help-0

[low forum]

No

[alpine sable]

what level math is this? High school, middle school?

[low forum]

Highschool algebra 2

But he just gave us this packet for honors

[alpine sable]

hmm.. my best guess is that theta prime refers to an angle that is congruent to theta

[low forum]

Oh I see but it is confusing since it just says given all the values of theta

yeahh

[alpine sable]

those angles would be 35, 35 + 360, 35 - 360, -35

[low forum]

woah whered u get that

[alpine sable]

and maybe there's a few more combinations

well, suppose you have an angle of 35 degrees, if the second hand of the angle goes all the way around itll end up in the same place

so angle 35 + 360 is congruent to angle 35

[low forum]

Ohhh

Yeah

[alpine sable]

and then if it goes all the way around the other way it's congruent again

so 35 - 360

35 - 720 would also work

35 + 720 not since it's outside of the given limit

[low forum]

Right because its saying 720 is greater than or equal to it okay

Do I make like a unit circle to show that

bc it wants me to show a diagram

[alpine sable]

yes, you can

I'm not sure if -35 counts though

might not depending on the definition of congruency for angles

[low forum]

Oh

It looks like it should

[alpine sable]

yeah, I think it's ok

so the answer would be 35, -35 and then adding and subtracting 360 and 720 from them

35 + 360 = 395, 35 - 360 = -325, 35 - 720 = -685
-35 + 360 = 325, -35 + 720 = 685, -35 - 360 = -395

these should be all the answers, unless I missed something

assuming the prime is referring to congruent angles

[low forum]

So like angle -35 tho where would I draw that like and i put 35 there

[low forum]

so whenever it says theta and then the ' sign like that it means prime?

[alpine sable]

that symbol is called prime

'

[low forum]

Oooh

[alpine sable]

because there's one

this symbol is called second ''

because there's two

' sometimes means minute as wel

positive angles go counter clockwise

so negative angles go clockwise

-35 will be at the opposite of 35 clockwise

[low forum]

Ohh minute as well wow

Oh okay

[alpine sable]

it's best to mark your angles with circle arcs to show where they start and stop

[low forum]

like that?

[alpine sable]

no

Like this

they start from 0

35 goes counter clockwise

-35 goes counter clockwise

and add these markers I drew as well to help you keep track which angle is which

[low forum]

Ohhh'

okay that makes sense

yeah

Omg that makes so much sense the problem was so weird at first

[alpine sable]

Maybe also add arrows like this

and then 35 + 360 will go all to 35 and then all the way around

like this

I think you can figure out how to draw the remaining ones

[lone heart]

@low forum Has your question been resolved?

[low forum]

oh to show the direction yes

[lone heart]

Channel closed

Closed by @low forum

Use .reopen if this was a mistake.

[low forum]

yes

omg tyy for all that help

[lone heart]

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[alpine sable]

Hello all. I had this shower thought yesterday and have been thinking about it ever since. Say you start with $1$, and there’s only two operations you can perform on $1$. You can double it or add one. Of course you can also apply those same operations to the result of whichever operation you applied before. What would be the most efficient (and by that I mean with less "steps"/operations) way to reach an arbitrary natural number $n$? The least efficient, of course, would be to iterate the $+1$ operation $n−1$ times. How would you approach this? Thanks.

[ocean seal]

3317

[alpine sable]

this sounds really interesting

[surreal sun]

agreed ahah, i'd like to see the solution

[alpine sable]

I'm not sure if there's a generalized solution

[surreal sun]

well you should probably iterate at the beginning at then double it no ?

but 'till when

[median dirge]

I think there is a generalized solution

[surreal sun]

maybe python could help visualize

[median dirge]

It should go this way

If n is 1

[alpine sable]

supose n is a prime number

you can't get to it by doubling

[verbal mural]

[median dirge]

You can double till a feasible number

[alpine sable]

so the last step has to be adding for example

[median dirge]

And start adding 1

Yea

[alpine sable]

so maybe determine multiple algoriths depending on the numbers divisibility

[surreal sun]

oh yes my bad

[verbal mural]

The issue is that adding 1 in the middle of doubling could be more efficient

[alpine sable]

Indeed.

[median dirge]

You have to do $2^m < n$ and then do $2^m+k(1) = n$

[ocean seal]

Muhammad Hussaini

[median dirge]

m and k can be 0

Thats it

[verbal mural]

Like to get to 6, I think is a counter example to your method @median dirge

[surreal sun]

This problem probably has a name no ? would bet some mathematician thought about this too ahah

[median dirge]

m = 2 and k = 2

[alpine sable]

Not sure, really. But who knows, right?

I feel like a tree diagram would work well to visualize this

[verbal mural]

1->2->4->5->6 is your way. But you can do 1->2->3->6 @median dirge

[alpine sable]

Exactly, so the second one is more efficient.

[median dirge]

Im just speaking generally

Actually then

We should compare two ways

[verbal mural]

I say this cause I guessed your way too at first

[median dirge]

Oh.

Wait we're seeing the most efficient way, right?

Hmm wait

[verbal mural]

Right

[alpine sable]

I thought maybe getting to the closest power of two that is also less than $n$ and iterating $+1$ from there could lead me somewhere, but there's probably some better approach.

[ocean seal]

3317

[median dirge]

I arrived at the same but yes there should be a better approach

[surreal sun]

would the best solution be to see if the number is even or odd ?
starting the problem by "the end"
if it's even, we know the last operation is double right ?
if it's odd we know the last operation is iterate 1

we can go like this until finding 1 probably ?

[median dirge]

$a + 2^b(c) = n$

[verbal mural]

What if you start backwards…half the number, then minus 1 to get to 1. That’s not the best but might be a more efficient method xD

[ocean seal]

Muhammad Hussaini

[surreal sun]

except small numbers

[median dirge]

This looks more generalized

[median dirge]

@verbal mural

This should fit into your assumption of 6 too

[alpine sable]

And if it's odd, the operation before that must be double, aye?

