not all agree on what to take as the starting point(s),
then we can never produce a proof of anything

[inner sorrel]

so we need to take some statements as givens

[pseudo basin]

the thing about the distributive law is that, in my experience, that is quite often taken as an axiom in its own right -- and number systems such as the real numbers are defined axiomatically (leaving separate the question of how to construct such a number system, and then how to verify that all the axioms are true in your construction)

[inner sorrel]

so stuff like distributive and associative properties must be taken as given s

because if we do not have distributive property as a given

I can’t properly prove to you why the product of two negative numbers is positive

I mean I could but not with the method I was using

[pseudo basin]

if we wanna go down to the basics basics, there's... peano axioms for natural numbers.

[unreal ivy]

Ok please explain

[inner sorrel]

we just explained everything

[pseudo basin]

no, i can yap.

[queen gazelle]

how is this for 5th graders?

[inner sorrel]

basic integer properties

[sinful siren]

even i dont know

[pseudo basin]

so if you wanna really start at the beginning, then we have to begin with the natural numbers: 0, 1, 2, 3, 4, ...

[queen gazelle]

LOL

i can help bri

bro*

[unreal ivy]

Is this group for fifth graders?

[queen gazelle]

yes

[pseudo basin]

this server has several hundred thousand people

[sinful siren]

13+

[pseudo basin]

it is not restricted to any grade level but you have to be 13 or older to be on Discord at all, not just here

[sinful siren]

i think 20000+

[queen gazelle]

alr so who was the one who put the question? whats the questiom? i can help bro

[inner sorrel]

so basically look here

[pseudo basin]

anyway if everyone else could please allow me to speak uninterrupted???

[sinful siren]

that means i can be in grade1

[inner sorrel]

ok mb

[pseudo basin]

a mathematician named Peano came up with the following axiom system to describe the natural numbers:

[inner sorrel]

guys let Ann explanation

[sinful siren]

gomenasai

[queen gazelle]

whats the question?

im the best at explain

[pseudo basin]

• there is a number called 0
• every number has a next number after it (called its "successor"; in arithmetical terms, it means adding 1)
• two different numbers never have the same successor
• 0 is not anybody's successor
• if some property is true for 0 and it propagates from a number to its successor, then the property is true for all naturals

[uneven coral]

Okay guys the channel is getting too congested, let's leave it to @pseudo basin and @inner sorrel , the rest of us should leave

[sinful siren]

lets leave

[unreal ivy]

Actually Ann I need a more mature answer like how a mathematician would answer

[pseudo basin]

i am answering like a mathematician.

[inner sorrel]

this is how a mathematician answer

bro

mathematicians said these statements word by word

[pseudo basin]

i mean i can barf ∀ and ∃ and other such shit at you if you really want to

[astral cargo]

this guy is obviously trolling

[pseudo basin]

but dont be surprised if you then do not understand anything

[inner sorrel]

I myself don’t understand those symbols

[unreal ivy]

No I heard that there is explanation in group theory and there are field axioms

[pseudo basin]

oh you brought up FIELD AXIOMS huh.

do you know what those are, or did you just hear funny math word w/o understanding its meaning

[unreal ivy]

I just googled and there were things like closure, commutativity and inverse

[pseudo basin]

yeah, so a "field" is fancy math speak for a number* system where you can do addition, subtraction, multiplication and division and they behave "as you would expect" (from school algebra)

distributive law is actually one of the axioms

so it doesn't have a proof

[inner sorrel]

so do you want us to explain these step by step

[unreal ivy]

Yes I would like you to

[inner sorrel]

or do you want the proof to your original question

[pseudo basin]

so you want a summary of field axioms

[inner sorrel]

ok so let us start out with closure

[pseudo basin]

eh idk maybe i can leave this to BKB since he seems so eager to help

[inner sorrel]

I need to go eat lunch mv

[inner sorrel]

It’s lunchtime for me, I need to eat

sorry

[unreal ivy]

No I want to understand that when we have 3×(-2), we interpret it as adding -2 thrice.
But when we have (-3)×2 how do we interpret it, like how can you take 2 and add it -3 times

[pseudo basin]

ok, so forget axioms entirely?

[unreal ivy]

Yes for now

[pseudo basin]

ok

i would be cautious about attaching yourself to "interpretations" too much, because at some point you'll have to let go of that anyway

but here is one that might work

are you familiar with the concept of a number line?

