help-0

[ocean seal]

huntifer

[shadow sparrow]

I reduced it down with the product rule to this point

$(t^3)(\frac{4}{t}) + (3t^2)(1+4lnt)$

[ocean seal]

huntifer

[shadow sparrow]

I'm confused at what I do at this point, I guess, I have an idea of what to do; but I'm not really that sure.

Factor out t^2?

$t^2(t\frac{4}{t} + (3)(1+4lnt))$

[ocean seal]

huntifer

[shadow sparrow]

?

#❓how-to-get-help

?

Wait are you asking a question, or

[placid zinc]

If your question is the second derivative, you're done here

[shadow sparrow]

Weirdly enough, it says to reduce further in the example?

Not sure if that isn't applicable to my formula

[placid zinc]

Okay, if you want to simplify we can. Pulling out the t² is a solid plan, but I'd also consider expanding fully to also work

[shadow sparrow]

See when I use the t^2, I end up here

$t^2(4t+(3+12lnt))$

[ocean seal]

huntifer

[shadow sparrow]

The only other thing I coud think of to do at this point would be to factor the t from 4t?

But that wouldn't work I think

Actually, wait

$t\frac{4}{t} = \frac{4t}{t} = \frac{4}{1}\frac{t}{t} = 4 \cdot 1$

[ocean seal]

huntifer

[shadow sparrow]

Is this true?

Which would make it instead:

$t^2(4+(3+12lnt))$

[ocean seal]

huntifer

[shadow sparrow]

which would then become

$t^2(7+12lnt)$

[ocean seal]

huntifer

[shadow sparrow]

I think the form of this equation is different when solved versus the example, no?

Because we're working with 3t^2 instead of like 3t

So you have to take the t^2 out

[trim sorrel]

I believe this is right

The example has different numbers

[shadow sparrow]

I actually have another similar question in terms of clarifying

I've found the derivative using the quotient rule, after finding R(x)

$\frac{50lnx-\frac{50x}{x}}{(lnx)^2}$

[ocean seal]

huntifer

[shadow sparrow]

Would 50x/x turn into 50?

Or basically I guess, would there be any more simplification done here before plugging in x=40?

Because the next possible step in my mind would be:

$\frac{50lnx-\frac{50}{1}\frac{x}{x}}{(lnx)^2}$

[trim sorrel]

What's marginal revenue

[shadow sparrow]

Nvm ignore the entire second half of that

[ocean seal]

huntifer

[shadow sparrow]

Marginal revenue is the derivative of the revenue function

[trim sorrel]

I see

How do you get revenue from demand?

[shadow sparrow]

And the R(x) in this instance is $R(x) = x \cdot p$

[ocean seal]

huntifer

[trim sorrel]

Ahh

Ok so R = 50x/(ln x)

[shadow sparrow]

Yes

[trim sorrel]

Right so u quotient ruled it

[shadow sparrow]

Yes

Which gave me the original

$\frac{50lnx-\frac{50x}{x}}{(lnx)^2}$

[trim sorrel]

Looks right

[ocean seal]

huntifer

[trim sorrel]

Just simplify from here

[shadow sparrow]

I guess the question is, would I reduce any further before plugging in?

[trim sorrel]

Well yeah you can reduce here

[shadow sparrow]

To this?

[trim sorrel]

To what?

[shadow sparrow]

$\frac{50lnx-50}{(lnx)^2}$

[ocean seal]

huntifer

[trim sorrel]

Yea

Looks good

[shadow sparrow]

I can't reduce any further, I think

So now it would be solve for x

[trim sorrel]

Solve for x?

Wdym

I think it just says put in the function

And then plug in x = 40 for the second part

[shadow sparrow]

Yeah

The function is hat

that*

[trim sorrel]

Yea

[shadow sparrow]

now I plug in x for 40

to solve

[trim sorrel]

Yea for the scond part

[shadow sparrow]

oh

huh

It did it like this

I guess this makes sense

[trim sorrel]

Where did 65 come from??

[shadow sparrow]

Will either equation give the same answer?

This is a different example

[trim sorrel]

Oh

I assume p = 65/lnx

[trim sorrel]

Right

[shadow sparrow]

I think they're equivalent

[trim sorrel]

Well yeah

[shadow sparrow]

this is just how they want it written

[trim sorrel]

65(lnx - 1) = 65lnx - 65

Its just distributive property

[shadow sparrow]

hmm... I wonder if it'd mark me down for not writing it this way

[trim sorrel]

No its fine

Im sure they would not care

Unless there is a specific rule

[shadow sparrow]

This is just a practice exam

So I'll put it in as is

and see if it marks it wrong

[trim sorrel]

If it does thats a terrible grading system

[shadow sparrow]

Yep

Computer grading systems be like

[trim sorrel]

Was it wrong

[shadow sparrow]

I wont know until I finish the entire exam's worth of questions lol

[trim sorrel]

Oh lol

[shadow sparrow]

Ty for the help! Was just mainly confirming the 50x/x into 50 thing lol

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[tiny terrace]

Stats & probabilities

[tiny terrace]

Im trying to find nCr

But i dont even know if its correct that its nCr(6,4)

Also trying to find samples too

<@&286206848099549185>

[stone oxide]

uh

hi

so what is the problem

im not sure i understand what the question is asking u\

[tiny terrace]

This is the full pic

Im tying to complwte the table

[stone oxide]

ok

i see

uhh

[shadow sparrow]

It would probably help to type it out, preferably using latex so people could more easily see whats going on

and what you've done so far

Latex is putting numbers and formulas between the $ sign

$4x+3$ like so

[ocean seal]

huntifer

[tiny terrace]

Complete the table

2. (2,2,2,5,3,4) sample size of 4

My mine problem is that it has repeating number

$2,2,2,5,3,4 sample size of 4$

[ocean seal]

Thoughts_In_The_Shower

[lone heart]

@tiny terrace Has your question been resolved?