We're getting somewhere.

Even -> last operation is to double
Odd -> before last operation is to double.

Idk

[surreal sun]

yhea

[verbal mural]

With this, you add 1 to get to c first, then double?

[surreal sun]

but it would work well in python but idk if it works mathematically

[median dirge]

Actually you should help me put some constraints on it

[surreal sun]

how to put it mathematically rather

[median dirge]

In your case of 6, c is 3

b is 1

And a = 0

Looks like

If we can derive a relation between

a b and c

We can probably create some constraints for the supergeneral equation we just made

$a + 2^b (c) =n$

[ocean seal]

Muhammad Hussaini

[verbal mural]

Ok yeah I kinda see that

[median dirge]

Yep

[verbal mural]

I think this is true

[alpine sable]

Is it an accurate representation though?

[median dirge]

Very general

[alpine sable]

I mean, yeah, it sure can represent every natural, but is it compliant with the process?

[verbal mural]

And we can put it in Muhammad’s form too maybe

But yeah

[surreal sun]

what number iterate when you add 1 in the middle of the process ?

[alpine sable]

So $a$ is the number of times $+1$ was used, $b$ is the number of times $\times2$ was used, but what about $c$?

[ocean seal]

3317

[median dirge]

c is if you started iterating before multiplying by 2

And then you multiply 2, b number of times

[alpine sable]

Ooh.

[median dirge]

So its like

Iteration before multiplication and iteration post-multiplication

That should include everything

Actually, you know what, we should discuss this question in a real good discussion channel

Is a great question!

[alpine sable]

Let us assemble a private group then.

Add me if you're interested.

[surreal sun]

let's say i want 10, i'd do
1 -> 2 -> **4 -> 5 ** -> 10, what number is interated in the bold section ?

[median dirge]

Oh wait.

[surreal sun]

should we move to #proofs-and-logic maybe ? i don't know where else

[median dirge]

Yeah I think so

This whole thing

[alpine sable]

WAIT WAIT

I just figured it out.

[median dirge]

Should have some constraints

[alpine sable]

Two simple rules.

Start from $n$, if it's odd, subtract one, if it's even, divide by two.

[ocean seal]

3317

[alpine sable]

Then you just deduce the optimal way..

[surreal sun]

we know

but how do you put that mathematically

[alpine sable]

Piecewise.

[median dirge]

Do I think it looks like the Collatz conjecture lol

[alpine sable]

Also reccurence relation

[alpine sable]

It does lol

[median dirge]

Oh well the only thing is, this can be put in a form of a conjecture

3317's conjecture

[alpine sable]

365->364->182->91->90->45->44->22->11->10->5->4->2->1

[median dirge]

Is it the shortest pathway?

[alpine sable]

It seems to be, aye.

[verbal mural]

I agree 🎉 but there’s still a lot more we can do

[median dirge]

So we cant arrive at a general form

But we can just write the steps

[verbal mural]

Yeah

[median dirge]

Because the general form will be way more complicated

[surreal sun]

and even if find the general form, would it tell us the optimal way to get n ?

[verbal mural]

It’s hard to like reverse it too lol. Idk what the rules going forward from 1 are

[median dirge]

It wont actually

[surreal sun]

yeah

[verbal mural]

I mean we can always get all the steps going in reverse but just the rules for going forward idk

[median dirge]

Since the more we make it bigger, the more variables we put in and more we complicate it

[alpine sable]

It's odd, so -1, it's even, so /2, it cloggs at 0

[median dirge]

Wait now

[alpine sable]

That's interesting.

[median dirge]

Lemme program this

Lemme program this and probably try to graph it.

[alpine sable]

May I add you, Muhammad?

[median dirge]

Do we have a good JS library for graphing?

You can

[alpine sable]

And Raspberry?

[alpine sable]

p5js

Or just use numpy.

[verbal mural]

I’m curious just how many steps it takes for each n, that would be cool to see

[alpine sable]

Or I can use C++ and SFML.

[median dirge]

Thanks!

[verbal mural]

Sure

[median dirge]

Yep

[surreal sun]

[median dirge]

Lol add Sam too if he wants

[alpine sable]

Seems like you're not accepting requests, lol.

[alpine sable]

Sorry, Sam, I just got lost in excitement, haha.

[median dirge]

There you go

They're open

[alpine sable]

Thanks.

[median dirge]

Welcome!

Hey guys Imma go for dinner! I'll be back

[alpine sable]

I'll be waiting, have a good meal.

[verbal mural]

enjoyy

[alpine sable]

Lol

[verbal mural]

i made this lol but it's only complete for the first 30 numbers

idk it's like the chains that you can go on

when you double

[alpine sable]

250716 -> 125358 -> 62679 -> 62678 -> 31339 -> 31338 -> 15669 -> 15668 -> 7834 -> 3917 -> 3916 -> 1958 -> 979 -> 978 -> 489 -> 488 -> 244 -> 122 -> 61 -> 60 -> 30 -> 15 -> 14 -> 7 -> 6 -> 3 -> 2 -> 1

[verbal mural]

so if n is on a color, you'd stay on that color going backward, until the very first one. then go to the left, and stay on the next color

[alpine sable]

Wow, this is amazing.

[alpine sable]

Oooh.

[verbal mural]

nice

[alpine sable]

I've coded it in C.

It'll test all unsigned long long int from 2 to (unsigned long long int)-1

[verbal mural]

i forgot how big that is but that's pretty big

lol

[alpine sable]

Literally 2 to $2^{32}-1$

[ocean seal]

3317

[swift robin]

hey, do you still need this, i think i may have an O(1) algorythm for this

[alpine sable]

Oh woww

Yeah, I do.

[surreal sun]

O(1) ?!

[alpine sable]

Any help is appreciated.

[swift robin]

okay so lets start by visualising it in binary

[alpine sable]

The 3317-Hussaini-Raspberry-Sam-Isus conjecture.