[unreal ivy]

Yes

[pseudo basin]

right.

so then, let's examine (number)×2 as "take that many steps of size 2 on the number line".

for these purposes you always start at 0

5×2 will have you walk two units forward (2) a total of five times, and you end up at the point 10.

4 × 2 = 8 (four steps of 2)
3 × 2 = 6 (three steps of 2)
2 × 2 = 4 (two steps of 2)
1 × 2 = 2 (one step of 2)

just positives so far, but i want to establish this pattern

so that makes perfect sense to you, yes?

[unreal ivy]

Yes

[pseudo basin]

ok

do you understand what 0 × 2 would be

[unreal ivy]

2 taken 0 times so 0

[pseudo basin]

yes

in other words, we take no steps of size 2.

[unreal ivy]

Yes

[pseudo basin]

so of course if we take no steps then we're left with 0

[unreal ivy]

Yes

[pseudo basin]

now (-1) × 2 is a little strange because we don't know what "take -1 steps" means.

[unreal ivy]

Yes

[pseudo basin]

but what you may notice is that every time we reduce the step count by 1, our destination shifts 2 units left.

it used to be 10, then it was 8, then 6, then 4, then 2, then 0...

if this pattern keeps going, where will we end up

[unreal ivy]

But this is just a pattern,

[mortal bison]

can u help me for this question i solved half but i didnt understand anything for left solution

[pseudo basin]

!occupied

[devout snow]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[pseudo basin]

you either get intuitive explanation or formal axiomatic proof. you cannot have both at the same time.

[mortal bison]

kırıcı

[pseudo basin]

there is nothing offensive about this, you're just intruding in somebody else's channel. claim your own. read #❓how-to-get-help for instructions.

[unreal ivy]

So basically (-3)×2 is assumed to be -6 by commutative property

[pseudo basin]

it isnt "assumed" to be anything, but whatever.

sure, you can invoke the commutative property if you would like.

to tell the truth i kinda don't know how to explain this to you if you dont like axioms AND also you dont like pattern-based explanations.

[inner sorrel]

ok I finished

[unreal ivy]

So could you also explain this that why product of negative and a positive number is negative

Please include axioms

[inner sorrel]

Ann I can take over from here if you want to

[pseudo basin]

sure, go ahead

i have better things to do right now

[inner sorrel]

So @unreal ivy

we use commutative property to explain your earlier doinf

[unreal ivy]

Doinf?

[inner sorrel]

Remember that commutativeness only applies to addition and multiplication and that it is a axiom

[inner sorrel]

Doubt my bad

so let us take two numbers, say 3 and 5

[unreal ivy]

Ok

[inner sorrel]

3 * 5 = 15

[unreal ivy]

Yes

[inner sorrel]

so let us change 3 to -3

-3 * 5 = ?

so multiplication is basically defined as repeated addition

so 3*5 = 3+3+3+3+3 = 15

so -3 * 5 = -3 + -3 + -3 + -3 + -3.

[unreal ivy]

And then for 3×(-5) we assume it to be same as -15 by commutativity

Is it right

[inner sorrel]

yes

[unreal ivy]

But why do need such a law like commutativity

[inner sorrel]

because

We need a reason to be able to change around the digits in some operations

Usually questioning such stuff like axioms is pretty bleh

[unreal ivy]

But how do we know that our answer is right after we change around digits in the operation

[inner sorrel]

that’s what commutativity says

it says that the answer will stay the same for addition and multiplication if we swap the two numbers position

[unreal ivy]

No but can we verify it

[inner sorrel]

not really

there are infinite such cases you have to verify to prove it

[unreal ivy]

Can you explain meaning of infinite cases?

[inner sorrel]

so there are so many cases you can’t count it

so it’s kinda impossible to prove the thing

[unreal ivy]

Infinite cases where?

You consider one problem

[inner sorrel]

there are infinitely many natural numbers

or integers for that case

so to prove commutativiry as a theorem, you would need to solve every single possible combination of integers

which is impossible

[unreal ivy]

No I was asking if we have a problem in algebra how do you check your answer

[inner sorrel]

by substituting your answer

in the equation

you solve, factorise or use quadratic equation and then you substitute your answer in the original equation

[unreal ivy]

Ok fine.