[tiny terrace]

<@&286206848099549185>

Damn

Its hopeless

[gleaming ridge]

So here you want to determine all the possible numbers with 4 digits

That you can construct with the set of numbers {2,2,2,5,3,4}

is that it??

[ornate ginkgo]

If that's the case it should be 6C4 × 4!

[lone heart]

@tiny terrace Has your question been resolved?

[lone heart]

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[amber plover]

f(x)=-2x^3(x-2)^2 is decreasing on the interval (0,2) is there a way to solve this without graphing or taking derivatives?

[amber plover]

this question is for advanced functions

[proven leaf]

solve what exactly?

[amber plover]

if that function is decreasing on that interval or not

increasing or decreasing i guess

[naive valley]

have you tried plotting it?

[proven leaf]

"without graphing"

[naive valley]

sure, but at least graph it first to see if you think it's true or not

before deciding whether it's something you need to prove or disprove

[amber plover]

its not true its only decreasing till 1.2

well on my exam i wont be able to graph it

on desmos or anything

[naive valley]

simplest way to show it without taking derivatives would be, find three points x < y < z where for which you do not have f(x) < f(y) < f(z) or f(x) > f(y) > f(z)

[amber plover]

uh i dont think i should be needing to take z

we didnt work with z on my course

would i just need to plot this down using transformations or something?

[naive valley]

without a third point, how will you know? With two points you always have either f(x) <= f(y) or f(x) >= f(y)

[amber plover]

i thought there was a way to solve it algebraically

[naive valley]

how about looking at the roots

[amber plover]

would doing transformations work?

[naive valley]

what sort of transformation?

[amber plover]

nah nevermind it doesnt work

[naive valley]

look at the roots and reason about how the function must behave between consecutive roots

it's a polynomial already in factored form, so this should be easy

[amber plover]

so the root is x=2 right

[naive valley]

that's one of them

what's the other

[amber plover]

and x=0

[naive valley]

yep

and conveniently you're interested in the interval (0,2)

the function is 0 at x=0 and x=2

and nowhere else

so, can it be monotone between 0 and 2?

[amber plover]

it had to increase and decrease by that point i think if both of those are x intercepts right

[naive valley]

yes

it can't only increase because then f(2) would be bigger than f(0)

for the same reason, it can't only decrease

[amber plover]

ok thanks

have a good night

.close

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[lone heart]

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[ashen lance]

https://math.stackexchange.com/questions/4722734/group-selection-problem-based-on-the-score-of-each-element-of-the-group?noredirect=1#comment10006256_4722734
Will anyone kindly read my question in the math exchange and help me by giving an answer?

[lone heart]

@ashen lance Has your question been resolved?

[lone heart]

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[hollow iron]

Ok I think I got the answer

[naive valley]

the first two lines are ok..

then not so much after that

[ashen lance]

No

I have encountered a problem that I collected and noted from an online source. However, I am unable to find the specific link for reference at the moment. The problem is described as follows:

A list of 500 student scores(out of 100) in a contest is given, represented as [92.77, 87.67, 83.03, 81.80, 80.24, 78.66, 78.48, 78.27, 77.98, 77.63, 77.36, 77.34, 76.37,...,68.19]. Note: scores are sorted in descending order.

An event is being organized for the participants, and the event organizer wishes to select students based on the n highest scores.

Please note the following:

The event organizers are selecting students solely based on their scores.

The selection criteria do not depend on a student's ability to pay for the event, the event's capacity, or any other factors.

Now, the question is: In how many ways can the event organizer select students based on the highest score? Do they need to choose the first 50% of the students, the first 40%, or are there other methods of selection? What are the available methods or statistical tools that can help in choosing the students?

I tried using the median, Took all scores present on the left side of the median value. But, in the end, it does not make any sense.

[lone heart]

@hollow iron Has your question been resolved?

[lone heart]

@hollow iron Has your question been resolved?

[rose temple]

well if the data is arranged in descending that means the first studets have higher number

now it depends how many studnets the organizer is looking for

is he wants maybe lets say 10 studentss that the 10 first studnets would be taken

and if he he says 10% than we would take 10% of 500 and that number of studnets would be taken

[tardy stag]

yall this is slump's channel

(I think; the channel name doesn't match either person)

[tardy stag]

remember you need to flip around inequalities when you multiply or divide by a negative number!

[lone heart]

@hollow iron Has your question been resolved?

[lone heart]

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[prime badge]

no, you multiply all the way

[tardy stag]

yeah it has that word and that threw me off for a second

but it's not using it in the same way as most, assuming it matters who is the president and who is the vice president

i'm assuming the trailers are all the same as each other

https://stats.libretexts.org/Courses/Las_Positas_College/Math_40%3A_Statistics_and_Probability/04%3A_Probability_and_Counting/4.04%3A_Counting_Rules/4.4.02%3A_Permutations_with_Similar_Elements

[lone heart]

@wind dagger Has your question been resolved?

[lone heart]

@wind dagger Has your question been resolved?

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[ashen lance]

So, that means there is no specific methods or tool with which I could determine how many of students organizer could select.

[mellow grail]

if u want to discuss an old question, pls use personal channels, dont use help channels. or post the question again

[rose temple]

yup but i do beleive question is missing detail here because there can only be a finite nmber of members in any event and it has to be speciefed how many is=f not specified than we should assume that organizer will take x% than we take the first x%number of students

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@ashen lance Has your question been resolved?

