help-33

[sharp harness]

and replace each number with half of it

i started with infinity/2 things and ended up with infinity things, therefore infinity isn't more than infinity/2

this is basically what i was talking about earlier

for it to make sense that "even naturals", "naturals" and "integers" all have different sizes, you need every object to have something that it inherently "is"

because even if all you do is take a set and replace every object with some other object that has identical properties, you can still change its "size"

[pastel storm]

you did not end up with infinity things

[sharp harness]

but i have all of the natural numbers...?

every natural number is half of some even number

so by replacing each even number with half of it i ended up with half of every even number, which is every natural number

[pastel storm]

not the ones bigger than half of the theoretical biggest natural

[sharp harness]

ah ok i see what the problem is

we're using different definitions of "natural number"

that's where all this confusion came from

the definition i normally use of the natural numbers is that they're defined by two constructors: 0, which is a natural number, and successor, which is a total function from the natural numbers to the natural numbers
and then recursion/induction, reflecting the fact that these are the only ways to make natural numbers, and also that for instance the successor of 0 isn't 0

so by this definition, there is no biggest natural number

if you did have a biggest natural number, then the successor of it would be bigger, which is a contradiction

[sharp harness]

...so by your definition of the natural numbers are + and * not always defined?

[pastel storm]

I mean I guess not

but like if I can name a number bigger then every number of the set then there must be a biggest number in the set

And obviously I can name a bigger number, infinity

[sharp harness]

...i don't get why that would be true at all

is it something to do with this notion of "number"? since for general partially ordered sets that's just false

[unborn chasm]

True, but can you name a natural number bigger than itself plus one?

[sharp harness]

the biggest gtbot-natural number just doesn't have an "itself plus one"

[unborn chasm]

Infinity is not a number

[sharp harness]

which... sounds like it just completely breaks everything? like how do you number theory when + and * are partial functions that sounds so annoying

[pastel storm]

It dose

its just not a natural

[sharp harness]

...oh, i see

so if we take the intersection of all sets that contain 0 and are closed under successor

by my definition this just is the natural numbers

and by your definition this contains the natural numbers and also some things that aren't natural numbers?

would infinity be in this set?

[pastel storm]

depending on how you defined "closed" that could either be the natural numbers or the positive omnific integers

[sharp harness]

i don't know what other definitions of "closed" i could use...?

"if the set contains x it contains x + 1"

[pastel storm]

(or the numbers up to half of the biggest natural but why would you do that)

[pastel storm]

but how many times are you allowed to to do that

[sharp harness]

uh... no.

the definition isn't phrased in terms of numbers at all

[pastel storm]

like are you allowed to do x+1 a tranfinte amount of times

[sharp harness]

it says that for every x, if the set contains x, it contains x+1

i can define any of the terms here if there's ambiguity about what they mean

(brb)

(back)

[pastel storm]

under your definitions of "closed" would this be able to reach transfinite numbers

[sharp harness]

depends what "transfinite" numbers are

if it can reach your "largest natural number" then it can reach that number plus one

it can't reach the ordinal "omega", which comes after all of my natural numbers, because the successor of any number less than omega is still less than omega

[pastel storm]

me when omega minus 1 /hj

[sharp harness]

that doesn't exist

[pastel storm]

me when surreal numbers /srs

[sharp harness]

ok yes it exists in the surreal numbers, but omega - 1 isn't an ordinal

[pastel storm]

but the "biggest natural number" is still only theoretical

its exists but like we can't reach it by doing a finite number of operations

[sharp harness]

...if you did n operations, for a natural number n, would you consider that a finite number of operations

[pastel storm]

I mean I wouldn't consider it infinite

Actually this makes me think if the biggest natural is inf-1

[sharp harness]

ok so start at 0, and do successor n times, where n is the largest natural number

[unborn chasm]

You keep saying "biggest natural number" but again that is not a thing.

Not even theoretically

[sharp harness]

also... you say "only theoretical" as if that's special
we're doing maths here, none of this is stuff we actually went and found out in the real world or something

[unborn chasm]

You are being very handwavey and you two keep going in circles because of that.

[pastel storm]

It must be or else there can't be things bigger than the naturals

[sharp harness]

...wait what
why?

[pastel storm]

like they have to end at some point for there to be things that are past them

[unborn chasm]

Define bigger

[unborn chasm]

No it does not have to "end". There is no maximal natural number.

[pastel storm]

then how can things be past them?

[unborn chasm]

Cantor's diagonalization argument proves there is no bijection from an infinite set to its power set.

[sharp harness]

...what

that's true but i don't see how it's at all relevant

[sharp harness]

i don't see what the problem is?

like take the ordering on the ordinals

[unborn chasm]

It's relevant because injections are a natural way to formalize "bigger"

[sharp harness]

in the ordinals there is no maximum natural number, the natural numbers just keep going and omega is after all of them

and i don't see why that would be a problem

you can literally just take something that goes on forever, add another thing and declare it to come afterwards, and you have something that never ends and something else that's past it

[sharp harness]

so essentially what you're claiming is just that everything is finite...?

[pastel storm]

how so?

[sharp harness]

well you can take anything that doesn't end and just add a new thing, "X", and declare that X comes after everything else

and then you have a thing that doesn't end with something after it

so if that's impossible, then things not ending is impossible

[pastel storm]

yes but X has to come after something

[sharp harness]

it comes after... everything else

[pastel storm]

infinity has to comes after the last real

[sharp harness]

there doesn't have to be a particular element of the ordering that X is the successor of

[pastel storm]

or else it would be a real

[sharp harness]

it doesn't have to come after any particular thing

an ordering isn't specified by each thing being the successor of something else, it's specified by a rule for evaluating "less than"

5 is after 2 even though there are numbers in between

omega is after 20 even though there are numbers in between

[unborn chasm]

The canonical example for "bigger" sets is the power set. You really aren't helping by dragging it to this "add an element" direction all the time.

He doesn't understand cardinal numbers and your explanations clearly cannot make him understand whatever you are trying to express.

@pastel storm If you are interested in ordinals, I would recommend looking up cardinal arithmetic or whatever the field is called.

You are making basic mistakes and I do not think this conversation will get anywhere even if you two continue for hours.

[pastel storm]

none of this is even relevant to the 2 questions that have been asked

[sharp harness]

i guess that resolves it: we've been talking for an infinite amount of time, hence the fact that anything that hasn't already happened by now never will, which implies that 2.5 hours is in fact the largest natural number

[unborn chasm]

It is because you brought up infinities / ordinal numbers in your question. Bluntly put, you not realizing that is also a basic mistake.

You two can continue talking in circles if you wish but I would be surprised if anything came out of this.

[pastel storm]

Do you guys not agree with anything said here?