I also have doubts in fractions

[inner sorrel]

ask

[unreal ivy]

Why when we have to find 1/7 of 1/3 we represent it by operation of multiplication like this- 1/7 × 1/3

[inner sorrel]

Yes

[unreal ivy]

I am asking why we do it

[inner sorrel]

because

$\fracxy of b$

oh

so x/y of B is asking

if you divide B into y parts, how much would x parts be

[unreal ivy]

No but why we use multiplication operation to represent it

[inner sorrel]

that’s because the multiplication operation and operation in question are virtually the same

[unreal ivy]

How are they same please elaborate

.close

[devout snow]

Channel closed

Closed by @unreal ivy

Use .reopen if this was a mistake.

[devout snow]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[kindred fern]

Hello, can someone help me with this problem?

[kindred fern]

I have to resolve the equation, n is part of N*

that's what I did so far

I don't really know how to continue or what to look for

[untold ravine]

on the right hand side you have the sum of 1/k

and on the left hand side you have the sum of 1/k+1

can things cancel

[kindred fern]

but if I'm adding things

how can they cancel

oh wait

nvm

I'm dumb

sorry

[polar chasm]

and then for x^2/(k(k+1)), try partial fractions

[untold ravine]

yes go on

[kindred fern]

1/2 - (1/1) + 1/3 - 1/2 + 1/4 - 1/3 and so on...

so there is left only -1, right?

[untold ravine]

yes, and one more term

what are the terms at the very end

[kindred fern]

1/k?

wait, I don't really know how to take it

[untold ravine]

take what

[kindred fern]

how to think

1/(k+1) will always be the last term

unless you decrease

[untold ravine]

k+1 has no significance, its all in terms of n

[kindred fern]

everything cancels except -1 and 1/k+1

is this correct?

[kindred fern]

the x^2**+k** in the first equation here is wrong, I separated the equations, don't take it into account

I think that's n, not k

or?

I'm very confused

[untold ravine]

-1 + 1/n+1

yes

[kindred fern]

yea, it was n, I'm sorry

so I get

[azure parrot]

Does it say find x so that for any n, the equation holds? Cuz ngl I don't think we can cancel out n fully

[kindred fern]

Let n be part of N*, Resolve the equation

I guess

I'm not really sure what can we do now

[azure parrot]

$\sum_{i=1}^n \frac{1}{i(i+1)}$

[woven radish]

Alexis_Fx

[azure parrot]

Is a telescopic sum

Well-used one

[kindred fern]

but we did cancel the terms

didn't we?

[azure parrot]

This sum solely has a closed form, you can factor out x^2 and find the closed form of 1/k(k+1)

[kindred fern]

so I continue from here?

[azure parrot]

Yeah

[kindred fern]

oh so I need to work on x^2/(k^2+k)

[kindred fern]

is this right?

[azure parrot]

Okay what, could you put brackets

[kindred fern]

I got 1/k(k+1) in 1/k - 1/(k+1)

and the result should be the same

like 1 - 1/n+1

right?

[azure parrot]

,w sum 1/(k(k+1)) k from 1 to n

[woven radish]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by yobi_fx.

[azure parrot]

Okay yeah

[kindred fern]

oh wait

that's 1/k(k+1)

not k/k(k+1)

it's not the telescopic

sum

[azure parrot]

You're doin it right

[azure parrot]

Where's the k come from

[azure parrot]

Oh wait

[kindred fern]

so the telescoping sum that cancels out

its k/k(k+1)

and can be written of as

[azure parrot]

Is the whole thing at the end for sum of x^2/k(k+1)

[kindred fern]

can be written like 1/k - 1/k+1

but the numerator is 1

and I applied this formula

for 1

[azure parrot]

No lmao

[kindred fern]

I'm confused

[azure parrot]

,w pfd 1/(x(x+1))

[woven radish]

Results provided by WolframAlpha

Click here to refine your query online
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Requested by yobi_fx.

[kindred fern]

oh

idk where the k came from

I must be schyzo

I'm sorry

so everything here is correct?

[azure parrot]

There won't be sum notation in the last step

[kindred fern]

because the result is 0?

and they cancel out

[azure parrot]

Because

[kindred fern]

wait

I rewrote everyhing

and I can write the first part of the sum as n/(n+1)

[azure parrot]

Okay so far so good, you should put brackets for what's inside the sum it won't be confusing

[azure parrot]

Yeah

[azure parrot]

Uh what

[kindred fern]

this were supposed to be normal brackets

[azure parrot]

x^2 is not mulitplied with (-1+1/n+1)

[kindred fern]

so the x is multiplied by the whole sum

isn't it?