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[open temple]

Help

[open temple]

.clear

[mellow grail]

help what

[open temple]

.close

[lone heart]

Channel closed

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[open temple]

.help

[lone heart]

Commands:
clopen: .close, .reopen, .solved, .unsolved
consensus: .poll
factoids: .tag
help: .help

Type .help <command name> for more info on a command.

[open temple]

.solved

.reopen

[lone heart]

✅

[mellow grail]

u have a question..?

[open temple]

How many digits are there in the solution to the following calculation:
2^3^4^5^6^7^8^9

How can I find the solution?

[mortal trellis]

you could try to find log_10(...)

[open temple]

Log_10 from?

[mellow grail]

$2^{3456789}$ or $2^{3^{4^{5^{6^{7^{8^{9}}}}}}}$

[ocean seal]

itzkraken.

[open temple]

The right lol

[mellow grail]

the first one suggests that u can do a^b^c^d = a^bcd

okay thats what i was expecting

i basically made a mistake like in my test, i comprehended the question as on the right but it was the left 😐

[mortal trellis]

log_10 of the number. but hmm doesn't look that nice either on second thought

[mellow grail]

take the ln 7 times?

that gets rid of the exponents

$ln(2^{3^{4^{5^{6^{7^{8^{9}}}}}}})$= $9*ln(2^{3^{4^{5^{6^{7^{8}}}}}}})$

[ocean seal]

itzkraken.
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[mellow grail]

i think u get what i mean, @open temple

[open temple]

No lol

[mellow grail]

😐

[mellow grail]

do u get what i did here

i used the following property: $ln(a^b) = b*ln(a)$

[ocean seal]

itzkraken.

[mellow grail]

a is $2^{3^{4^{5^{6^{7^{8}}}}}}$, b=9

[ocean seal]

itzkraken.

[open temple]

Okay, you mean the first steep would be $ln(2^3^4^5^6^7^8^9) = 9*$ln(2^3^4^5^6^7^8)

[ocean seal]

redstoneking18
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[mellow grail]

u have to add {} around every number

[mellow grail]

but yes

can u try doing this once again?

[open temple]

Bruh why?

[mellow grail]

i didnt make the latex bot 😐

[ocean seal]

redstoneking18

[open temple]

9*$ln(2^{3^{4^{5^{6^{7^{8}}}}}})$=$ln(2^{3^{4^{5^{6^{7^{8^{9}}}}}}})$

[mellow grail]

uhh how do u get that?

no its not equal to anything

remove =1, theres no way thats true

[ocean seal]

redstoneking18

[mellow grail]

Yes

[open temple]

You mean this?

Okay, and then?

[mellow grail]

now ln both sides again

keep doing it tell there are no exponents left

[open temple]

$ln(9*$ln(2^{3^{4^{5^{6^{7^{8}}}}}})$)$=8*$ln(9*$ln(2^{3^{4^{5^{6^{7}}}}})$)$

[mellow grail]

i recommend u do this on paper

and give me the final result as a photo

or just the final result

its easier

[ocean seal]

redstoneking18
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[open temple]

You mean it like this?

And then again and again?

[mellow grail]

yeah\

i think i have something that might work

[open temple]

Okay, and can the calculator calculate this?

[mellow grail]

i think so

lemme do it myself once

the idea is in my mind

not on paper

[open temple]

3*$ln(4*$ln(5*$ln(6*$ln(7*$ln(8*$ln(9*$ln(2)$)$)$)$)$)$)$

[mellow grail]

yeah this

remove 3 4 5

[ocean seal]

redstoneking18

[mellow grail]

perfect

[open temple]

What is ln?

[mellow grail]

log with base e..

u havent been taught that?????

[open temple]

e is 10?

[mellow grail]

no

e is eulers number

[open temple]

What is a eulers number?

[mellow grail]

this

I have a feeling u havent been taught that

[mellow grail]

2.718 numerically (approximation)

[open temple]

Okay, how can I calculate the thing now?

[mellow grail]

open up ur calculator, the ln function should be there

[open temple]

There is no ln..

[mellow grail]

use the logarithm with some other base (greater than 1)

this is the windows one, can usee the ln in the bottom left?

[open temple]

It doesn't work..

The calculation is wrong:

I did the test with 2^3^4

[mellow grail]

mad..?

marked u mean?

[open temple]

did sry

If you want to get the length of the result from 2^3^4

[mellow grail]

oh i forgot to ask, have u been taught any formula

[open temple]

You would type 3ln(4ln(2))

[mellow grail]

no

4ln(3ln(2))

,calc 4ln(3ln(2))

[ocean seal]

The following error occured while calculating:
Error: Undefined function ln

[open temple]

You told me that this would be the solution

[open temple]

Why now 4 before 3?

[mellow grail]

since its 2^3^4

oh crap

ur right

sorry my brain malunctioned

[open temple]

Okay, what would be the calculation to get the digits from the solution of 2^3^4?

[mellow grail]

2^3^4 is 2^81, yes?

[open temple]

is 4096

Yes

[mellow grail]

let me think..

[open temple]

I'm wrong lol

Solution would be 2417851639229258349412352

[mellow grail]

the answer is $log_{10} {a} + 1$

[ocean seal]

itzkraken.

[mellow grail]

where a is your number

[open temple]

Which number?

[mellow grail]

2^3^4^5^6^7^8^9

we need to remove the expoentiation

till atleast 3

[open temple]

Okay, why you don't use ceil?

[mellow grail]

ceil? u mean the ceiling function? how would that help?

[open temple]

Idk?

[mellow grail]

@open temple is the answer 9876543*1.301

,calc 9876543*1.301

[ocean seal]

Result:

2.3605344e+5

[mellow grail]

bruh 😐

[open temple]

Its wrong

[mellow grail]

236053

[open temple]

For?