[verbal mural]

hmmmm

[wary stream]

Sounds very similar to the Collatz Conjecture, if that was your goal

[alpine sable]

Wasn't, lol. But it's similar in some aspects, yea.

But this one wouldn't be hard to prove though.

[verbal mural]

doubling in binary is like multiplying by 10 in decimal

[alpine sable]

Exactly.

[verbal mural]

i didn't realize that until now lol

[swift robin]

there are two actions you can do to your binary number, you can multiply it by 2 (which is adding 0 to the end), or you can add 1
Now it makes no sense to add 1 if you already have 1 on the end because you could have achieved this easier by multiplying the numbers by twwo (we can prove this later on)

[alpine sable]

But I think he meant we could just represent any number as a sequence of bits. As if 0 meant "add one" and 1 meant "multiply by two", I guess.

Yep.

[swift robin]

So lets say you have number 6
in binary that is 110
you had to multiply by 2 two times and add 1 one time
So the solution should be = number of digits - 1 + number of 1s -1

[verbal mural]

with these two steps it's really like you just kinda write out the number, like a typewriter

[swift robin]

exactly

[verbal mural]

cool

[swift robin]

and this is always optimal because to get a 0 by adding 1s you have to do it at least twice with the exception of a starting 1

so this should be always optimal

[surreal sun]

this is well done

[swift robin]

Yayy competetive programming has paid off

[alpine sable]

I'm just a noobie. 😦

[swift robin]

Dont worry, everyone is at some point

Practice makes it perfect

[verbal mural]

weirdly none of these go to 3

at the end

[alpine sable]

Indeed

[verbal mural]

actually that makes sense

[surreal sun]

yeah the fastest way to go to big number is probably to double up directly

[verbal mural]

when you start at 1, what you do first affects the digits on the left in binary. so the biggest digits

[surreal sun]

instead of adding one, except for power of 3s

[verbal mural]

something similar to that yeah

[swift robin]

they dont go to 3 cause their binary representation should start with two 1s for that, no numbers here have that property

[median dirge]

Thanks, I'm back

[verbal mural]

rightt and i imagine the first digits won't change bc he's in the 9000s

[median dirge]

Actually

I rethought everything

You know

We can write the Collatz conjecture

[verbal mural]

wbb

[median dirge]

Not only for the 3n+1 and n/2 rule

But also for n+1 and n/2 rule

[alpine sable]

That makes sense.

[median dirge]

And the whole thing ends in a 2->1 loop

Instead of the normal 4->2-> 1 loop

Actually

This made me conclude

That we can maybe make every number end in a certain loop

[alpine sable]

||Proving it is cursed though...||

[median dirge]

By just altering one number in the Collatz conjecture

the 3

[alpine sable]

Yeah, it's just 1n+1 if you generalize it to an+b

[swift robin]

my man thats a wild way of solving this task xd

i would love to see it

[median dirge]

That is actually the best way at hand

So basically

The whole thing is not gonna have a pattern

[swift robin]

why do you think that?

[median dirge]

And how do you even arrive at the Converse of the Collatz conjecture

[median dirge]

The Collatz conjecture doesnt follow a pattern

[alpine sable]

Aye, but the point was just to find the shortest possible path from 1 to n, and that was accomplished

Maybe not with formality, but yea

[median dirge]

Yeah

[alpine sable]

Works for me

[median dirge]

But it left us with a greater question that we're gonna discuss in your server or in any other place haha

[verbal mural]

i think you're right our methods work to kinda "solve" the n + 1 and n/2 rule (but they won't for the collatz problem)

[alpine sable]

Yep, and if anyone is interested, just DM me.

[median dirge]

So basically

Lemme work out a little formula

For modifying the Collatz conjecture

[swift robin]

hold on what are we trying to solve now xd (im lost)

[median dirge]

Into a general conjecture for all natural numbers

We're done solving the help-question

Now what Im doing is modifying one of the results of the question using the Collatz conjecture

The purpose of the help channel is fulfilled I guess

[verbal mural]

i meant we can prove a "modified collatz conjecture" with an n+1 instead of 3n+1 rule and it's similar to what we just did

[alpine sable]

It is, yes.

[median dirge]

Yeah

[alpine sable]

And I can't thank you guys enough for helping me on this.

[verbal mural]

but still it won't work for the actual collatz conjecture (as far as i can see anyway)

[median dirge]

Lets do one thing, Raspberry...

Lets switch channels.

And free the help channel

[alpine sable]

Aye, thanks all.

.close

[lone heart]

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[lone heart]

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[alpine sable]

This converges to 0 right?
sin(-n)+sin(n)
-sin(n)+sin(n)=0
and
sin(0)=0

[alpine sable]

And it does so for any sin(n)^m where m is a positive odd integer right?

I think I'm right but not 100% sure

[fervent timber]

hmm but are you allowed to do that?
by the same reasoning the integral of x dx from -inf to +inf is 0
but I heard somewhere that's not true
idk

[alpine sable]

It doesnt work with negative numbers since theres a 1/0 and it doesnt work with even numbers since it'll obviously diverge

So if does it work with positive odd integers how would I prove if it works with all other positive odd numbers or not?

[lone heart]

@alpine sable Has your question been resolved?

[last ether]

This doesn't exist

It doesn't converge to anything I think

Think of it as this

$$\lim_{(a,b)\to (-\infty, \infty)}\sum_{n=a}^{b}\sin{(n)}$$

[ocean seal]

Umbraleviathan

[last ether]

If you try to take the limit of sin(n) as x approaches any infinity, it's undefined as it oscillates between -1 and 1

[lone heart]

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[brazen berry]

wondering if anyone can verify my work

[bitter vault]

it's fine but you want the point on y = ln x, not the equation of the tangent

[brazen berry]

yeah, i just asked my prof, and he also said i cant use the same variable x in my m

i guess the x in the point slope cant cancel with the x in the slope

since the slope x is a constant not a variable

😭

[sudden nacelle]

Help

[bitter vault]

it's actually fine, but sure, you can use a different letter.
let (a, ln a) be the required point.
then the tangent at a is
y - ln a = 1/a (x-a)
The point (0,1) is on this, so
1 - ln a = 1/a(0-a) = -1,
i.e ln a = 2 etc etc

[brazen berry]

oh so i did the right thing

and got the right answer

just used the wrong notation?