[sharp harness]

well i'm up for continuing to talk in circles, although if we're going to continue this forever i may need to sleep at some point

[sharp harness]

i do not agree with any of that

[unborn chasm]

Our point is that the question itself does not make sense.

[still temple]

pless help

[sharp harness]

all of those sets are countably infinite by my definitions, you haven't defined what "infinity" is or what you think "natural numbers" are

[unborn chasm]

We see what you are trying to get at in the initial question, but you phrased it in a way that does not make sense mathematically. It is not well-defined and as a direct result, the rationals case is completely intractable.

The further conversations did not help either.

[sharp harness]

...that's a pretty good question actually
what are the natural numbers, according to you? how are they defined?

[still temple]

cna someone help meeeee

pls

[pastel storm]

!help

[marsh citrus]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[sharp harness]

no, we've been debating for, apparently, literally forever, about infinity

[still temple]

thx

[pastel storm]

https://arxiv.org/abs/2311.09951
There is this similar idea (but not fully the same) to what I am talking about

[sharp harness]

i'm more interested in whatever it is that's exactly what you're talking about than something that's "similar" and that might turn out to just not have all of the things i'm really confused about and therefore fail to resolve anything

so anyway
what are the natural numbers?

i've already given my definition earlier (0, a total function "successor", recursion and induction (i can expand on what this means if you want))

[pastel storm]

THe natural numbers are all of the non negative real finite numbers that are separated by a distance of 1 starting from 0

[sharp harness]

...ok well now i have more questions

what are real numbers?

[pastel storm]

a number on the same line as 1 and -1

[sharp harness]

...what's a line

(i should maybe ask now, do these definitions actually end up at logical primitives at some point or are we going to end up going in circles if i try to chase the definitions all the way back)

[pastel storm]

If you keep asking for me to define things this channel is going to become a dictionary

[unborn chasm]

@pastel storm What level of math are you at? This is cutting edge.

[sharp harness]

i mean i'm fine with this channel becoming somewhat of a dictionary
probably we're only going to go a few more layers before either we end up at something that's clearly useful or i give up

[unborn chasm]

New research is not the place to learn about ordinals / cardinals if you do not already know them well.

That article seems intuitive but it is most likely hiding a lot of complex stuff.

[pastel storm]

If thats cutting edge then I'm a genius for having a similar idea or it's very easy to make a cutting edge in math

[unborn chasm]

Your idea is not the same

[pastel storm]

no but its similar

[unborn chasm]

It really isn't

[sharp harness]

having a "similar" idea is far less difficult than making it all actually work

[unborn chasm]

you haven't even managed to phrase your initial question in a mathematically sensible way

[pastel storm]

my initial question?

the formula question?

[sharp harness]

(brb)

(back)

[unborn chasm]

Anyways, GTBOT, the curiosity is a good thing but you really need to get a better understanding of the basics of cardinals / ordinals instead of diving into this.

The article defines m(N) = $\omega$. That is natural - $\omega$ is commonly defined as the cardinality of the natural numbers. However, they do not try to extend that to Q at any point from what I see.

For why it doesn't make sense to try, I again point you to how there are infinitely many rationals in (0, 1). So just the magnum of $Q \cap [0, 1)$ should be $> \omega$. And there are infinitely many such subsets (upto translation in Q) so $m(Q) \gg \omega$.

Assuming the sensible behavior the article seeks, the screenshot I sent earlier on Euler's totient function would suggest they want $m(Q) \approx \frac{3}{\pi^2} \omega^2$. However, the theory is simply not there.

[elfin berry]

chencking

[unborn chasm]

To re-emphasize, they are trying to build up the magnums and laying the groundwork for that. However, I do not see any claim they did. I also do not see them suggesting a magnum for the rationals at any point.

If they flesh out the system it may turn out like that, but that seems to be far off even if they finish their theory.

[sharp harness]

tbh i think the issue is less cardinals/ordinals in particular and more... the general way that maths works?

like i can't think of any sensible system where the natural numbers are defined as a subtype of the real numbers, which are defined in terms of geometry

and i didn't get around to doing it but i feel like if we had kept defining terms it would have either gone around in a circle or led to something where you just can't define it at all, and neither of those lead to defining things you can actually reason about

[unborn chasm]

Yeah. I am guessing he hasn't taken real analysis yet, so he is just mathematically immature. Which plays into many of the handwavey arguments.

[sharp harness]

...it is really unhelpful how so much of the education system and i guess just society in general acts like maths is about numbers or something

[unborn chasm]

😅

True, but it can't be helped either. Math majors struggle with abstractions; I doubt the general public would understand even if given an explanation.

[sharp harness]

they could at least know that they don't understand it instead of... whatever it is that they think right now

i genuinely don't know what most people's model is of what a mathematician actually does

like do they think we just spend all our time doing random calculations in the complete ordered field or something

[pastel storm]

this is why I started with asking for the formula and not just staring with what I ws truly looking for

[sharp harness]

...yeah this did kind of get a bit out of hand didn't it

well the tl;dr is

i think you... haven't really understood how maths works

...and also you're using a set of really nonstandard definitions that makes communication harder and that don't seem to actually be well-defined at all

[pastel storm]

You guys started to disprove this even though thats not what I asked for

[sharp harness]

...well you didn't define the thing you were asking for and it doesn't look like you can

the thing is

maths isn't arguments that "sound right"

if you have a "definition" but you can't continue being pedantic about it until you've obtained something so precise a computer can reason with it, then what you have isn't a definition

a proof either splits into smaller pieces, or it's something that you assume without justification because it's something really basic like "everything is equal to itself", or it's invalid

[pastel storm]

I could continue being pedantic about It but I shouldn't need to revent all of math to have a conversation

[sharp harness]

in the proof "infinity is greater than all natural numbers, therefore there is a largest natural number", i don't know how the "therefore" step splits, and i don't know what you mean by "natural number" well enough to tell a computer about it

[sharp harness]

in most conversations about maths the people involved both already know what a natural number is, so they start at the first thing where they don't have the exact same information

in this conversation, "what is a natural number" is the topic, because you are making claims about "the natural numbers" that aren't true about the inductive type generated by 0 and successor

[pastel storm]

no

the topic is the amount of rationals

[sharp harness]

ok well that was the question you originally asked, but that's not a topic we can have a conversation about yet

because we don't have a common definition of what rational numbers are or what an amount is

the same way that we can't currently have a conversation about (infinity,1)-categories because you don't know what those are, if i wanted to start a conversation about those i would have to define them first

[pastel storm]

well sure

but I would never ask you to define every word you use

[sharp harness]

i would expect that you would ask for definitions for any word you didn't already know the definition of...?

which would not be all of them because the way i use a word like "and" is probably the same as how you've seen it used before

but if i started claiming that there aren't any real numbers you would probably at some point ask what exactly i thought a "real number" actually was

[pastel storm]

Unless you don't understand how I got these numbers, you are being pedantic

[sharp harness]

i am being pedantic, because mathematics is pedantry

and in that sense i don't understand how you got those numbers, because i don't know what i could tell a computer about assigning "sizes" to sets that would produce exactly those numbers

i understand the psychology of it but i don't understand the underlying maths

i don't know what definitions and fundamental assumptions you made that made these deducible conclusions, just that they must be different in some way from mine, which make "all of these sets have cardinality aleph 0" deducible instead

and in that sense i don't know what constraints a suggested value for the "amount" of rational numbers should satisfy, because i don't know what's a valid deduction in this unknown system

[pastel storm]

well you haven't tried

like if you chose not to understand that every other natural being odd means that there are inf/2 odd numbers then your choosing to not help

[sharp harness]

(...what do you think i've been doing for the past 4 hours then...?)