I thought -1+1/n+1 was part of the sum

[azure parrot]

It's not lol

[kindred fern]

that's a photo from a few steps back

[azure parrot]

This's what you wrote earlier

Yeah

[kindred fern]

oh so the last things don't become a part of the first sum?

😭

[azure parrot]

Bruhhh

[kindred fern]

I'm sorry

just started using sigma

and I get confused sometimes

so the x multiplis just with the n/n+1

like what I put into parantheses

[azure parrot]

Yeah

Okay so it's trivial now

[azure parrot]

Okay so n+1 can't be 0

[kindred fern]

the problem says n is N*

[azure parrot]

Ykw you should try first

[kindred fern]

so n can't be 0

so n is >=1

I have no idea

lol

[azure parrot]

That equal 0 right?

You can get rid of the denom

[kindred fern]

wait

can I combine -1 + 1/n+1

so it's a single expression

[azure parrot]

Ofc... Why did you ask that

[kindred fern]

maybe it's more useful

in this form

idk

😭

I'm a bit slow

[azure parrot]

Get rid of the denom, then show me what you get

Do cross multiply or smth

[kindred fern]

-(n+1/n+1) + 1/n+1 = -n/n+1

right?

I hope

I'm so sorry I'm this slow

x^2 * n = n

[azure parrot]

Nahh

[kindred fern]

so x is +- 1?

[azure parrot]

It's alright

[azure parrot]

Yeah, if what we did are all correct

[kindred fern]

yea!!!

I checked the answers

x is {-1, 1}

thanks a lot!

[azure parrot]

Np

[kindred fern]

have a great day!

.close

[devout snow]

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[jaunty kettle]

Example 10,11,12,14 -

[final tulip]

!show

[devout snow]

Show your work, and if possible, explain where you are stuck.

[jaunty kettle]

I have no idea how to solve and stucked in the question obviously

[pseudo basin]

,rccw

[final tulip]

start by drawing a figure

[deep abyss]

10 can be done using mid point theorem

[jaunty kettle]

I have created

These black once. But they are not much useful to show

For some extent i understood 10th question but not other ones

[fossil locust]

if you draw 11, 12, 14 you should be able to guess those

say for q11, you already know that it must be a kind of parallelogram

[deep abyss]

How about u name the points? Like label the figure idk how to say

[fossil locust]

so either it's a square or it's a rhombus

(think about why that shape isn't necessarily a rectangle)

[devout snow]

@jaunty kettle Has your question been resolved?

[jaunty kettle]

As in the notebook. I have already drawn a rectangle.

But how can I guess the figure? I can't just guess

[fossil locust]

are those interior angles 90 degrees?

[jaunty kettle]

Which one you are talking about?

[fossil locust]

the shapes inside the rectangle (q11) and a parallelogram (q14)

both of those

[jaunty kettle]

There is nowhere written that interior angle is 90°

Ok ahm, forget these questions.

I have another question. It would be better if focused on that first

Example 18 -

[fossil locust]

,rccw

[jaunty kettle]

Diagram

[fossil locust]

oh, you should know that the (long) diagonal of a rhombus bisects the angles

hmmm

yes and that's not true for a general parallelogram

you can only conclude angle SPR = angle PRQ for a parallelogram (SSS)

so you would need SP = PQ and SR = RQ, the condition for a rhombus

then the diagonals of a rhombus bisect each other at right angles

[jaunty kettle]

Oh ok

Thanks

.close

[devout snow]

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[devout snow]

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[vagrant spire]

How to understand indefinite integral, definite integral and antiderivative using a geometric interpretation?

[vagrant spire]

I'm confused about the relation between antiderivative and area below the curve

From my understanding, indefinite integral finds the function that uses the given function as its rate of change, which when summing all the previous rate of changes before a certain bound gives the area under the curve in the given function from 0 to the bound

[supple knot]

Khan academy has videos on all of this

[vagrant spire]

Can you link please?

The method to calculate definite integral is by finding the difference between the antiderivative at the 2 bounds (F(x1)-F(x0))

Which sounds intuitive, as it's subtracting a lesser figure at the lower bound from greater figure to the higher bound

However, F(0) can be non-zero (e.g. e^x), but the area should always be 0 at 0 since the figure is a flat line

[misty crest]

do you know how to use google?