[bitter vault]

um.. the right answer is the point on y = ln x, and this is nowhere to be found in your work

[brazen berry]

oh

is e^2 not the x point

and i just gotta find y?

[bitter vault]

yes, give the point (x,y) so give y-coordinate too

[brazen berry]

okay, do you know if my x point was correct?

[bitter vault]

it is

[verbal mural]

i think your notation is correct

your prof may not have understood it

what you wrote in the pic you sent is correct anyway

[bitter vault]

it is slightly sloppy, which is why his prof had some misgivings with it

[verbal mural]

i also agree with @bitter vault that you should indicate the final answer as the point but that's separate

[brazen berry]

can anyone explain the notation (a, ln(a))

[bitter vault]

point with x coordinate a and y coordinate ln a

[brazen berry]

oh!

[verbal mural]

there's nothing wrong with the use of x in parts 1, 2, 3 imo

i would agree it's not the clearest method

but like i just want to say that it's not actually wrong to treat x as a variable in it

[bitter vault]

sure, but his professor has an issue with it. our opinion as strangers is worth nothing compared to his professor's opinion. his prof is the one going to be marking his work

[brazen berry]

Do y'all mind if i keep this channel open and send in another version of my answer?

with your advice in mind

[verbal mural]

fakesies described the prof's complaint as

i guess the x in the point slope cant cancel with the x in the slope
since the slope x is a constant not a variable

there was nothing wrong with the way he did this tho

it was fine to treat both x's on equal footing

[bitter vault]

gee, i'm not criticising you. let's leave it at here

[verbal mural]

i'm not offended

[bitter vault]

#help-0 message

i did mention "it's fine, but ok sure use a diff letter"

[verbal mural]

i'm just kinda suspicious that the prof is taking off points bc of the prof's own mistake

[bitter vault]

sussy prof 🕵️

[verbal mural]

er, idk if he took off points i guess, but still, i think he might've made a mistake

[brazen berry]

ah shit, lectures started

thanks for the help, y'all

wont have time to finish this one

glhf!

.close

[lone heart]

Channel closed

Closed by @brazen berry

Use .reopen if this was a mistake.

[verbal mural]

oh i didn't mind lol but gl with lecture anyway

[brazen berry]

ty!

[lone heart]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• 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.

[alpine sable]

Desmos is easy to use on mobile, sorry

Limit as i approaches infinity

Is any of this bullshit or is my work correct?

[thick yoke]

These limits aren't the same

[alpine sable]

Limit as i approaches infinity (at the top of the caption)

[thick yoke]

yes ofc

its still not the same

You have to use 2 different variables for the top and the bottom.

[alpine sable]

👍

Is flipping the bounds bs or is that fine?

[thick yoke]

wdym by flipping the bounds

[alpine sable]

3rd last step

From 0 to -i, to negative (i to 0)

[thick yoke]

yes that's fine

[ocean seal]

Ariel1300

[thick yoke]

This is the correct limit

[alpine sable]

Ah damn

Then it just turns into an indeterminate form minus an indeterminate form

Or infinity minus infinity

Im too sleepy to figure out which

Thank you anyways @thick yoke, I'll sleep on it

.close

[lone heart]

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Remember:
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• 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!

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[alpine sable]

I love the reaction of "ah fuk" to opening a help channel lol

Terms?

at the begining

a1 = 1

a2 = 2 * a1 = 2 * 1

a3 = 2 * a2

and so on

[wary stream]

Do you mean $$a_1 =1$$
$$a_n+1=2 a_n$$?

[ocean seal]

dldh06

[alpine sable]

it's just a progression

a1 = 1

your first term

n+1 is the next term

so for n = 1

a2 = 2 a1

a3 = 2 a2

a4 = 2 a3

etc

[wary stream]

n is the term you are on

So if $n = 1$, then it's $$a_{1 + 1}=2 a_1$$

[heady pollen]

whats the question btw?

[ocean seal]

dldh06

[wary stream]

As mentioned, n is the term

So you want n = 1 to n = 5

And for each n, relies on the previous n term

It gives you $a_1 = 1$ as the initial term

[ocean seal]

dldh06

[wary stream]

So you now want n = 1

Plug in 1 for all the n's

[wary stream]

You get this

Simplified, $$a_2 = 2 a_1$$

[ocean seal]

dldh06

[wary stream]

The $a_{n+1}$ is just notation for the next term

[ocean seal]

dldh06

[heady pollen]

the formatting is hard to understand

do you mean this?

so yeah, what formatting did you mean?

this?

is it related to the former question?

k is something like n

a variable

do you have more info about c?

or just post your homework here xD

would be easier then

[wary stream]

Can you post the full question? Just that statement is not enough to make a conclusive decision

[heady pollen]

what language?

ah no sry

[proud pier]

Hi

Can someone help me

[heady pollen]

like this?

[heady pollen]

open a ticket pls

[proud pier]

What u mean open a ticket

[heady pollen]

go into #help-22
and ask your question there

@alpine sable
you still there?

so do you have more info?

[proud pier]

No

[heady pollen]

and is this the term you had?

[high rapids]

sab are you a fan of fidel castro?

[heady pollen]

then what is the task?

[high rapids]

martin very nice profile pic

[heady pollen]

thx^^

so what is the qustion? xD

ahhhh

ok

so

you know this

and now look at what c can be

you will notice it is either 1 or -1

now think about when is c positive and when is it negative

equal?

you mean even haha

yes if n is even then c is 1

if n is odd then c is -1

now look at your n

n=2k+1

you know that k is in [0,1,2,3,...]

that makes 2*k an even number

adding one to an even number makes it odd

thus 2k+1 is odd

which is your n

so c_2k+1 is -1

you're welcome ^^

[lone heart]

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[supple fiber]

Could someone help? The problem is in green according to the answer on the textbook. I just don't know what I did wrong

Got it

.close

[lone heart]

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• Be polite and have a nice day!