[pastel storm]

nothing to do with rational numbers

[sharp harness]

...ok yeah i really think you're not understanding what this is like for me

i understand that somehow this is a valid inference - that through some sequence of steps you can derive that a set of natural numbers with asymptotic density 1/2 has a "size" of inf/2

but i don't know what any of that means

what is a size, is it just a primitive notion or is it defined in terms of something else

[unborn chasm]

That is incorrect though. The article you linked about magnums was trying to establish a function other than cardinality. It was assigning different values to sets of the same cardinality.

They spent 30 pages just building a basic notion of what that would mean.

[sharp harness]

and does that deduction break into pieces or is it a single axiomatic step here

[pastel storm]

The article I linked was only to show that I'm not crazy like yall are acting like I am

[sharp harness]

is this true for numbers other than 2? i can guess it might be but i don't know, and if i have to make a thousand guesses like that i'm not going to be right on all of them

[unborn chasm]

You're not crazy but you are not expressing yourself in a mathematically understandable way.

[pastel storm]

because to you guys the only mathematically understandable way is a bunch of fancy set theory or other equally not fully relevant things

[sharp harness]

(i'm not a guy)

[unborn chasm]

Those notions are not fancy. They are basic. The fact we keep running into issues on basic notions are why it keeps going in circles.

Like Bee said, mathematical conventions exist so mathematicians can communicate without defining everything. But you are either unfamiliar with or breaking them. So your statements end up incorrect.

For example, one example we kept circling back to is that infinity is not a natural number.
If you add $\omega$ to the natural numbers you get a different set.

[elfin berry]

chencking

[unborn chasm]

You can work with that new set, but you cannot use any of the conventional results for the natural numbers because it is something else.

So at that point, you would have to redefine / prove everything upwards of the natural numbers.

As an immediate example of things breaking, that new set is not closed under addition or multiplication.

[pastel storm]

no one said that it was

[sharp harness]

also... i don't really think you need "a bunch of fancy set theory"?

the thing i've been looking for this entire conversation is just any set of rules

like, you can describe the natural numbers perfectly by just saying that 0 and successor exist, that 0 is not the successor of anything, that if two numbers have the same successor they're the same, and that for any proposition P, if P(0), and P(n) implies P(S(n)) for all n, then P(n) for all n

and maybe you also want + and * but you can define those by just saying that x + 0 = x, x + S(y) = S(x + y), x * 0 = 0, x * S(y) = x * y + x

and after declaring all those rules to be just assumed to be true, you can do stuff like prove 0 + x = x like this:

1. 0 + 0 = 0
2. if 0 + x = x, then 0 + S(x) = S(0 + x) = S(x)
3. therefore by induction on natural numbers, 0 + x = x is true for all x
   and each step is justified by one of the assumed rules (and in the case of step 3, it's also justified by steps 1 and 2)

(to be clear for this example i'm assuming that logic-y ideas like "if", "for all", and "=" already have known meanings and i'm mostly skipping over the reasoning with them, because i expect everyone here can fill it in)

[pastel storm]

ok

all I wanted was a formula I could plug infinity (or infinity-1) into

[sharp harness]

...what kinds of formula can you plug infinity into

also wait no that isn't all you wanted, you also wanted the formula to mean something specific

otherwise you could have come up with an expression like "infinity + 5", which is a formula with infinity in it, without our help

[pastel storm]

bruh thats not what I meant

Im talking about the pinned question I started with

Is there a formula for the number of unique rationals that you can make only use integers from 1 to x

[sharp harness]

depends on what a "formula" is

[pastel storm]

Don't you dare ask me what a formula is

[sharp harness]

well if i'm not going to ask you what a formula is, which i'm not, i don't know what i should be doing...?

[pastel storm]

well what I want is the formula (closed form) if it exists

[sharp harness]

...and how do i find a formula without knowing what a formula is

[pastel storm]

how are you "helpful" on a math sever if you don't know what a formula is

[sharp harness]

well firstly a lot of people ask questions that i do understand

and also from what i can tell you're using "formula" to mean a particular sort of expression that it's valid to put infinity into within this system i don't understand

usually if someone is asking for a closed form for a particular thing they're looking for a form that's just like, easy to work with, which even if it's not a precise mathematical criteria is something i do somewhat understand

[pastel storm]

I am simpley asking for a formula

[sharp harness]

you're not "simply" asking for a formula

you're asking for a formula

is that a simple thing to do? i have no idea because i don't really know what you mean by the word "formula"

[pastel storm]

I am asking for a closed form formula to find the number of unique numbers in form p/q where p and q are only integers from 1 to x

[sharp harness]

saying the word "formula" over and over is not going to make it clearer

...also like, unless you've just decided that you really like the word "formula" and just want to acquire them for no particular reason, you're not actually looking for just "a formula", this is some step of a plan for which you think having "a formula" will be useful

[pastel storm]

https://letmegooglethat.com/?q=What+is+a+math+formula%3F

[sharp harness]

at this point i honestly can't tell if you're even reading what i'm saying

whatever you want to use this for probably does not operate on whether random websites you can find on Google consider a particular string of symbols to be "a formula"

knowing whether random websites on Google consider a particular string of symbols to be "a formula" does not give me information about your actual goals

[pastel storm]

as far as you should be concerned what I want is this formula if it exists

what I do with it is irrelevant

[sharp harness]

well there's this

[pastel storm]

thats not closed form sadly

[stark trail]

why don't we have -0

[pastel storm]

because that is equal to 0

[stark trail]

why

[pastel storm]

because it is?