2^3^4^5^6^7^8^9?

[mellow grail]

number of digits

[mellow grail]

yeah

[open temple]

Wrong

[mellow grail]

what is it then???

u have options...?

[mellow grail]

54618?

[open temple]

The length should be this...

Told me a other person lol

[mellow grail]

ur answer is incorrect

its highly likely that ur answer is a decimal

when it should be a whole number

[jolly widget]

?

[mellow grail]

this is from wolfarmalpha

i just checked myself

[open temple]

Yes

[mellow grail]

it doesnt make sense tbh

but well u have ur answer

[open temple]

The other person use wolfram alpha?

[jolly widget]

wdym?

[mellow grail]

who is the other person

[open temple]

A person from ouer school?

[mellow grail]

Yeah well iam 99% sure

[open temple]

btw. 54618?

Or 236053

[mellow grail]

this

[open temple]

To less?

[mellow grail]

eh its lesser than wolfarm alpha

but its the only thing that makes sense mathematially

[open temple]

Hm

[mellow grail]

since log(2^3^4^5^6^7^8^9) = 9!/2 * log(2)

it took me wayy too long to realize that

[open temple]

Oh, wolframalpha can't calculate lol

Because the count of digits for 2^3^4 would be ceil(3^4 * log(2)) = 25?

But the AI isn't smart...

[mellow grail]

yes which is 81*log(2)

81*0.301

[open temple]

Stop

Haven't used 10 logarithm

[mellow grail]

okay 😐

[open temple]

So Wolfram alpha is fine...

[mellow grail]

$log = log_{10}$

[ocean seal]

itzkraken.

[open temple]

Look

For 2^3^4^5 would the solution be ceil(3^4^5 * log(2)) = 1'049'626'694

[mellow grail]

uh huh

Oh

its not 9!

[open temple]

Or is my calculation wrong?

[mellow grail]

its 3^4^5^6^7^8^9

[mellow grail]

no its right

[open temple]

Okay, so your solution maybe is wrong?

[mellow grail]

but..okay i dont think i can help

i have to go anyways

sorry 😔

but someone should come come eventually

[open temple]

Thx anyways lol

[lone heart]

@open temple Has your question been resolved?

[open temple]

Help pls

[lone heart]

@open temple Has your question been resolved?

[lone heart]

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[marsh echo]

if the p.d.f of X is given by $f(x) = (1-p)^{x-1} p, 0 < p < 1$, where x is all real positive numbers, determine the cumulative function of X.

[ocean seal]

Big Chicken

[marsh echo]

pls

[ashen lance]

,I see, Actually this is the whole question. Thank you for giving time to my problem though.

[teal plaza]

Just integrate this pdf to get the cdf

Also this is a geometric distribution if it matters

[marsh echo]

i cant integrate im afraid

as we havent learnt that yet inclass

also what a geometric distribution

[teal plaza]

Ohhhh shit I'm sorry

[marsh echo]

algs

[rose temple]

👍

[marsh echo]

<@&286206848099549185>

[teal plaza]

Honestly I don't know lol, how to get Cdf by pdf without integration

[marsh echo]

oh

[teal plaza]

Maybe it's asking you to recognise it's a geometric distribution and just memorise it's cdf

[marsh echo]

what about if its from a probability distributiont table

[lone heart]

@marsh echo Has your question been resolved?

[teal plaza]

Wdym?

[marsh echo]

,rotate

[teal plaza]

Cdf table would be

0.4 0.7 1

[marsh echo]

how did u figure that uot

wait nvm i get the cumulaitve part

but im confused with the restrictions for the picewise functions

[lone heart]

@marsh echo Has your question been resolved?

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[haughty valve]

I think uh to solve it you use triangle similarity and lining AF?

But idk whats next

[marsh echo]

the fomrula

for DC

is 1/2 (a+b)

where a and b are the bases

the entire thing is a scalene trapezium

[haughty valve]

Okay thanks

.close

[lone heart]

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[compact hawk]

I have a doubt :
I am currently studying trigonometry again (for passion) and i was thinking about the first fundamental relation of trigonometry which is : sin^2(a) + cos^2(a) = 1 [F1]
Which formula comes from the equation (or the definition) of "x^2 + y^2 = 1" [F2] (that's what's written in my school book.

So my question is :
How can i identify this equation as?
I mean, the formula F2 could be both a Pitagora equation as the number 1 can be also be written as 1^2 (and so this equation represent a number but with x,y parameters) or the equation of a circle (a unit circle to be more precise).

So, what i am thinking is : 'Aren't these both formulas representing (or sharing) the same equation but with different meaning?
I mean, when can i assume to use that formula to represent a circle and when to represent a number with Pitagora?

(I am italian, so the translation might not be so good)

[mortal trellis]

well in general the equation for a circle of radius r is x^2+y^2=r^2 and it comes from pythagoras

https://i.imgur.com/qxDCAN5.png

[compact hawk]

ok, but so, doesn't it mean that a circle equation means also a number?

so, what concept do i have to attribute to that formula?
That formula represent a circle or a segment?

[mortal trellis]

the point (x,y) is on the circle if it satisfies x^2+y^2=r^2 which comes directly by considering the triangle I drew

[compact hawk]

or, do i have to specify also the Domain (if it's so called in english) of which number r can be?
I mean : 1 is the square or 1, but, for example, there is no number which square gives 2. this means that, now that the r is 2, the formula represent a circle

[mortal trellis]

sqrt(2) is such a number

r should be positive but that is a reasonable demand cause r is a length which should be positive

[compact hawk]

but sqrt(2) is an infinite decimal number and that shouldn't be really a number. i mean : 1.41... is not a number until we consider a limit of decimal points

[mortal trellis]

sqrt(2) is a real number

[compact hawk]

for example : (1.41142135)^2 doesn't not give exactly the number 2

[mortal trellis]

so?