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[alpine sable]

how?

[zenith pagoda]

sum of squares? difference of squares?

[alpine sable]

how would that work?

[steel sorrel]

do you know the formula $p^2-q^2=(p-q)(p+q)$?

[ocean seal]

JanekKwadrat

[zenith pagoda]

this is the difference of squares

[steel sorrel]

try applying it to the denominator

and see what you get

also notice that the nominator is divisible by x^3

[wary stream]

I wouldn't use the term divisible like that

[steel sorrel]

my bad

[wary stream]

I would say it has a GCF of x^3

[zenith pagoda]

well, it is divisible tho right

[wary stream]

Divisible makes it seem like you would divide all the terms by x^3

[zenith pagoda]

I guess divisible might imply it's an integer multiple of x^3

but all the same

[alpine sable]

x^3(x^3+a^2y)

(x^3+a^2y)*(x^3-a^2y)

im not sure abt this

[zenith pagoda]

yes

[wary stream]

Looks right

[alpine sable]

what do I do with that?

[zenith pagoda]

you gotta

[wary stream]

Now notice how some things can cancel out

Because it's the same in numerator and denominator

[alpine sable]

you mean cancel out the (x^3+a^2y) part?

[steel sorrel]

yes, that's probably what they meant

[alpine sable]

then ig it would be x^3
-----
(x^3-a^2y)

[steel sorrel]

that's right

[alpine sable]

ok thanks

[steel sorrel]

it doesn't seem you can do much more with it

[zenith pagoda]

,w xxx/(xxx-aay)

[ocean seal]

Results provided by WolframAlpha

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

Requested by mixolydian mode#3126

[zenith pagoda]

seems to be as simplified as it can get

[wary stream]

,w (x^6 + a^2 * x^3 * y)/(x^6 - a^4 * y^2) simplify

[ocean seal]

Results provided by WolframAlpha

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

Requested by dldh06#2659

[wary stream]

Yes, the goal of my statement was to lead you to that observation without clearly stating it

[alpine sable]

Ah, but it isnt that hard to find if you know you can cancel out something, but thanks for not instantly saying the right answer

[wary stream]

You did ask for the next step, when the next step was to cancel stuff out

[alpine sable]

i found another interesting problem im not able to solve are that too many or is it ok?

[wary stream]

When you do simplifying problems like that, the steps are normally, factor when you can, then cancel

[wary stream]

Post the question

[zenith pagoda]

I guess the idea is solve for y

or..?

[wary stream]

There's going to be an infinite amount of possible answers

I believe

[zenith pagoda]

log_2(615 + xx) integer solution

,w log_2(615 + xx) integer solution

[ocean seal]

Results provided by WolframAlpha

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

Requested by mixolydian mode#3126

[wary stream]

What is xx?

[alpine sable]

as far as I understand the website claimed that there are only 2 solutions

[zenith pagoda]

xx = x^2

[wary stream]

You can type x^2

[zenith pagoda]

it's faster to type xx

and no need to press the shift button

[wary stream]

,w 615 + x^2 = 2^y

[ocean seal]

Results provided by WolframAlpha

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

Requested by dldh06#2659

[wary stream]

Yep, two solutions

[alpine sable]

im not quite sure if I understand WolframAlpha's "explanation"

can anyone explain?

[zenith pagoda]

hmm

well it seems like you need to find a power of 2

since that's the only way that log_2(something) can be an integer

so x^2 + 615 is a power of 2

so we know powers of 2 go like 256, 512, 1024, 2048, 4096 etc

so then we essentially just need to check integer squares one by one until we find one that is exactly 615 less than a power of 2

[wary stream]

Interesting to decide to start at 2^8

[zenith pagoda]

well, might as well start close to 615 in size

so we check: 1024 - 615 = 409
2048 - 615 = 1433 etc

those are not square numbers, so we keep going until we find out that:
4096 - 615 = 3481

3481 is a square number, since 59^2 = 3481

4096 is the 12th power of 2, thus y = 12, x = 59

[alpine sable]

thanks, but I have a question

how did u know that 3481 = 59*59 did u check with calculator or did u calculate in your head?

[zenith pagoda]

I already knew the answer by looking at Wolfram Alpha, of course

but the idea is to check the square root of each answer until you find an integer

[alpine sable]

oh ok, I would have found it quite impressing if you had calculated in your head that 3481 = 59*59, but the explanation is quite impressing too thanks

[zenith pagoda]

haha I know the squares but not that high

[alpine sable]

can you help me with a little more different problem?

[zenith pagoda]

of course

well, not me, I gotta do something rn

[alpine sable]

oh when are you back?

[zenith pagoda]

just ask and someone else will answer :)

[alpine sable]

but I want your opinion abt that

[zenith pagoda]

fine I'll be back in like 10 minutes

[alpine sable]

ok thanks

[zenith pagoda]

@alpine sable well thats exactly 10 minutes

[alpine sable]

nice accuracy

so

I did the problem with dldh06 and i just wanna see if u get a different answer so you in?

[zenith pagoda]

which one?

[alpine sable]

its about creating a functional equation

[zenith pagoda]

fine by me post the problem

[alpine sable]

but it's not a problem from the intenet its a problem I found on my own so I have to explain a bit

so you have 4 variables
co
cl
hp
cn

co = definable
cl = definable
hp = 30
cn = F(co, cl)

its seems strange wait a sec

[zenith pagoda]

why are the variables not a,b,c,d?