0*x=0

0-0=0

[flat raft]

aren't there 4 * infinity + 1 integers

suppose we take a natural number (defined as whole numbers bigger than or equal to 1)

[pastel storm]

natural numbers include 0

[flat raft]

we generate the integers by 2n-1, 2n, -2n, and -2n + 1

okay then there are 4*infinity - 3

like suppose we use 1 to generate the numbers 1, 2, -1, and -2

and then we use 2 to generate 3, 4, -3, and -4

so therefore there are 4 integers per natural number, except 0, which only has 1

[pastel storm]

those would not all be integers iirc

[flat raft]

they are

[pastel storm]

no

[flat raft]

which ones aren't integers

[pastel storm]

the numbers generated by "the set of half the naturals such that they are the biggest"

[flat raft]

which natural numbers

name one

[pastel storm]

I can not

[flat raft]

every natural number makes 4 integers

so why should a set of them suddenly make a non-integer?

[pastel storm]

every natural number dose not make 4 integers

I just said that

[flat raft]

wait by this we can conclude 4*infinity - 3 = 2 * infinity - 1, which means 2 * infinity = 2, and therefore infinity = 1

[flat raft]

which natural number doesn't make 4 integers, besides 0?

[pastel storm]

I give up

[flat raft]

this makes a lot of sense because 1 is a bunch of infinite parts

[pastel storm]

.close

[marsh citrus]

Channel closed

Closed by @pastel storm

Use .reopen if this was a mistake.

[marsh citrus]

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.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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• 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.

[stuck marsh]

hello

[stuck marsh]

What does it mean to find an upper bound of |f(x, y)| using a norm?

[stark trail]

this function is from R2 -> R

so I think the norm you use is just the normal absolute value

so you want to find something which bounds this function from above and below maybe

which doesn't actually seem possible

could you show the question even if it is in spanish?

[stark trail]

french?

[stuck marsh]

yes

so i use the absolute value

[azure sky]

I don't know whether this uses a norm but you could try finding a minimum of the inverse.

(Multiplicative inverse)

1/f(x,y)

[stuck marsh]

why not immediately f(x,y)?

[azure sky]

Try it! It gets simpler

[stuck marsh]

ok$

[stuck marsh]

1/x?

[azure sky]

I meant you could try to minimise 1/|f(x,y)|

[stuck marsh]

$\frac{1}{{f(x, y)}} = \frac{{x^2 + 2y^2}}{{xy^2}} \geq \frac{{y^2}}{{xy^2}}$

[elfin berry]

lilisworld

[stuck marsh]

or by minimum you meant what?

[azure sky]

I meant a number L such that 1/|f(x,y)|>L for all x and y

Then 1/L is an upper bound of f.

[stuck marsh]

a number that doesnt contain x or y?

[azure sky]

A constant, yes

[stuck marsh]

0 then

uhm no

[azure sky]

Do you know whether you are supposed to find the supremum of |f(x,y) (i.e. the smallest upper bound) or just any upper bound.

[stuck marsh]

yes

but why not immediately a supremum of f(x,y) then

[azure sky]

Yes to what?

[stuck marsh]

i need to find an upper bound srry

yes upper bound

[azure sky]

Okay

[stuck marsh]

it's just the norm part that i dont understand

[azure sky]

Hmmm, actually no such bound should exists.

[stuck marsh]

you mean a constant?

or just any other function

like for example |f(x, y)| <= |x| i think

[azure sky]

I'm not sure

[stuck marsh]

but when you said no such bound exists, you mean no such constant or no such function?

[azure sky]

A constant

Sorry

[stuck marsh]

ah ok

[azure sky]

But thats probably not what's required then.

Honestly I don't know

[stuck marsh]

i just need to show that it's continuous

if we ignore that part then how should we do it then

@azure sky

@stark trail

[azure sky]

Sorry, I don't think I understand the problem

[stuck marsh]

like i mean

if i asked you to prove that f(x,y) is continuous, how would uou do it

[azure sky]

Oh

Well, assume |x| and |y| are less than ε

Then do two cases: x>y and y>x. In both cases, |f(x,y)| is less than ε/2.

Thus f is continous at 0,0 and it's clearly countinous everywhere else.

*ε/3

[stuck marsh]

wait idont understand how |f(x,y)| < ε/2 implies it s continuous at (0,0), did you use the limit definition?

[azure sky]

Yes

[spark otter]

There are multiple good norms for R²

[stuck marsh]

shouldnt we show that that limit is 0 tho

[spark otter]

You can choose any valid norm on R² btw, they're all equivalent

[stuck marsh]

absolute value?

ok

[spark otter]

Absolute value only exists on R

[stuck marsh]

oh

norm2

[spark otter]

On R², you have 2 coefficients

[spark otter]

Sure

[unborn chasm]

This isn't bounded, is it? Taking $x = y$ gives $x/3$. That is clearly unbounded.

[elfin berry]

chencking

[spark otter]

Norm1, norm2, norm-p,... infinite norm

[stuck marsh]

so i need to bound with norm2

norm2 of (x,y)

?

[spark otter]

You want to show continuity on which point?

[stuck marsh]

0

(0,0)

[spark otter]

Ok

So the exercise hints you at showing |f(x,y)| <= something in terms of ||(x,y)||

[stuck marsh]

ok

[spark otter]

(Hint : the denominator is bigger than the second term)

[stuck marsh]

$\frac{xy^2}{{x^2 + 2y^2}} \leq \frac{xy^2}{{x^2 + y^2}} = \frac{xy^2}{{\lVert (x,y) \rVert}_2}$

[elfin berry]

lilisworld

[stuck marsh]

uhm no wait i forgot sqrt

[spark otter]

Ah this is maybe not the best way

[spark otter]

Try something like this

[stuck marsh]

ok so x²+2y² > y² so $\frac{xy^2}{{x^2 + 2y^2}} \leq \frac{xy^2}{y^2}$ only?

[elfin berry]

lilisworld

[stuck marsh]

so |x| remains

[spark otter]

Yep

Put it in absolute value too

|f(x,y)| <= |x|

(You can check this is also true when y = 0)

But |x| <= ||(x,y)||

[stuck marsh]

ok so i just say the lim of ||(x,y)|| is (0,0) when approaches (0,0)

so its continuous on (0,0)

[spark otter]

Yep

Et voilà

[stuck marsh]

and this one?

show that f(x,y) is not continuous at (0,0)

[spark otter]

Haha et merde

Ok so you might have a result that arctan(a) ~ a when a->0

[stuck marsh]

so arctan(xy) ~ 0 when (x,y)->(0,0)?

or what

[spark otter]

no, arctan(xy) ~ ...

[stuck marsh]

xy

[spark otter]

Yes

[stuck marsh]

but xy is 0?

oh nope nvm

ok

[spark otter]

So

Now our task is to prove that xy/(x²+y²) is not continuous at 0

[stuck marsh]

ok

[spark otter]

Maybe show that you can find (at least) two ways to approach (0,0) such that f(x,y) doesn't approach the same quantity

[stuck marsh]

do i need the sequential characterization?