[compact hawk]

so, what sense does it do to consider a segment as sqrt(2)

let's see it :
x^2 + y^2 = sqrt(2)

What does this equation represent?

[mortal trellis]

https://i.imgur.com/b023B7y.png

[compact hawk]

a segment or a circle?

[mortal trellis]

what is the length of that third side

[compact hawk]

the sum of x and y squared both

[mortal trellis]

and then the sqrt of that

[compact hawk]

under sqrt

yes

[mortal trellis]

which is?

[compact hawk]

r, the radius

[mortal trellis]

I mean the actual number

[compact hawk]

sqrt(2)

[mortal trellis]

ok. and do you now want to tell me that its nonsense to have this triangle

cause sqrt(2) is not a "proper" number in the sense that you want?

[compact hawk]

hmm, this makes sense but then i cannot 'render' that idea. i mean : how can i draw a segment which continues to the infinity of 0? (i mean the infinity decimal numbers)

the segment shouldn't have an 'end' point

while a segment of value of 2 has a start and an end point

[mortal trellis]

this also has a start and endpoint. the vertices of that triangle

[compact hawk]

(in this 'discussion' i don't want to consider approximations of decimal nunmbers)

[compact hawk]

ahahaha

as i said, i cannot render the idea of representing an irrational number on the cartesian plane

i mean, how can you draw a point which actual x (let's say y = 0) value has an infinite decimal numbers?

[teal plaza]

Bruh

[compact hawk]

i mean, once i drew a point on (1.4114, 0), then i can draw a more precise point on (1.411421, 0)

and so on

[mortal trellis]

well there is a whole field which concerns these so-called "constructible" numbers which can be constructed with just a compass and straightedge

[teal plaza]

We use approximtion

[mortal trellis]

which is more complicated and not all numbers can be done constructed like that

[compact hawk]

i said that i don't want to consider approximations

[mortal trellis]

but that said, to draw (sqrt(2), 0), you draw a triangle like I did and then just take that length and rotate it to the x-axis

[compact hawk]

i surely have no idea on how these concepts works as i just know 4th high school math concepts but i wanna understand these

[worn fox]

A circle of radius 1 has area π, does that also trouble you?

[compact hawk]

yes

i just don't understand how can an infinite number represent a finite area

[mortal trellis]

1 is also an infinite number. 1.000000000

[worn fox]

It's not an infinite number, π is less than 4. It just has an infinite decimal representation

[compact hawk]

yes but no (i think), as 0s do not mean nothing

[mortal trellis]

are you fine with stuff like 1/3 = 0.3333...?

[compact hawk]

these are periodic number and are less complicate than irrational numbers

[mortal trellis]

still infinite

[compact hawk]

however, let's go back over the first question :
x^2 + y^2 = 2

What does that equation mean?
Does it represent a circle in the cartesian plane or a segment?
In case it represent both (as the segment is the radius of the circle), then what does this mean?
Does it mean that the radius is the circle?

[acoustic beacon]

Circle?

[compact hawk]

why?
Why cannot it represent a segment because of Pitagora?

Pitagora says that the hypotenus (a segment) is equal to: x^2 + y^2 = i

[acoustic beacon]

Oh hypotenuse.

[mortal trellis]

segment is a different thing btw in english

[compact hawk]

? what does it mean?

[mortal trellis]

well the equation is related to the triangle. sure

[acoustic beacon]

Yeah it represents both. But the implicit equation for values of x and y represents a circle.

[mortal trellis]

but as a general equation it represents the circle

[compact hawk]

so, with parameters this equation represent a circle and by inserting the parameters values it represent a lenght?

[mortal trellis]

https://i.imgur.com/5Ui5T7l.png

[acoustic beacon]

Yes.

[acoustic beacon]

They meant hypotenuse.

[mortal trellis]

yes I know

and then they asked what segment means

[acoustic beacon]

Oh.

Gotcha.

[compact hawk]

i still cannot idealize.
i mean, i am not acepting that fact

[acoustic beacon]

Imagine a triangle with a fixed hypotenuse and change the values of the other 2 sides and plot one of the vertices that's with the hypotenuse, you'll get a circle.

[compact hawk]

i don't understand what you wrote

[acoustic beacon]

Now on changing those x and y values, you will see it makes a circle.

[compact hawk]

Wait, i think i have understood my main question :
The equation x^2 + y^2 = 2 represents both the circle on the Cartesian plane and the radius of that circle, as it represent a circle once i do not 'input' the parameters of the equation and it represents a radius lenght once i input these parameters.
Is that right?

[compact hawk]

i don't want to think in a graphic way but only with an analytic way, with formulas

[mortal trellis]

but at the same time you arent ok with "analytically drawing" something with length sqrt2

[compact hawk]

oh, that's also thinking in an analytic

[acoustic beacon]

I didn't understand what you meant here.

[compact hawk]

because you are changing x and y, while in the graph you are likely changing the B point along (?) the O point

[compact hawk]

it's hard to explain what i am doubting about...

i mean :
x^2 + y^2 = 2
represents both a circle and a lenght of a segment (which is the radius of the circle)

[acoustic beacon]

2 is not the length, it's sqrt(2).

[compact hawk]

because the circle exists when i do not input the x and y parameters, and the lenght exists once i input these parameters.
Am i right?

[acoustic beacon]

Not exactly.

You have already defined the length.

[compact hawk]

so, i am defining the radius and the circle at the same time with the same formula?

this means that a circle is also a lenght?

[acoustic beacon]

Radius = length
General form of circle: x^2+y^2=r^2
Where r is the length of radius.