[alpine sable]

because I have them like that in a programm, they stand for co(coins_old), cl(clicks), hp(health_points) and cn(coins_new) but you can use A,B,C,D

so every cl, hp gets subtracted by 1 and every 30th subtraction, when hp = 0 I want hp to be reset to 30 and have a counter of how many times hp was set back (cl:hp(30))

and that counter does co*2^amount of times hp was reset

so if I have this function the problem is cl:30 can equal to = 2.5 if we have 75 for an example and is it possible to get the amounts of absolute resets without a floor function

so that the outcome would be 2^2 because hp reset would only been done 2 times so 2^number of hp resets

[zenith pagoda]

well instead of resetting the HP you can count the amount of damage done total

the HP is then equal to 30 - (damage modulo 30)

[alpine sable]

but saying you'd do 40 damage then you'd have negative 10 hp

[zenith pagoda]

no, because you use the modulo function

[alpine sable]

how would that work?

[zenith pagoda]

that basically ensures it goes 0,1,2,3...28,29,0,1,2,3

this is the sequence of integers modulo 30

every time it reaches 30 it subtracts 30

that's what modulo does essentially

61 modulo 30 = 1

[alpine sable]

so the output would be 1 in 61 modulo 30?

[zenith pagoda]

,w 61 modulo 30

[ocean seal]

Results provided by WolframAlpha

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

Requested by mixolydian mode#3126

[zenith pagoda]

61 damage dealt ==> 1 damage dealt

[alpine sable]

but wouldnt i want the output to be 2 then because the hp of the enemy went down to 0 2 times

[zenith pagoda]

huh?

[alpine sable]

wait a sec

[zenith pagoda]

honestly I'm not exactly sure what the problem actually is

or why you would want to avoid the floor function

[alpine sable]

cause im trying to get that equation without any function

i want it to be written in math that anyone would understand without the knowledge of needing to know certain functions

the design looks terrible i know so we have 4 coins

[zenith pagoda]

well there's no way to write it without floor or modulo I'm pretty sure

[alpine sable]

if i press the kill button 30 times, coins will be multiplied by 2 for each time im pressing the kill button 30 time

is it possible to implement that into an equation

without any external functions?

[zenith pagoda]

it's either 2^(total damage/30) with rounding or 2^((total damage - total damage modulo 30)/30)

[alpine sable]

look (i clicked the click button thats why 5*2=10)

the 1 was reset to 30 because it gone to 0

and the coins were multiplied by 2

[zenith pagoda]

those are the two solutions as far as I'm concerned

[alpine sable]

ok thanks and can you check if my math is right for the passive income section?

[zenith pagoda]

um, which math specifically?

[alpine sable]

basic multiplication and division

[zenith pagoda]

oh I see

[alpine sable]

important is cps, cpsr and mg.after(500)

so after 500ms the current coins get multiplied by 1.25

and I wanted to calculate the cps(coins per seconds)

how would I do that?

or have I done it right?

[zenith pagoda]

I have no idea what im looking at

uhh, what's ppp?

[alpine sable]

random variable

ppp = 10

forgot to mention sry

[zenith pagoda]

it would seem to be right but I can't say I fully understand whats going on

[alpine sable]

wait i found a mistake let me change the code then analyze it again pls i need help

[zenith pagoda]

it's likely more appropriate to ask in a Python-related discord

[alpine sable]

its not about the code only the math

[zenith pagoda]

alright

[alpine sable]

ok now

[zenith pagoda]

it still seems like you're multiplying them right away

I do not see what passi is even doing in there

it's not like you're using that variable

is the idea that it runs the function after 500 ms?

if so then you should do mg.after(500,aftercheck) right there in the passive() body

[alpine sable]

thats why its darkened out, its for the future if i want to cancel that 500ms loop, i can only do it by mg.after_cancel(passi)

yep every 500ms aftercheck() is being ran

[zenith pagoda]

I see

[alpine sable]

and every 500ms coins are multiplied by 1.25

[zenith pagoda]

and this only happens after you get 10 thousand coins?

[alpine sable]

no thats how much the passiveincome ability costs, its the trigger

[zenith pagoda]

aha so when you get it a second time it will multiply it by 1.25^2 every 500ms

[alpine sable]

but my problem is I cant just multiply that by 2 so I have every second times 1.5 cause the coins * 1.25 are different coins after they were being multiplied

no do you see the line
passiveincome["state"] = DISABLED
it makes the button inaccesible after being purchased

[zenith pagoda]

I see

[alpine sable]

imagine it like this

you have x
x is being multiplied by 1.25 every 500ms
how can you define how much x per second you would get

[zenith pagoda]

how can I define how much x per second I would get?

[alpine sable]

i think i got it
is it cps = coins / ppp if ppp = 4?

for an example if x = 6

i mean 8

[zenith pagoda]

oh, so you're calculating the expected result of the next call

[alpine sable]

then what would 8 * 1.25 be 10 right?

oh no

[zenith pagoda]

or you're calculating the exact amount you would get every second, including the stuff you already get

since this is a Cookie Clicker-type game

[alpine sable]

coins gets updated after every multiplication so every 500ms a different value of coins get multiplied

[alpine sable]

yes

not fully im calculating the expected add value (x + expectation)

[zenith pagoda]

aha

well, it's (1.25coins - coins)/interval

so that's 0.25coins/interval

the interval, is 500 ms I guess?

[alpine sable]

so cps = coins * 0.25 / ?

[zenith pagoda]

this is the amount of coins per second you would expect to add

though you need to convert ms to seconds

[alpine sable]

which is not directly possible due to the update

[zenith pagoda]

0.25c/500ms = 0.25c/0.5s = 0.5c/s

hmm, is that right?

[alpine sable]

no

imagine this
coins = 8

after 500ms you would have 10

and after another 500ms (1 second in total) you'd have 12.5

8 * 1.5 = 12

12.5 != 12

[zenith pagoda]

because of the rounding, yes

[alpine sable]

what rounding?