[vagrant spire]

Yet to find anything that solves my inquiry

I sincerely apologize if I'm being slow

[misty crest]

literally just look up khan academy calculus and go to the appropriate unit

[vagrant spire]

Summary:
Why are derivative and indefinite integrals inverse operations of each other, in the sense that definite integrals are area under the curve? (Explain me like 5 why is fundamental theorem of calculus)
What are antiderivatives in the sense that definite integrals are area under the curve?
What does F(x0)-F(x1) mean in geometric sense?

[devout snow]

@vagrant spire Has your question been resolved?

[sand pumice]

https://youtu.be/FnJqaIESC2s?si=Th1m_p52dLB87DTS a great video

[devout snow]

@vagrant spire Has your question been resolved?

[devout snow]

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[glass hare]

ok so I have a test in 4 hours and I want to cram the material, can someone go over the material with me?

[timber pebble]

did you have any specific questions you wanted to look at?

[glass hare]

eah

yeah

sorry for the wait I’ll send them

@timber pebble

[devout snow]

@glass hare Has your question been resolved?

[supple knot]

You just do the problems and if you get stuck just ask for help here

No one's gonna do 30+ problems for you

[glass hare]

wdym for me

I want to be walked through them

I’m still gonna try to do it

[supple knot]

Yea exactly. No one's gonna do this for you

[glass hare]

welp

[supple knot]

Then do them and if you get stuck show your attempt here

[glass hare]

ok

[glass hare]

Ok

[glass hare]

I need help with 44

<@&286206848099549185>

[supple knot]

Show your work for 44 like it says here and then you can ping helpers

[tender wharf]

Hm.. take duration of train ride t

So bus = total time -t

Then form eqn

[glass hare]

what

[tender wharf]

Let time for which distance travelled by train be t

Total time is 5.5 so time while traveling by bus = total - t

[glass hare]

actually

I’m gonna do them 1-50

then you tell me

what I got wrong

is that cool

[tender wharf]

Which one you on?

[tender wharf]

If not me someone will be there

[glass hare]

I’m about to start

[devout snow]

@glass hare Has your question been resolved?

[devout snow]

@glass hare Has your question been resolved?

[devout grotto]

that's not how this works-- if you get stuck on a problem come talk to us

[devout snow]

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[devout snow]

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[unkempt harness]

For this question, why can't we just rewrite the equation with negative exponents and then use the power rule to integrate?

[iron sun]

Well how would you rewrite it as you say?

Also, about that

[unkempt harness]

e^-2x - 2e^-x - 3 ^-1

[iron sun]

So you’re saying 1/(all that) = what you wrote above?

[unkempt harness]

I believe so

What makes that different from something like 1/(x^5) being x^-5?

[iron sun]

Well it’s unfortunately not true; take something simple as 1/(1+1) ≠ 1^-1 + 1^-1

[unkempt harness]

Is it because of the multiple terms being added

[iron sun]

Mhm

[unkempt harness]

ohh ok

What if you did it in parentheses

like 1/ (1+1) as (1+1) ^-1

[iron sun]

Yes

1/x is x^-1

So that’s true

With x = 1+1 here

[unkempt harness]

So you could technically rewrite the original function above with a negative exponent right?

(e^2x - 2e^x - 3) ^-1 so e^-2x - 2^-1e^-x - 3^-1?

[iron sun]

Mhm but about that

You would not be able to use the power rule

[unkempt harness]

Im guessing thats just harder to integrate then

Yeah that makes sense

[iron sun]

It’s the same infact

Same problem but different notation

[unkempt harness]

Ok that makes sense, thank you

.close

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[woven radish]

HugoL

HugoL

[idle lynx]

Apologies for bad formatting but hopefully it's clear when I'm talking about 3-vectors vs 4-vectors etc

In the full context of my question, it's written here, and I'm getting a minus sign before the summation term in the second line (angled brackets mean time ordered vacuum expectation value)

[willow helm]

just a tip for latex, when you have this much text, it's much easier to read if math is in blocks instead of inline

[idle lynx]

apologies i wasn't entirely sure how to do it with this bot

[short hare]

this also doesn't look very help-channels-friendly perhaps search in the 'advanced mathematics' category starting from #get-advanced-access for the appropriate channel.

[idle lynx]

can I use \begin{equation}?

[willow helm]

Ah, just double dollar signs do the trick

[idle lynx]

ah thanks

[supple knot]

Try uhhhh #diff-geo-diff-top or #advanced-pdes

[acoustic leaf]

the bot literally compiles a latex document, it has all of the same features

[idle lynx]

right I see thanks

[idle lynx]

Is it ok if I leave this here or should I close my question and move there?