[acoustic beacon]

The radius of the circle is the length.

[compact hawk]

so that's right.

[acoustic beacon]

No, the length always exists.

[compact hawk]

i define a lenght once a circle exists. This means that a lenght is correlated to a circle. means that if a lenghe exists, then a circle exists. means that the formula (pitagora and circle formula (which is defined trought pitagora)) represents both a circle and a lenght

[acoustic beacon]

Yes.

You got it.

[compact hawk]

So a lenght exists only if a circle exists and a circle exists only if a lenght exists or does the circle exists even if the lenght doesn't exists?
This should mean that the equation represents both a circle and a lenght as a circle cannot exists without a lenght and a lenght cannot exists without a circle. am i right?

[acoustic beacon]

Can a circle exist without a defined radius?

No.

[compact hawk]

well yes

but it's a point

it's a degenerated circle

[acoustic beacon]

The radius is zero in that case. I am talking about something undefined.

https://www.desmos.com/calculator/npckjkftdq Try this graph.

[compact hawk]

0 means that the radius doesn't exists. has the 'undefined' word the same mean of 'unexiststance'?

[compact hawk]

what do i have to try?

[acoustic beacon]

Yes but I meant you can't make a circle without being given the knowledge of the length of the radius.

[compact hawk]

this means that the circle exists even if the lenght doesn't exists but this point is a specific concept of the definition of a circle

[acoustic beacon]

You can see how triangles are related to circles.

[compact hawk]

right. but as the radius is 0, then the radius doesn't exists. we are talking about the existence of a circle with and without the existence of a radius. we are not talking about 'defining' the circle

what a brain fk i am having

i think i lost the point of what we are talking about

[acoustic beacon]

A circle can't exist without a given radius. You need to know the radius to define ( make it come into existence) a circle.

[compact hawk]

we are going into philosophy now

[acoustic beacon]

Happens. I suggest you familiarize yourself more with analytical geometry and trigonometry.

[acoustic beacon]

This is math, sir.

[compact hawk]

a radius of lenght 0 means that is defined but it doesn't exists

[acoustic beacon]

Language barrier is really fricking us over.

[compact hawk]

that's my intent for this summer (not only these 2 subjects)

[compact hawk]

yes

[acoustic beacon]

Yeah but if you don't know the radius how will you even think of the circle?

I am not talking about radius = 0

By undefined radius I mean a radius that we don't know.

[compact hawk]

a circle of radius 0 is by definition a point.
But this allows us to say that a circle exists even if a radius doesn't exists. but this is likey a 'special' definition to explain that : 'Can a circle exists without the existence of a lenght (radius) and viceversa?'

[compact hawk]

and what's the point of that?

[acoustic beacon]

It satisfies x^2+y^2=r^2 so it does exist in technicality.

[acoustic beacon]

You asked that can a circle be possible if we didn't know the length.

[compact hawk]

i am asking that

[acoustic beacon]

Actually let's start over.
Ask your question in a precise manner.

I'll try to answer/

[compact hawk]

normally a circle exists if exists a lenght (radius). but if a radius exists, this means that a circle also exists.
This is not true in the special case in which the lenght of the radius is 0 which means that the radius exists (is defined) and so the circle exists but the radius graphically doesn't exists but so, in this case, a circle can exists even if the radius doesn't exist

[compact hawk]

yes, that's the best way but to discuss about it we had to discuss about the smallest doubt

[acoustic beacon]

They both exist in technicality. But both can't be represented on a graph. The circle also doesn't exist in this case because if the radius doesn't exist then the circle doesn't exist and vice versa. They exist in technicality only. It's trivial to be concerned with this circle because it's only zero.

[compact hawk]

my question is :
**What does this equation mean?
x^2 + y^2 = r^2

By Pitagora it is the implicit way to express a value which represents the lenght of a segment (with a start point A and an endpoint B).

By the definition of a circle it represents a circle in the Cartesian plane. This definition (by the way) has been extracted from Pitagora.

So : Does this equation represent both a lenght of a segment and a circle or only one of them?
**

[acoustic beacon]

It represents both.

But to define a circle using the pythagoras theorem, you need to know the radius.

That's it.

[compact hawk]

that's then what i firstly tought at the start.
NOW, my question is : When do i have to consider that equation as a lenght and when as a circle?

I mean : if someone asks me : draw that equation on the Cartesian plane.
What do i draw?
A circle or a CASUAL lenght of a segment (which could be the radius)?

[acoustic beacon]

Ah. You draw the circle because the radius would be given in the equation.
x^2+y^2=4
Here radius = 2

[compact hawk]

Ok, then i have not clear what 'inserting any value instead of x and y to get R' means.
For example, 4 + 9 = R.
What does this mean in the graph?

[acoustic beacon]

This means you know the x and y value of a point on the circle.

[compact hawk]

isn't it the lenght of the radius?

[compact hawk]

oh, this means that the radius has lenght from (0, 0) to (2, 3)? (as x^2 = 4 and y^2 = 9)

[acoustic beacon]

x and y represent the sides of a triangle and also a point, now you know pythagoras's formula so you can calculate for the length or radius, they are the same.

[acoustic beacon]

Yeah.

[compact hawk]

my brain is now likely trying to not accept what we have discussed now.
I feels like it has nonsense but it has sense

[acoustic beacon]

You can see.

2 and 3 are lengths of sides.

Which can be represented by (2,3) on the circle.