[spark otter]

Sure this will come in handy

[stuck marsh]

so it's obviously not (1/n, 0) and (0, 1/n)

[spark otter]

Well at least you found f(1/n, 0) that converges to 0

So let's find something else that doesn't converge to 0

[stuck marsh]

(n, n)?

converges to 1/2

[spark otter]

I know what you meant with (n,n) but perhaps rewrite it to be correct

[stuck marsh]

the sequance such that $x_n = (n, n)$ converges to 0

[elfin berry]

lilisworld

[spark otter]

When n goes to infinity?

[stuck marsh]

ahhhh no

srry

lol

[spark otter]

Its ok xd

[stuck marsh]

then (1/n, 1/n)

[spark otter]

Yep

[stuck marsh]

so it's still 1/2

the value of f(x_n)

[spark otter]

Yes

So two sequences that converge to 0 have f(...) not converge to the same thing

Which means that

[stuck marsh]

it's not continuous at (0,0)

[spark otter]

Exactly

[stuck marsh]

ok thank youuuu 🥲

.close

[marsh citrus]

Channel closed

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[spark otter]

Pas de problème

[marsh citrus]

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[ruby mulch]

!status

[marsh citrus]

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

[still temple]

1

[gleaming pecan]

use de l'Hospital's rule, then this limit solves itself practically

[ruby mulch]

i don't think that's the first step.

[gleaming pecan]

it is

[gleaming pecan]

a hint, additional one:

$\frac{d}{dx}\left[ \int_{a}^{g\left( x \right)}f\left( t \right)dt \right]=g'\left( x \right)f\left[ g\left( x \right) \right]$

[elfin berry]

Joanna Angel

[still temple]

i see

well i tried something right now

[gleaming pecan]

ok, sure

[still temple]

considered root (t) as K

then used by parts and integrated

then applied limits and got answer 2

[gleaming pecan]

I disagree, but it is your exercise 🙂

[still temple]

uh okay

[gleaming pecan]

$\lim_{x \to 0} \frac{\int_{0}^{x^{2}}sin\left( \sqrt{t} \right)dt}{x^{3}}=\left[ \frac{0}{0} \right]\overset{H}{=}\lim_{x \to 0} \frac{2xsinx}{3x^{2}}=\lim_{x \to 0} \left( \frac{2}{3}\cdot \frac{sinx}{x} \right)=\frac{2}{3}\cdot 1=\frac{2}{3}$

[elfin berry]

Joanna Angel

[still temple]

hmm okay

.close

[marsh citrus]

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[valid kestrel]

I'm stuck on part c and part d of the question

how would I do part c

because no transformation I try works

[hidden plaza]

hint: (\wrb{8 = 2^3})

[elfin berry]

Pure

[valid kestrel]

How does that help?

Ohhh

is it f(2x)

?

no that wouldn't work

[grizzled bobcat]

why wouldn’t it work?

[valid kestrel]

because it gives y=2x^3-6x^2+16x-24

which is not the function

[hidden plaza]

It doesn't give that.

[\wrb{(2x)^3=2^3x^3}]

[elfin berry]

Pure

[valid kestrel]

oh

yeah

ur right nevermind I interpreted it wrong

thanks

.close

[marsh citrus]

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[storm fern]

translation: given a geometric sequence consisting of positive terms, if ... = 1/9, then ... = ?

[storm fern]

i only found the ratio of the sequence

3

[marsh citrus]

@storm fern Has your question been resolved?

[storm fern]

.close

[marsh citrus]

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[brazen flame]

Hello, can someone help me in understanding how to find an angle between 2 rights, also, to find the angle between CT and NO

[past frigate]

CT and NO are on two diff planes

[brazen flame]

I can see that

But I don't know what to do from here

[marsh citrus]

@brazen flame Has your question been resolved?

[marsh citrus]

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[marsh citrus]

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[west nest]

Is this an error?

[west nest]

Should one of the t's have a negative sign?

[marsh citrus]

@west nest Has your question been resolved?

[obsidian python]

no, they are equal

x2=x1=t

[west nest]

So is this incorrect?

Here they put a -???

[marsh citrus]

@west nest Has your question been resolved?

[marsh citrus]

@west nest Has your question been resolved?

[long osprey]

yes, there is an error in this one

[marsh citrus]

@west nest Has your question been resolved?

[marsh citrus]

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[wary bluff]

1. is true right

and so is 3)

but what abt 2) and 4)

[sacred kestrel]

as far as im aware you can orthogonalize any basis. about 90% sure on that

[wary bluff]

so 2) is true aswell?

[sacred kestrel]

i would consider (2) true unless someone else has a better explanation

[wary bluff]

what about 4)

[long osprey]

(2) is indeed true as given a basis, you can orthogonalize

[sacred kestrel]

(4) is definitely true to me, because the orthogonal (normal) vector of a plane is the nullity/kernel/0 eigenspace

its the orthogonal complement to the row space

[wary bluff]

wait wait what

i dont get what u mean here

[sacred kestrel]

any multiple of the vector n is outside the plane U

think about it

what direction would a normal vector face from a plane?

directly away

[wary bluff]

right

[sacred kestrel]

T top of the t is the plane, the stem is the orthogonal vector n

[wary bluff]

I didnt understand what u meant by "is the nullity/kernel/0 eigenspace"

[sacred kestrel]

U being the row space, n defines the null space.

[wary bluff]

oh wait ur just talking about the null space right

[sacred kestrel]

null space is orthogonal to the row space.

[copper raven]

why would 1) even be true

[wary bluff]

oh shit mb i didnt know what kernel was I thought u were dividing them or something

[sacred kestrel]

kernel = nullity

[wary bluff]

alr

[wary bluff]

its not?

[copper raven]

span {n} is dimension 1

U is a plane so it's dimension 2

[wary bluff]

oh

[copper raven]

no way they're equal

[sacred kestrel]

assuming thiss is R^3, U is a 2d space in R^3, n is the missing dimension (null space)

[copper raven]

yes exactly

[wary bluff]

alr

ty

.close

[marsh citrus]

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[eternal sequoia]

How do I find a equation for X, Y and C in this problem? I have atan2(x,y) but that's a piecewise function. So then, how do I even solve this?

[eternal sequoia]

There are 6 different possible values for X,Y, and C in the end given atan2 has 6 possible values

[marsh citrus]

@eternal sequoia Has your question been resolved?

[marsh citrus]

@eternal sequoia Has your question been resolved?

[marsh citrus]

@eternal sequoia Has your question been resolved?

[opaque widget]

<@&286206848099549185>

this dude needs help

[unborn chasm]

I'm sorry, but could you explain your question in more detail?

I see $f(x, y) = x^2 + y^2 - 4$ at the top. What are you trying to do with this f? What's all the stuff below it?

[elfin berry]

chencking

[marsh citrus]

@eternal sequoia Has your question been resolved?