[short hare]

It's not prohibited to use help channels for advanced mathematics, It's just suggested to use the intended channels.

so perhaps... do as you please! but yeah, I doubt you'll get help with allllll of that anytime soon

so perhaps closing the channel is a good option.

[idle lynx]

ok thanks for the advice

[woven radish]

HugoL

[idle lynx]

it turns out i'm profoundly stupid

apologies all

.close

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[devout snow]

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[frigid rain]

Can somebody give me the steps / hint about how to do this? I recall learning it in the past but am currently reviewing, so i have forgotten, and searching online isnt helping much.

[gritty terrace]

Use polar form

[frigid rain]

Okay i found its polar form being radius of 2 and angle of 5pi/6. if i remember correctly, from here, its something to do with demoivre's theorem, yes? i will try that.

[gritty terrace]

Mhm

[frigid rain]

okay so im looking at some stuff rn but finding myself stuck on exactly what to do with this polar number. and how exactly to use it in regards to the demoivres theorem? the example in my textbook is using the complex number of just 8 (screenshot) which im not exactly sure how to translate over to this problem. would i have my problem being z^3 = 2, which is its modulus?

[devout snow]

@frigid rain Has your question been resolved?

[frigid rain]

Okay so i tried something and will explain my logic to check if im right. I ended up getting the correct answers (screenshot from the answer sheet) and my logic is like. for example, w0 = (nth root) modulus of z, cis original angle / 3. so in this case, 5pi/18. then for w1, you keep the original part of nth root modulus, add 1 loop around over n (so, 120* or 36 / 3 = 12pi/18) cis that. so 3rd root 2, (5pi/18 + 12pi/18, or 17pi/18)

then for the final one, w0, we keep 3rd root 2, and do the same as w1 just with 2 loops around (720 degrees or 72pi/18), that over 3 for (240 degrees or 24pi/18). plus the original angle for (5pi/18 + 24pi/18, or 29pi/18)

is there any part in my logic which wouldnt work in every circumstance? is what im wondering now.

and something that concerns me is that the fact it was to the fourth power in the original question didnt seem to come up. did i miss something and just get lucky or is that mostly irrelevant or something?

[lunar harbor]

you used the fourth power stuff at the start

and yes this is generally the logic you use to find all of the nth roots of unity

and hence solve stuff of the form z^n = w

[devout snow]

@frigid rain Has your question been resolved?

[frigid rain]

i fear i didnt do that consciously, at least. i found the polar form without regard to the fourth power (-root3)^2 + 1 = 4, making the radius 2, and finding the angle easily enough with that. at that point i lowkey forgot about the fourth power part and just operated without it, which is why im confused that i got it right

eh lowkey i got it right so i dont really care that much

.close

[devout snow]

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[quick osprey]

What is Logarithm

[solar goblet]

the exponent the base must be raised to get a certain number

[jaunty bane]

a logarithm is kind of like the inverse or opposite of an exponent.
see this image on how they relate.

[solar goblet]

so 2^x = 8, you call x is the log base 2 of 8

i.e. 3

[quick osprey]

Ok

.close

[devout snow]

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[devout snow]

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[dapper quail]

I don't understand how to set up calc 2 volume problems

[devout snow]

Please post images (such as PNGs or JPGs) of the question rather than other filetypes such as PDFs which have to be downloaded. Non-image downloads can potentially contain viruses or other security risks.

[ionic birch]

hm

lets start with A first

$x + y = 3 and y = x^2 - 4x + 3

.

Texit stop

$x + y = 3$ and $y = x^2 - 4x + 3$

[woven radish]

zeppelin

[ionic birch]

ok good

so we have the above binding the region

let’s make this easier identify what the curve and what the region are first

@dapper quail do you know how to find either?

[dapper quail]

i dont think i do

i know its the area under between them i guess?

[ionic birch]

yes

exactly

with that in mind we need a graph

you’re gonna have to plot all this

(in order for me to show you what to do obviously)

[devout snow]

@dapper quail Has your question been resolved?

[dapper quail]

sure sorry i went afk

sorry if this is rly slow

[ionic birch]

this is fine

[dapper quail]

ok so i have sort of an idea its going to be 0 to 3 and its between that line and area underneath

[ionic birch]

yes

do you see the X-point where both lines meet?