[acoustic beacon]

Also it's not R it's R^2

[compact hawk]

so, let's resume up.

x^2 + y^2 = r^2
and
a^2 + b^2 = i^2

are not the same thing, as the first one represent an equation to represent the circle on the Cartesian plane, while the second one represent the lenght of the hypotenusa.
I mean that the first equation has 3 parameters that change and are not constant, while the second equation has 3 parameters that can be taken in input and represent a lenght and a and b are constant

[compact hawk]

yes

[compact hawk]

cry me. i still do not understand what's the difference between Pitagora formula (a^2 + b^2 = i^2) and circle formula (x^2 + y^2 = r^2).
FINALLY. That's what i meant to ask

[acoustic beacon]

Yeah, you're pretty much right. They're the same thing, they can be understood in different contexts.

[compact hawk]

OOOh, finally. that's what i mean and i wasn't able to express that out of my brain

[acoustic beacon]

Yup.

Gotta go eat dinner now!

See ya.

[compact hawk]

thank you. see you

how do i close that 'ticket'?

!close

[subtle birch]

.

[tacit arch]

.close

[lone heart]

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[alpine sable]

Hello. Problem is as follows.
Given n+1 boxes, with box i containing n-i+1 black balls and i-1 white balls. Aka box 1 has n black balls and 0 white balls; box 2 has n-1 black balls and 1 white balls, etc
So, given a random box, the probability to pick 3 white balls

[alpine sable]

@surreal crystal srry i was afk and bot closed channel

[surreal crystal]

are you familiar with the large product notation?

[alpine sable]

i know this is

P(3 whites | box k) = P(box k) * P(box k | 3 whites)

P(box k) = 1/(n+1)

but idk what is P(box k | 3 whites)

i know for k = 1, 2, 3, it is 0

cuz there arent enough white balls

on box 1 u have n black and 0 white
on box 2 u have n-1 black and 1 white
on box 3 u have n-2 black 2 and whites

So on box 4, n-3 black and 3 whites

so the probability here is i dont know XD

but on box n+1 u have 0 blacks and n whites, so the prob is 1

[surreal crystal]

i believe you should be able to simply multiply the probabilities of each box

nevermind

[compact hawk]

is that a high school problem?

[alpine sable]

not rlly

[surreal crystal]

the total balls in each box is n

[alpine sable]

ye

[compact hawk]

cause i did probability but i understand quite nothing. or at least i should spend some time to understand what the problem means

[surreal crystal]

so we must choose 3 balls from n

so nCr(n,3) is 3 balls from n

[alpine sable]

what is that exactly? i know that nC3 means

[surreal crystal]

n choose 3

[alpine sable]

but in our problem, what does this represent exactly?

[surreal crystal]

the probability of choosing any three of any of the balls

[alpine sable]

this number is always bigger than 1

so it cant be a probability

[surreal crystal]

we arent done

we have to choose 3 white balls from them, which is i-1 whites

so the probability of choosing 3 white balls from a box is C(i-1,3)/C(n,3)

[rose temple]

its a very interesting case of pythogoras where the hypoteneous is the radius which remains the same but with increasing angle the perpendicular and base change ther values to satisfy it

[alpine sable]

??? this channel is already occupied. Previous topic no longer belongs here

[alpine sable]

okey, thats the solution but i dont quite see it

is this Laplace thingie?

[surreal crystal]

now we sum i=1 to n+1, P(box i) * P(3 white balls from box i)

[alpine sable]

must be but i cant see at all

i see nC3 is all the possibles scenarios of picking 3 balls out of n

[rose temple]

ohh

[alpine sable]

but why the succeds are i-1C3?

[surreal crystal]

because it is defined that there are i-1 white balls

and we want to choose 3

[alpine sable]

aaaaaaaaaaaaaaaaaaaaaaaaaa

we are only interested on picking white balls

and doesnt matter the order or something

just picking 3 of them

aaaaaaaaaa

[surreal crystal]

P(box i) = 1/(n+1), P(3 white balls from box i)= (C(i-1, 3) / C(n, 3))
this should give you the answer you want

[alpine sable]

yeah well

last step would be

1/(n+1) * sum from k=4 to n+1 (k-1)C3 / nC3

to sum this thing XD

which im not sure how to

[surreal crystal]

k-3?

[alpine sable]

1 srry

[surreal crystal]

yeah

[alpine sable]

nC3 can go outside, cuz does not depend on k

so how do u sum k-1 C 3

[surreal crystal]

$nCr = \frac{n!}{r!(n-r)!}$

[ocean seal]

Cycadellic

[surreal crystal]

use this to simplify

then, since n is not specified, i doubt the summation must be simplified

[alpine sable]

well

[surreal crystal]

but if you necessarily have to, i think you can use the binomial theorem to get a polynomial of n

[alpine sable]

the answer is 1/4

so idk XD

but idk how is the sum done

P(Ck) is the box

and P(B3) is 3 whites

this is what u said

so far so good

i gtg rlly quick, but ill read u after. Thanks very much for u help tho ^^

[surreal crystal]

np

you just need to show that $\sum _{k=4}^{n+1}nCr(k-1,3)=nCr(n+1,4)$

[alpine sable]

yeah, but idk how to xD

[ocean seal]

Cycadellic

[surreal crystal]

write out the summation, im sure the pattern will jump out

binomial theorem

this one

with finite bound

set a=1, b=1

$2^n = \sum_{k=0}^{n} nCr(n,k)$ follows

[ocean seal]

Cycadellic

[lone heart]

@alpine sable Has your question been resolved?

[lone heart]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[lone heart]

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• 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.