[eternal sequoia]

Oh the context of this is basically just computing light reflections. I have a vector field and I broke it down into its components. I basically multiplied some constant c by the vector and set the product equal to the surface vector function. So basically, I am trying to find at what point x and y, the reflected ray will hit the surface again to compute a secondary reflection ray

basically,

R(x,y) = <R_x , R_y, R_z>
V(x,y) = <x , y, x^2 +y^2 -4>
cR(x,y) = V(x,y)

c*R_x = x
c*R_y = y
c*R_z = x^2 + y^2 -4

that atan2 really throws me off

[unborn chasm]

Is that handwritten part how the problem defines the ray's position function?

It doens't move linearly?

Is $a\tan^2$ here $(\tan^{-1})^2$ or $\tan^2$ multiplied by a coefficient $a$?

[elfin berry]

chencking

[unborn chasm]

What is $\tan(-x, -y)$ supposed to be?

[elfin berry]

chencking

[marsh citrus]

@eternal sequoia Has your question been resolved?

[marsh citrus]

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[ebon depot]

What is the general rule for when I can put x = 0 when handling lim?

[twilit grove]

when the denominator doesn't equal 0

[ebon depot]

Oh ok! So the moment that happens then I can put x=0?

[twilit grove]

yes. that's why they "irrationalize" the denominator

[ebon depot]

Aaah I see!! Tysm ❤️

.close

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[wooden grotto]

Could someone help me with this problem: "Let F(x,y) = <-y,x>. Find the line integral of F(x,y) along the quarter of the unit circle in the second quadrant oriented counterclockwise from (0,1) to (-1,0)"

[gleaming pecan]

$\int_{\frac{\pi}{2}}^{\pi}\left[ -y\left( t \right)\cdot x'\left( t \right)+x\left( t \right)\cdot y'\left( t \right) \right]dt,\text{ where}\text{ }x\left( t \right)=cost,y\left( t \right)=sint$

[elfin berry]

Joanna Angel

[marsh citrus]

@wooden grotto Has your question been resolved?

[marsh citrus]

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[wooden grotto]

.reopen

[marsh citrus]

✅

[wooden grotto]

Could we go through the entire process of this problem? I'm deeply lost.

[gleaming pecan]

Have you ever calculated a line integral along an oriented arc of a vector field?

[wooden grotto]

I've calculated line integral using the fundamental theorem of calculus for line integrals...but that's about it.

[gleaming pecan]

there is a formula, I omit specialised assumptions since all of them will work :

$\text{If }\text{ }\Gamma=\left{ \left( x,y \right):x=x\left( t \right),y=y\left( t \right),t\in \left[ \alpha,\beta \right] \right}\text{ }\text{ then}\\\int_{\Gamma}^{}P\left( x,y \right)dx+Q\left( x,y \right)dy=\int_{\alpha}^{\beta}\left[ P\left( x\left( t \right),y\left( t \right) \right)\cdot x'\left( t \right)+Q\left( x\left( t \right),y\left( t \right) \right)\cdot y'\left( t \right) \right]dt$

[elfin berry]

Joanna Angel

[gleaming pecan]

F = [P(x,y), Q(x,y)] sia a vector field, = vector function

gamma arc is a line of integration, it is oriented arc due to parametrisation given in yoru task

in yoru case, that is arc of an unit circle, you said,

from point (0,1,) to (-1),0) , anticlockwise

unit circle parametrisation is :x = cost, y =sint, these are known equations to represent circle

plz sketch the graph , situaitonal on yoru paper, to obsrve what i am talkign ab

[marsh citrus]

@wooden grotto Has your question been resolved?

[wooden grotto]

Is there a simpler way to approach this? never worked with this formula before.

<@&286206848099549185>

[marsh citrus]

@wooden grotto Has your question been resolved?

[marsh citrus]

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[marsh citrus]

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[naive echo]

unfortunately my question is in Turkish, but I'm hoping that people abroad have learnt these topics too 😭 I'm a high school freshman, and I've been stuck on this question for the past few minutes. (It says that X, Y, and Z are whole numbers, and that A=B. It's asking for the answer of x multiplied by y multiplied by z)

There's nothing I've really been able to try, because I'm not sure how to process the question in general. I'd be glad if someone could help in any way ★

[somber holly]

Are A and B sets by any chance?

[naive echo]

I believe that's what it's supposed to be in English, yes.

[jolly sand]

Is the answer -15?

[naive echo]

How did you work it out if I may ask?😭

[jolly sand]

So we are given sets

And they are equal

The first element of A is 1

If A=B that means there must be 1 in B as well, doesnt matter which element of B is it (cause its sets)

So B has 3 elements

Lets check

Is 1 = 9?

[naive echo]

No, but I think I understood your solution from you saying it didn't matter which element it was. Thank you so much!!

I was wondering how 1 was able to equal 9.

[jolly sand]

Haha all good. Pay attention that you have whole numbers, there is a trap if you dont pay attention and you get that the answer is 0

Gunaydin or whatever ya all say in turkish

Gule gule actually

[naive echo]

It's currently night time so you say iyi geceler, haha

[somber holly]

By that logic y is - 5 right?

[naive echo]

i believe so

[jolly sand]

-1

X is 5

[somber holly]

But y is a whole number, it cannot be negative

That's what we follow here, do they include negative integers as whole numbers in turkey?

[jolly sand]

Whole numbers can be

[naive echo]

Yes.

[jolly sand]

Whole means -5 0 1 6 -1783 etc

1.4 is not whole

[dire spire]

i don’t think so

i’ve also been taught that whole numbers cannot be negative

[jolly sand]

Hmmm

[naive echo]

I've been taught the opposite

[dire spire]

interesting

[jolly sand]

I think nikolais just made a poor translation

Ur right whole numbers are 0, 1, 2...

[dire spire]

maybe the question says integers

[jolly sand]

That problem wont have a solution otherwise so yeah

[naive echo]

Could be, I'm sorry if so!! I haven't really heard the term integer before but thanks for the information.

[jolly sand]

I got whats the problem

[naive echo]

Yeah, it was a poor translation. I'm sorry for the mixup!

[jolly sand]

From turkish when you translate

It literally means whole numbers

Its just language problem

I translated to my native armenian it said whole numbers too but meant integers

[naive echo]

Yes, your term "whole numbers" means "doğal sayilar" in Turkish which literally translates to "Natural numbers", I'm sorru for the mixup

[somber holly]

Well

This was a funny incident

Tells you how important a good translation is

[naive echo]

yup

I'll shut the channel then, thanks again!!

.close.