[dapper quail]

yes

[ionic birch]

that’s our limit

x=0 and x=3

[dapper quail]

yup

[ionic birch]

and you see the two lines touching on the y axis?

[dapper quail]

yup

[ionic birch]

that’s our rotation

y=-3 and y=3

wait oof

[dapper quail]

wait why y=-3?

[ionic birch]

it’s part of the curve

which helps us get our answer

that’s where this comes in:
$(3-x) - (x^2 - 4x + 3) = x^2 + 3x = x(3 - x)$

[woven radish]

zeppelin

[ionic birch]

since we know are points we form an equation

and THIS specific equation is what will help us find our top and bottom curve

[dapper quail]

im still not sure how we figure out the rotation is on this

[ionic birch]

oh i’m sorry i went and skipped everything

i’ll go over that now

[dapper quail]

heres an image version of the problem btw

just to have for reference in the chat

[ionic birch]

so you know how find curves right?

that’s the basic way for me to explain it properly

it’s basically what we went over and what we went over

and what you plotted

[dapper quail]

sort of

areas between curves right

[ionic birch]

yes

you basically look at the points that meet the X-points , and rotate them to find the y

you should get y= -3

[dapper quail]

sry i hate to make you wait, i have to go but ill come back at some point, should i close the help channel

[ionic birch]

sure

just ping me when available again @dapper quail

[devout snow]

@dapper quail Has your question been resolved?

[devout snow]

@dapper quail Has your question been resolved?

[dapper quail]

@ionic birch lmk when ur back

im back

[devout snow]

@dapper quail Has your question been resolved?

[brave junco]

i can help u

[dapper quail]

alr

[brave junco]

okay so whats the issue

[dapper quail]

i dont understand how to set up the problem

[brave junco]

this corrrect?

[dapper quail]

mhm

[brave junco]

ok so you have the line

y = 3 - x

[dapper quail]

yes

[brave junco]

then you see the parabola y = x^2 - 4x + 3

set them equal to find intersection x values

[dapper quail]

ohh

(3-x) = x^2-4x+3 = (x-3)(x-1)

not sure here

@brave junco

[brave junco]

ok so the factoring is fine

but u only factor after u move everything to one side

and set the equation equal to 0

so start with 3 - x = x^2 - 4x + 3

[dapper quail]

ah yeah that makes it easier

[brave junco]

and move everything to one side so subtract 3-x from both side

[dapper quail]

0 = x^2-3x = x(x-3)

[brave junco]

?

[dapper quail]

oops

so x = 3

and 0

[brave junco]

yes

so ur points rae 0,3 and 3,0

[dapper quail]

yes

[brave junco]

ok

now rotation

rotate the horizontal line y = -3

use washers with vertical slices

and integrate with respect to x

[dapper quail]

how do you do the rotation step

and decide vertical vs horizontal slices

[brave junco]

which is easier with thefunctions u have

both curbes are written as y = f(x)

that makes vertical slices easier

because u do not have to solve for x in terms of y

if the slices are perpendicular to the axis of rotation

disks/washers

if slices are parallel

cylindrical shells

y = -3

vertical slices are perpendicular to horizontal line

so use washers

then you have to find radii for washers

each vertical slice at some x b etween 0 and 3 turns into a washer when rotated around y = -3

distances are measured vertically from the line y = -3

[dapper quail]

I see

[brave junco]

tp curve: y_top = 3 - x

bottom curve: y_bot = x^2 - 4x + 3

[dapper quail]

Right

[brave junco]

then r(x) = distance from axis to top curve

R(x) = y_top - (-3) = (3 - x) - (-3) = 6 - x

then r(x) = distance from axis to bottom curve

r(x) = y_bot - (-3) = (x^2 - 4x + 3) - (-3) = x^2 - 4x + 6

[dapper quail]

Then is it ytop - ybot

[brave junco]

no, thats the height of the region

for washers about y = -3

u need radii

which are distances from the axis

[dapper quail]

Oh ok

[brave junco]

then u just write the volume integral

v = pi * integral_a^b [ R(x)^2 - r(x)^2 ] dx

amd a = 0 here and b = 3

[dapper quail]

Right

[cloud remnant]

wouldnt it be easier to transform the coordinate axes to y=-3 and then use integral pi y^2 ?