[shrewd ruin]

I need help with Matrix questions

[alpine sable]

I can try my best to help

whats the question

[tacit arch]

[shrewd ruin]

ok

I have 3

part a, b, and c

[alpine sable]

ok

what part a

[shrewd ruin]

this is the first one

[alpine sable]

ok looks just like subtraction

so each element is P gets subtracted by each corresponding element in R

I think

[shrewd ruin]

ok?

so how will

i do it

what's P and what's R

[alpine sable]

the top left in P is 4 right

and the top left in R is 1

[shrewd ruin]

wait wait

that's not how it works

[trim sorrel]

I dont think thats it

[shrewd ruin]

because

P

is a matrix

[trim sorrel]

P is a separate matrix from (4 6 3 -2)

[shrewd ruin]

that projects vectors onto 2;3

[trim sorrel]

Its being multiplied

[shrewd ruin]

right right

[alpine sable]

yeah im not a linear algrbra fan

idk

[shrewd ruin]

ok

[trim sorrel]

How can you use this to help

Where AB = C

[shrewd ruin]

huh

what is C

and what is A and B

[trim sorrel]

All generic matrices

[trim sorrel]

If AB = C where A B and C are all matrices this holds

How can you use that to help

[shrewd ruin]

i mean

i think

i know how to do

matrix multiplications

it's just

I don't know how to get P and R

[trim sorrel]

You don't need it

That's the point

[shrewd ruin]

huh

[trim sorrel]

Try to do it without finding P and R explicitly

[shrewd ruin]

how will i do that

[trim sorrel]

Just knowing what P and R do is enough

[trim sorrel]

Using this

We know that Px = y where y is the projection of x onto (2 3)

[shrewd ruin]

right

[trim sorrel]

For vectors x, y

Each column of a matrix is a vector

[trim sorrel]

Now use this

[shrewd ruin]

should we like

use a generic matrix for P?

[trim sorrel]

Wdym

[shrewd ruin]

like

for P

[trim sorrel]

We want to find $$P \begin{pmatrix} 4 & 3 \ 6 & -2\end{pmatrix}$$

[shrewd ruin]

should we set it as like

[trim sorrel]

Don't set it as anything

[shrewd ruin]

oh ok

[trim sorrel]

You don't gotta find P

Sorry bad formatting

[shrewd ruin]

its ok

so

[ocean seal]

wakebloom

[shrewd ruin]

yea

P projects vectors onto 2; 3

[trim sorrel]

Let this product be the matrix Q

[trim sorrel]

Find each column of the matrix Q using this

[trim sorrel]

And this

[shrewd ruin]

ok

oh wait

wait x is a vector tho

not a matrix

[trim sorrel]

^

[shrewd ruin]

oh

well

i kinda get it

but then i still don't know how

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What r u thinking

What r u trying

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i mean

Px = y

P(the matrix) = Q

that's what we set right?

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Yea

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and y is the projection of x to 2,3

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Yup

How can you find the first column of Q

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With this

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matrix A multiplied by column of B

right?

wais so here

the matrix is A

wait no is that right?

is P same as A here?

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Yes but whats A and B

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Yes

Because P (the matrix) = Q

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right

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And AB = C

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so the matrix would be B

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So A is P and B is the matrix

Right

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right

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So the first column of Q is what?

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P(4, 6)?

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Yup!!

And what's that

Using the definition of P

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ohhhh

so it's the projection of

(4,6) onto (2,3)

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Yesssssss

You got it

Now just apply that logic to each column

And then do the same for R

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ok

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Then you subtract and youre done

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is that just 4, 6?

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Yea

You see why?

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yea yea

so that's the first column

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Ya

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and the second is 0, 0?

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I believe so cause its orthogonal

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ok

oh wait

R is like

a reflectio

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Right

Youll be able to do it with the columns tho

They made them pretty simple

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ok

lemme try

i don't really know how to do that

does that mean

we reflect (1, 4) across a line with direction (1,4)

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Yea

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and we reflect (0, 0) across a line with direction (1,4)

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Yea

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i don't know how to do that

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Try to visualize it

Visualize the line

And then visualize each vector

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is the first one just 1, 4

?

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Where r u getting that from?

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i mean

we have the vector

1, 4

and then the line is also 1, 4

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Yup

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so wouldn't that just equal itself?

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So its not being reflected at all

Yes

So it is just 1, 4

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so it would be (1, 4)

ok

thanks

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What about 0, 0

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then for 0, 0

0,0 again?

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Yup

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oh

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Cause its also on the line

Pretty simple

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right

so do we just

subtract

?

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Subtract the resulting matrices

Cause youve found P(4 6 3 -2) and R(1 4 0 0)

So yea

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ok

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Was it right

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yep

it is

i have 2 more

question

is that ok to ask?

they are similar i think but then different

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Sure

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ok so that's the question

i think

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How would you approach this using the same method

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i have like a vague idea

will it be like

we calculate

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Remember, matrix multiplication is associative

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B(-4, 1, 1, 4)

that first

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Yup

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then A?

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You got it

Thats it

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ok

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Find this matrix

And then do A * that matrix

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right

how do i work out A

cus B is like the same as the last problem

A says rotate clockwise by a right angle

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I think you can visualize it and find it intuitively

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let's say we get (a, b, c, d) after we do B(-4, 1, 1, 4)

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Rotation by right angle is not hard

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ok

we will do like rotating (a, c)

by clockwise 90 degrees

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Right

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and (b, d) by 90 degrees right?

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Right

Just using the same logic as before

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will it be

c, -a?

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I think thats correct yeah

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ok

well that problem's good

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i don't get

what dilating vector by factor of 3 means

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Dilate means scale

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and how an inverse of that works

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How do you scale a number by a factor of 3

Like any old number

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just multiply 3?

to everything?

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Rit

Right

Whats the inverse of that

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divide by 3?

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Right

Basically the same thing but in matrix form

In fact you can find what D is very trivially

And D^-1

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so i got

E(matrix)

can i just

divide each number by 3?

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Wdym

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i did like the first part

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Like you got the resulting matrix from this?

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yea

(0, 9, -3, 24)