.close

[marsh citrus]

Channel closed

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[somber holly]

Anytime

[marsh citrus]

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[fossil grail]

How do I solve: if n∈N\{0} and U is a set with n elements, find the number of elements in {(X, Y ) | X, Y ⊆ U & X ∩ Y = ∅}

[fossil grail]

Is it true that the answer is 3^n?

looking at the possibilities, it should be can be included in X, Y, neither, both but not in both

[proud ice]

Do you mean "if $n\in\bN_0$"?

ah

[fossil grail]

the damn discord cringe markup

sorry

[proud ice]

If $n\in\bN\setminus{0}$ and $U$ is a set with $n$ elements, find the number of elements in $Z={(X, Y) : X,Y\subseteq U \wedge X\cap Y=\emptyset}$

[elfin berry]

SWR

[fossil grail]

yessir

[proud ice]

!status

[marsh citrus]

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

[fossil grail]

I think I solved it

I just am asking if I'm right

[proud ice]

In short, you're trying to find the set of all disjoint subsets of U?

[fossil grail]

uhh

no

the number of elements in the class above

for the sets given

[proud ice]

err... the cardinality of the set of all disjoint pairs of subsets of U

[fossil grail]

I think so

idk

it's not the same language I have

[proud ice]

3^n seems right, but how did you get it exactly?

[fossil grail]

well

I just visualized it

it has to be in X, in Y, in neither, but not in both

[proud ice]

"it has to be in X"? What is "it"?

[fossil grail]

the n elements

[proud ice]

ah

[fossil grail]

my lecturer used Newton's binomial formula

[proud ice]

So you iterate through each of the n elements, say "it belongs to exclusively X, exclusively Y, or neither", giving you 3 choices. Thuse 3^n total outcomes for the n elements

Clever approach.

Yeah I agree with your reasoning

pog indeed

Only other way I could think to solve it would be through induction, which sounds like a nuisance.

[fossil grail]

oh nah

this is combinatorics

My lecturer used Newton's binomial formula

which confused me a little

it

says

C(n, 0)2^n + C(n, 1)2^n-1 + ... + C(n, n)1=(1+2)^n

idk how

[proud ice]

yeah I have no idea either. No clue what the 1 and 2 are doing here.

[fossil grail]

that's alright

[proud ice]

btw, I never took combinatorics. Do you treat 0^0=1?

[fossil grail]

thanks for the help

uhh

yeah

g2g

[proud ice]

wild

[fossil grail]

buh bye

.close

[marsh citrus]

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[still temple]

So for this I tried an alternative approach by using the joint hypergeometric distribution: [
\m\P{Y_1 = y_1, Y_2 = y_2, \dots, Y_k = y_k} = \f{\binom{m_1}{y_1}\binom{m_2}{y_2} \dots \binom{m_k}{y_k}}{\binom mn}, \q (y_1, y_2, \dots, y_k) \in \N^k
]

[still temple]

but I don't get the same answer

so what's wrong

[runic temple]

Some of these numbers look sus, gimme a sec

[still temple]

agreed

I think my teacher is being sussy

[runic temple]

Did you make the table or did they

[still temple]

they

professor just wrote that the other day but I was kinda sceptical in class

[still temple]

just thought about it again because my calculations thru this don't land the same answers

[runic temple]

Actually it checks out

[still temple]

yeah so I'm wrong

uhh lemme see

oh I see lmao yeah it's okay

thabks

.close

[marsh citrus]

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[marsh citrus]

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[marble glacier]

def secant(f, a, b, tol):
    x0 = a
    x1 = b
    n = 0
    while abs(x0 - x1) > tol:
        n = n + 1
        x2 = x1 - f(x1)*(x1 - x0)/(f(x1) - f(x0))
        x0 = x1
        x1 = x2
        print(x0)
    return x2, n

[marble glacier]

i have this secant method

for root finding

1.1 1.0319286204529856 1.0319286204529856
1.0319286204529856 0.9929544004363596 0.9929544004363596
0.9929544004363596 1.0004259265358206 1.0004259265358206
1.0004259265358206 1.0000054550936772 1.0000054550936772
1.0000054550936772 0.9999999957379389 0.9999999957379389
0.9999999957379389 1.0000000000000426 1.0000000000000426
1.0000000000000426 1.0 1.0
1.0 1.0 1.0

However

as you can see it reutns 8 iteratiohs, even thought it does it in 7

Thats because my condition while is only base on x0 and x1

so i was wondering which is the correct ammount ofiteraiotns

would it be when all of them are 1?

[crystal lintel]

“correct amount of iterations” depends on your desired error

[marble glacier]

print(secant(f, 0.8, 1.1, 1E-15))

1E-15 is my tolerance / error

But i just dont understand if its 7 or 8 iterations

is it when x0 reaches the root, or when all 3 of them reach the root

if it helps my f(x) is :

def f(x):
    y = (x-1)*(x-2)*(x-3)*(x-4)
    return y

and my interval is a = 0.8 b = 1.1

[crystal lintel]

can you post a screenshot of the output? it doesn’t display very well on my phone screen

[marble glacier]

https://gyazo.com/b933a2441769752090b2b2f73e5865b3

this is x0, x1, x2 in that order

[crystal lintel]

thx, this wasn’t quite doing it

[marble glacier]

xD

Sorry

But yea on my 7th iteration

x1 and x2 are on the root (1) but x0 is not

so has it found the root

or not?

[crystal lintel]

lol dw blame phone ui

i think it’s 7?

[native frigate]

it never actually finds the root, but it finds it within your tolerance

you dont actually need to do the extra step to make x0 also the root, but its always going to be exactly one step so computationally irrelevent (unless you have the worlds most complex function)

[marble glacier]

so

would

def secant(f, a, b, tol):
    x0 = a
    x1 = b
    n = 0
    while abs(x0 - x1) > tol:
        n = n + 1
        x2 = x1 - f(x1)*(x1 - x0)/(f(x1) - f(x0))
        if f(x2) == 0:
            return x2, n
        x0 = x1
        x1 = x2
        print(x0, x1, x2)
    return x1, n

me adding the

f(x2) == 0:
return x2, n

be better

fuck it

im just gonna secant method 'bounces'

[native frigate]

its marginally better, but harder to read

i wouldnt add it

[marble glacier]

Well

[native frigate]

def secant(f,x0,x1,tol):
  while abs(x0-x1) > tol:
    x2 = x1-f(x1) * (x1 - x0) / (f(x1) - f(x0))
    x0, x1 = x1, x2
    # print(x0, x1)
  return x1

def f(x):
  return (x-1)*(x-2)*(x-3)*(x-4)

secant(f, 0.8, 1.1, 1E-15)

This also has the one extra step, and is how I wrote it for testing

[marble glacier]

Yea

so

you think i should just leave the extra iteration

blame it on some floating point approximation

[native frigate]

i geniunely think adding the breakout to save the interaction will make the program slower

if you were going to optimize for time there are better ways to do it

ie not call f(x1) twice

[marble glacier]

Im not too fussed about speed or optimisation

I just want to know which number of iterations is correct

7 or 8

[native frigate]

7 is within your tolerances, and 8 is slightly better

does that answer your question?