[brave junco]

u can do that

but u still do the washer formula

shift up by 3 units

y = y + 3 then axis y = -3 becomes y = 0

[cloud remnant]

yea

[brave junco]

top curve: y_top = 3 - x, y_top = Y_top + 3 = (3 - x) + 3 = 6 - x bottom curve: y_bot = x^2 - 4x + 3, Y_bot = y_bot + 3 = x^2 - 4x + 3 + 3 = x^2 - 4x + 6

then roate around Y = 0

the corss section at each x still a washer

[dapper quail]

I see

[dapper quail]

So for shells it would be for x=-3

[surreal matrix]

guys so you wondereing algebra is the easiest method

i meant

uhhh

physics

[atomic narwhal]

bruh

[scenic thorn]

What

[scenic thorn]

wait that means
(R(x)=(3-x)-(-3)=6-x) ?

[woven radish]

HOT LESBIAN SERJ TANKIAN

[dapper quail]

I guess

[devout snow]

@dapper quail Has your question been resolved?

[regal fiber]

do you want me to verify or you’re done ?

If so close ticket once done

[devout snow]

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[devout snow]

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[bitter echo]

does anyone have a good source of open problems in graph theory that's undergraduate level? looking to explore the field a bit more.

[bitter echo]

I found the following online, but it appears outdated and it looks like it may be very outdated.
https://dimacs.rutgers.edu/~hochberg/undopen/graphtheory/graphtheory.html
http://www.openproblemgarden.org/category/graph_theory

<@&286206848099549185>

[devout snow]

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[civic umbra]

Find the curve such that the initial ordinate of any tangent is less than the abscissa at the point of tangency by 3 units

[civic umbra]

<@&286206848099549185>

[pseudo basin]

wow this handwriting is unreadable

but ok

so let's say you are looking at y=f(x) and its tangent at x=c

the tangent should have y-intercept equal to c-3

[pseudo basin]

can you write the equation of this tangent in general form

[civic umbra]

y'=m(x'+x)+y ??

[pseudo basin]

that looks extremely sus

[civic umbra]

x' and y' for tangent coordinate?

[pseudo basin]

... i think you're trying to overdo it or just messing up the symbols real bad

[civic umbra]

??

[pseudo basin]

y - f(c) = f'(c) (x - c)

[pseudo basin]

this is what i expected to see

line passing through (c, f(c)) with slope f'(c)

[civic umbra]

It's a differential equations question

Answer: y=Cx - x lnx - 3

I want to know how to get to it?

[pseudo basin]

was trying to take you through that

and give you the starting pt

[civic umbra]

Can you explain ?

[pseudo basin]

line passing through (c, f(c)) with slope f'(c)

do you know point-slope form for straight lines?

[civic umbra]

Ooh...

So I can frame the diff. equation like:
y - x (dy/dx) = x-3 right

[pseudo basin]

mmm

yeah ig you can

[civic umbra]

Ok perfect, wasn't that tough, I was dumb this time🥲

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[steep topaz]

Help

[rain summit]

!original

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Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[rain summit]

send in your question

@steep topaz

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[molten spindle]

This is the formula to rotate p around an arbituary axis using quaternions. When constructing q do I first normalise the rotation axis and then construct q or do I first construct q and then normalise q?

[silk crystal]

is this occupied?

!help

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[molten spindle]

.close

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[willow helm]

[steel sage]

You find the unit vector v, representing the axis, the angle of rotation θ you want, then q should be cos(θ/2)+sin(θ/2)a. no idea how order or operations comes in.

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[molten spindle]

And what is vector v is not unit vector. Do I first normalise it and then construct q or construct q and then normalise q

[steel sage]

You normalize your v, then you get your q

[molten spindle]

Aha, got it. Thanks

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[wise wren]

It says if t > c, we reject the null hypothesis but on the other slide, we reject it anyways even though t < c.

[wise wren]

Which one is it?

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[young spade]

okay

this is more of a generalization

You usually take t and c to be positive when you say "t > c we reject H0", but for this case we have the inverse.

You can rather look at the bounds of the probability dist.

We are conducting a left tail test here, since we are testing at μ<5.
Our critical value (C) is at the bound between the shaded and unshaded region.

You can see that t falls in the shaded region

Which for statistics, means we should reject H0

Hypothesis Test actually allow for a lot of ways to be set up, you can do inner, left, right, two tail, etc...

And you have to know how to read critical values vs estimators.

[wise wren]

yeah

I think I understand now thanks

.close