[zenith pagoda]

10 * 1.25 = 12.5 but you round down

[alpine sable]

8 * 1.25 -> 10 * 1.25 -> 12.5
8 * 1.5 -> 12
no rounding done

different outputs

know imagine bigger amounts

lets say

[zenith pagoda]

indeed but why would you multiply by 1.5

thats not how compound interest works

1.25 * 1.25 is not 1.5

[alpine sable]

0.25c/500ms = 0.25c/0.5s = 0.5c/s (that's your statement)

0.5c/s = + c * 1.5

thats the thing being added converted in multiplication

the add value

[zenith pagoda]

oh I see

hmm

[alpine sable]

we have to predict the outcome of the first update
we have coins
and coinsupdate ig

and then multiply that coinsupdate value by 1.25 to get the real prediction

[zenith pagoda]

well the coins per second is getting updated every half second

[alpine sable]

yes

[zenith pagoda]

it would be correct the way I wrote it then

otherwise it's 1.25^2 coins - coins / 1 second

[alpine sable]

let me import the thing you mean in python and check if it works

can you write it again?

[zenith pagoda]

0.5c/s ?

[alpine sable]

ok wait let me import that

[zenith pagoda]

the way I'd calculate it abstractly it is of course using 1000(factor - 1)coins / interval

where factor = 1.25 and interval is in milliseconds

the 1000 to convert it into seconds

[alpine sable]

this is the output if coins = 10

you can check this

im not quite sure if I understand the output tbh

oh now I see

and see I was right

nvm

i dont understand it

[zenith pagoda]

well turn the interval and factor into a variable then use this

[alpine sable]

what do you mean?

[zenith pagoda]

factor = 1.25; interval = 0.5

[alpine sable]

like this?

[zenith pagoda]

yeah

[alpine sable]

100% same output let me check

[zenith pagoda]

well of course

it's just for bookkeeping

[alpine sable]

oh

ok xd

[zenith pagoda]

what output were you expecting?

if you want it to calculate two function calls ahhead

(since it goes every half second)

[alpine sable]

i understand know and its the problem i told you look at this

ignore cps: 7.8125

so after one run 10 * factos is obviously 12.5

factor*

[zenith pagoda]

then just do (1.25^2 - 1)coins

this is the prospective amount of coins you will get after a second

[alpine sable]

so based on that cps definition coins after two runs has to be coins(10) + first cps(6.25) which would be 16.25 but its 15.625

you mean like this?

[zenith pagoda]

no no

only in the calculation of cps you do it twice

factor = 1.25

(factor^2 - 1) coins

this is the cps

(assuming the interval stays 500ms otherwise you have to make it more complicated)

[alpine sable]

so like this?

[zenith pagoda]

use parentheses

coins times (factor factor -1)

[zenith pagoda]

minus one goes into the parens

and obviously you still need the * to multiply

coins * (factor*factor - 1)

[alpine sable]

To check if it works you have to do
10 + first cps = 2nd coins output

[zenith pagoda]

hmm

[alpine sable]

17.0 != 15.625

[zenith pagoda]

I guess I got something wrong

hmm

coins * (factor*factor) - coins

but that clearly is the same thing

[alpine sable]

i think im close to it see what i got

now cps predicts the new coins value and it works

but I want to predict the add value

did it like this if it helps

created a new variable called upcoins(updated coins)

[zenith pagoda]

coins + cps = coins * 1.25 * 1.25

cps = coins * 1.25 * 1.25 - coins

I don't really wanna spend more time on this

it's clearly more of a coding problem than a math problem

[alpine sable]

you cant give up on me 😦

[zenith pagoda]

I gotta do my own assignments too :)

[alpine sable]

we've been doing math for almost an hour now

assignments wdym?

[zenith pagoda]

for school, yknow

[alpine sable]

this is not an assignment its jff

what grade are you?

[zenith pagoda]

university

[alpine sable]

oh wow which?

[zenith pagoda]

thats a secret to everyone

[alpine sable]

oh ok

are you studying maths?

[zenith pagoda]

compsci

[alpine sable]

you're studying computer science? Isnt the whole thing we were doing the past 1 hour computer science

[zenith pagoda]

haha yeah but I dont wanna be debugging stuff lol

[alpine sable]

what direction are you going for?

[zenith pagoda]

idk something like AI or maybe more math-related

[alpine sable]

ai without debugging xd?

[zenith pagoda]

not for free for strangers on the internet :3

[alpine sable]

are you able to program ai yet?

stranger? im malik

[zenith pagoda]

it's been fun but Im gonna go now

[alpine sable]

ok one last question

are you studying in america, asia or europe?

[zenith pagoda]

europe

thats the last one!

[alpine sable]

you're not speaking german are you?

[zenith pagoda]

no more details I'm going

[alpine sable]

only that one

[zenith pagoda]

have a nice day

[alpine sable]

have a nice night

[clear stump]

Good morning

[alpine sable]

hey whats up

[clear stump]

Im good

hbu

[alpine sable]

also good just discussed a problem with a helper which he could not solve sadly

[clear stump]

What problem

[alpine sable]

math

problem

[clear stump]

show

[alpine sable]

wait

[clear stump]

?

[alpine sable]

nono

[high rapids]

Mallik are you a boy or girl?

[alpine sable]

boy

sadly im not kylie jenner

[high rapids]

From where?

[alpine sable]

germany

[high rapids]

What do you study?

[alpine sable]

nothing im going to 8th grade

[high rapids]

Based

[alpine sable]

wdym by based?

[high rapids]

Very cool

[alpine sable]

this was the problem btw

[alpine sable]

oh and you?

[high rapids]

I am from Sweden

[alpine sable]

this was the problem

[clear stump]

We won in hockey hehe

[high rapids]

Cool

[alpine sable]

oh nice and what do you study?

[high rapids]

12th

[alpine sable]

oh

anyone know how to solve that?

cps(coins per second)

*this is the problem

basically every 0.5 seconds coins should be multiplied by 1.25 and I want to predict the next value added to coins every second

indirectly

any1 have an idea

[clear stump]

hmm

1/2t=1,25c?

[alpine sable]

what is t what is c

[clear stump]

t is second c is coin

[alpine sable]

so 0.5seconds = 12.5 coins?