[marble glacier]

Yea

Thanks

.close

[marsh citrus]

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[light wren]

quick linear algebra II question: if we want to find the diagonal matrix of a matrix A, then do we just find the orthogonal matrix of A using the gram-schmidt process and then we do (Q^T)AT to get D, which is the matrix we want to find?

[light wren]

Q^T because the transpose of and orthogonal matrix is its inverse

[marsh citrus]

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[frank tusk]

Is it A

[lament stag]

What is the LCM of 24 and 18

[frank tusk]

72?

@lament stag

[lament stag]

Thanks!

[frank tusk]

So uhh is it A

..

Oh nvm I think you were just asking

<@&286206848099549185>

[marsh citrus]

@frank tusk Has your question been resolved?

[marsh citrus]

@frank tusk Has your question been resolved?

[void lagoon]

hi we arent interested in doing your homework for you, but if you have a question on how to solve the problem we could potentially point you in the right direction

[marsh citrus]

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[marsh citrus]

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[final copper]

Im having a little trouble wiht the set up with this problem

[final copper]

i get to

3x^2 + 6xh+3h^2 - x - h - 3x^2 + x - 2

after cancelling out im left with

6xh+3h^2 -h -2

/ h

[distant peak]

recheck your calculations

[final copper]

am i setting it up incorrectly?

3(x+h)^2 - (x+h) - f(x)?

holy crap

im missing the +2

🤦‍♀️

lmao sorry for that

have a good one

.close

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[wide flare]

I tried using this and put 2 where N was but it didnt work

or is that wrong

[marsh citrus]

@wide flare Has your question been resolved?

[wide flare]

<@&286206848099549185>

whatever man help channels dead

.close

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[thorny jungle]

How do we approach b?

For just considering say one suit, there is 12 consecutive cards of sequence

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[final copper]

would this be 50?

[final copper]

or am i trippin

[wide acorn]

What do you think ?

[cunning basin]

yes 50

[final copper]

in my logic

i figured since even function its linear

so -2 to 0 would be half

and it asks for [0,2]

5*10=50

thats how i approached it at least

[wide acorn]

That’s it

[final copper]

preciate it

.close

[marsh citrus]

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[wooden grotto]

Could someone help me with this problem?

[wooden grotto]

Specifically B and C....I wa given comments but don't understand how to address them. Look:

[cunning basin]

there is no integer n such that f(a)<n<f(a)+1 is a contradiction so you don't need to explain further

[wooden grotto]

But I was told I must?

[cunning basin]

oh I meant the hence part is not needed

[wooden grotto]

Okay. Do you understand How I'd be able to explain the last sentence more clearly?

<@&286206848099549185>

[marsh citrus]

@wooden grotto Has your question been resolved?

[wooden grotto]

<@&286206848099549185> please?

[marsh citrus]

@wooden grotto Has your question been resolved?

[marsh citrus]

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[heavy vortex]

Hai can someone help how to solve this please? Thank you

[still galleon]

are those two seperate lines?

[heavy vortex]

It's Quadratic Simultaneous Equations

two seperate yeah

[still galleon]

what do you have to find? where they meet?

or just x and y?

[heavy vortex]

It only says this

[still galleon]

ah okay I see

[twilit grove]

try getting both as expressions of y and equating them

[heavy vortex]

these are the examples

but unfortunately in my class we don't have a great teacher so-

lol

[heavy vortex]

I don't know how it's my first time

[twilit grove]

it's the same procedure as that first example

[still galleon]

dw my teacher taught me this today 😂

[heavy vortex]

😂

[still galleon]

you wanna set them so they give a value and = to 0

[twilit grove]

make y the subject of your first equation, then set it equal to the second

[heavy vortex]

yup-

so first the y= or?

[twilit grove]

do you know what "make y the subject of the first equation" means?

[heavy vortex]

No.. it's my first time-

[twilit grove]

get the first equation in the form of "y = ..."

[heavy vortex]

y = 2x^2 -x -6

??

[heavy vortex]

?

[twilit grove]

the first equation

[heavy vortex]

2y = or

y =

?

[twilit grove]

I stated above what it should be

[heavy vortex]

uh-

[still galleon]

y=x+5.5?

[marsh citrus]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[heavy vortex]

Bruh this server is different

[still galleon]

u dont say

[heavy vortex]

nevermind I'll probably ask the other server then since this server probably won't help-

[still galleon]

in short I just made it simpler for you

[heavy vortex]

yeah-

[still galleon]

yk cause I divded them all by 2

[heavy vortex]

damn

so everytime theres those equations for example 2x at first it always divide by 2 or?

[still galleon]

i mean it looked like it could be divided by 2 so I did it hahah

[heavy vortex]

thanks hahah 👍

[still galleon]

cause then i guess you'd have 1y=1x+5.5

and then you have your y value, which you'd replace in the other equation

[heavy vortex]

true yeah 👍

[still galleon]

and then you should be able to get your x value by just isolating x

and then replace that x value into your y value to get your y value haha

makes sense?

[heavy vortex]

I'll try just sec the other server replied

sorry for bothering

[still galleon]

no worries brother

I learned that literally today haha

[marsh citrus]

@heavy vortex Has your question been resolved?

[marsh citrus]

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[orchid oracle]

$$\int \frac{1}{\sqrt{x} + \sqrt[3]{x}} dx$$

[elfin berry]

rainy

[orchid oracle]

How would you solve this?

i was thinking u substitution

but what should i take as u

[craggy needle]

hmm

interesting question

just a quick qquestion

what level of maths are you t

[orchid oracle]

uuh

idk how to tell u exactly im not from us

its uni

calc

[craggy needle]

neither lol

first year?

second?

[orchid oracle]

its complicated 😂

lets say first

[craggy needle]

okay no worries

what integration techniques do you know?

im trying to work out how they would want this done

[orchid oracle]

uuh i think just u subst atm

i mean were supposed to do it like that for this one

[hidden plaza]

sub it (\wrb{u=\s[6]x})

[elfin berry]

Pure

[orchid oracle]

ok but why

how do you see that

[craggy needle]

oh my god wtf

the solution is so bad

yea its exactly that

[orchid oracle]

yes its a long one lol

[craggy needle]

its the common root kinda

like

like

x^1/2

and x^1/3

[hidden plaza]

well we wrte

[\wrb{\int \f 1{x^{1/2}+x^{1/3}}}]

[elfin berry]

Pure

[craggy needle]

dude why is ur latex so cool

[hidden plaza]

so the idea is if we have u^6 as x then the powers will cancel and we'll get integer powers

™️

[craggy needle]

also the other thing to see that as

is to divide the top and bottom by the difficult powers

i.e

divide everything by x^1/3

[hidden plaza]

Actually @orchid oracle

try this integral