[clear stump]

well each half a second the coins multiply by 1.25

[alpine sable]

yup

but dont forget it updates, so the coins multiplied by 1.25 change every time they're multiplied

im going to bed, this is the solution i came up with bt

w

[clear stump]

How do we get the previous amount in there

[lone heart]

@alpine sable Has your question been resolved?

[lone heart]

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

[frail rivet]

hello, i cant figure out how to prove that C^1 is not a banach space. the tip ive been given by scapeprof is that f_n = sqrt(1+x+1/n) but im clueless

[lone heart]

@frail rivet Has your question been resolved?

[lime bobcat]

$f_n=\sqrt{1+x+1/n}$

[ocean seal]

jnkena

[lime bobcat]

Calculate the limit under the norm and see that limit does not belong to C_1.

We guess $f(x)=\sqrt{1+x}$.

[ocean seal]

jnkena

[lime bobcat]

We want to see that $\lim_n ||f_n-f|| =0$.

[ocean seal]

jnkena

[lime bobcat]

$(f_n-f)(x)=\sqrt{1+x+\frac{1}{n}}-\sqrt{1+x}$

[ocean seal]

jnkena

[frail rivet]

Oh so instead of the absolute value we use the ||.|| norm

[lime bobcat]

$\sup_{x\in (-1,1)}(f_n-f)=\sqrt{2+\frac{1}{n}}-\sqrt{2}$

Yes! Think that $(\mathbb{R},| |)$ is just a special case of normed space.

[ocean seal]

jnkena

[lime bobcat]

I think the supremum is √(2+1/n)-√2 isn't it, I mean, at 1, isn't?

[frail rivet]

yea true

[ocean seal]

jnkena

[lime bobcat]

Clearly it tends to 0 when n goes to infinity!

And lim_n f_n = f

What does that imply?

If a sequence in a normed space has a limit, it implies it is...

[frail rivet]

it iiiis

[lime bobcat]

It is a Cauchy-squence,n

!

Elements of the sequence get closer to one another because they all get closer to the limit! so convergent sequences are easily proven to be Cauchy ones!

Now... what remains to prove?

[frail rivet]

Oh i see now

that the cauchy sequence does not converge to something thats in our set

[lime bobcat]

That's it

[frail rivet]

wait but it converges to 0

[lime bobcat]

No

Look what we have proved:

This $\lim_n ||f_n-f|| = 0$ means $\lim_n f_n = f$.

[frail rivet]

Ok and we got this by just lim n->inf right?

or what do you exactly mean by guessed

[lime bobcat]

oh fuck a deja vu haha

[ocean seal]

jnkena

[frail rivet]

wait but why even 0

im sorry im so clueless

Oh wait i think i get it now

you subtract f from both sides so it has to equal 0??

[lime bobcat]

$\lim_n ||f_n-f|| = \lim_n \sup_{x\in (-1,1)} (\sqrt{1+x+\frac{1}{n}}-\sqrt{1+x})=\lim_n (\sqrt{2+\frac{1}{n}}-\sqrt{2})=\sqrt{2+0}-\sqrt{2}=0$

[ocean seal]

jnkena

[lime bobcat]

Do you understand this?

[frail rivet]

yea

let n->inf

[lime bobcat]

Yes

It is always infinity, it's a sequence

[frail rivet]

Ok and let x=1 because of the supremum

[lime bobcat]

Yes

[frail rivet]

by that we prove that f_n is a convergent sequence which is also a cauchy sequence

[lime bobcat]

And 1/n→0 because of n→∞

Yes

And we prove the limit is f

By that we prove $\lim_n f_n = f$

[ocean seal]

jnkena

[lime bobcat]

What remains to prove?

[frail rivet]

that this does not hold for every f_n?

[lime bobcat]

? no

You said earlier..., we have to see that f does not belong to C1

[frail rivet]

yea

[lime bobcat]

So, do you have any idea for that?

[frail rivet]

does this have to do something that it converges to a non differentiable function?

[lime bobcat]

That's exactly that

What you have to prove is that f is not differentiable in (-1,1)

[frail rivet]

so sqrt(1+x)

but it is differentiable right??

Oh no the derivative has to be differentiable right?

nvm the derivative has to be continuous thats it

[lime bobcat]

Yes mmm

I dunno really

[frail rivet]

Can i use f_n = sqrt(x^2 + 1/n) instead? because then it would be like the absolute function which isnt differentiable?

[lime bobcat]

I mean, shouldn't it be C¹[-1,1]? It depends on your definition of C¹(-1,1)

[frail rivet]

wdym?

[lime bobcat]

The intervals

[frail rivet]

yea its ]-1,1[

[lime bobcat]

Is that a close interval?

[frail rivet]

No its like (-1,1)

[lime bobcat]

okaj

[lime bobcat]

Yes, I think you can!

No you can't

Be careful, functions must be in C1

[frail rivet]

because its not in the set right?

ok thats what i thought

[frail rivet]

but i dont know how to prove this

[lime bobcat]

I know whats happening here

C1 is the set of continuously differentiable functions

[frail rivet]

that means: that the derivative is continuous right?

[lime bobcat]

$\frac{1}{2\sqrt{1+x}}$

[ocean seal]

jnkena

[frail rivet]

o.O

[lime bobcat]

If the interval were [-1,1] then it would not be continuous at -1.

But since the question text has a mistake (differentiable instead of continuously differentiable) it could have two mistakes maybe XD

[frail rivet]

Oh i see

but minus and minus = plus

[frail rivet]

but why would it not be continuous at -1?

[lime bobcat]

What?

[lime bobcat]

Better said, it would not be defined at -1

[frail rivet]

sqrt(1 + x) = sqrt(1 + (-1)) =0?

[lime bobcat]

The derivative

[frail rivet]

oooh

i see now

then this would be a proof true

[lime bobcat]

$C^1[-1,1]$ is not a Banach space.

[ocean seal]

jnkena

[lime bobcat]

✔️