help-13

[digital cliff]

with that youre done

[dusky linden]

so function 1 = f(g) = 40x8.05
function 2 = f(x) = 0.05(x-1000) + g right?

[digital cliff]

iffy notation but yeah
f(x)={ 40(8.05) 0<=x<=1000
40(8.05)+0.05(x-1000) 1000<x

[dusky linden]

how would inverse relation work for the first function if it doesnt have a x value

[digital cliff]

what would we normally do?

starting with y=322

[dusky linden]

i would switch y and x

[digital cliff]

so x=322

thats it, though we cant call it a function

[dusky linden]

oh i get it

so its a vertical line now

[dusky linden]

but if she earned 420 a week

that would be

for the first function it would be f(x) = 40x - 420

and the second function will become f(x) = 0.05(x-1000) + 420

right?

[digital cliff]

the wording feels confusing though that could just be me

could i see the bits under it?

[dusky linden]

wdym

oh wait

im stupid

[digital cliff]

ah expression, never mind

well, when we made an inverse we changed things a bit

so now x is her pay and y is her sales

for x=420, then we're using the second expression's inverse since its larger than this

[dusky linden]

i think its not the inverse one

[digital cliff]

the inverse relation gives us

sales in relation to earnings

which is what it asked for no?

[cedar kiln]

@dusky linden Has your question been resolved?

[dusky linden]

im assuming

its based on the original

where f(x)={ 40(8.05) 0<=x<=1000
40(8.05)+0.05(x-1000) 1000<x

[digital cliff]

you could, sure, but it would just involve a bit of extra working

we'd set f=420

the first expression doesnt accomodate so it must be the second one

then we solve for x, which is pretty much just doing the inverse

[cedar kiln]

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[normal snow]

I don't know where to start

[loud topaz]

it is basically just asking when y is greater than 0

so just go to each x point and see if its corresponding y value is greater or equal to 0

so like for -1 it would be true because the y value is 7

but for -9 it isnt because the y value is -2

[normal snow]

do i just list the numbers

[loud topaz]

the coordinate pairs yeah

[normal snow]

its telling me inqualities using x??

[loud topaz]

correction sorry, it would be every x value with a positive y

so -4, -3, etc.

[merry stag]

You can't list infinite numbers, but you can list the intervals that represent all those numbers.

or should i say you can't hope to write them all down...

[normal snow]

but how do i write the inqualities

[merry stag]

I mean a (closed) interval is typically denoted [a, b] meaning: All the reals between a and b including a and b themselves.

[normal snow]

it shows this on the bottom

[whole nexus]

Like if there is a function that's always valid for greater than 2, you don't have to state everything, you can just write f(x)>2

[cedar kiln]

@normal snow Has your question been resolved?

[vagrant moat]

It wants intervals in the form a ≤ x < b

[cedar kiln]

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[vernal shell]

how is this wrong

[cedar kiln]

@vernal shell Has your question been resolved?

[vernal shell]

what do i do for the direct substitution? my prof didnt go over this part

do i literally just directly subs then differentiaitierte

[ancient lodge]

Yes

[ancient lodge]

!show

[cedar kiln]

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

[cedar kiln]

@vernal shell Has your question been resolved?

[cedar kiln]

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[chrome agate]

i went onto google & it said to calculate LW, LH, 2LH, WH, 2WH then add them together

LW = 4x^2-24x+19
LH = 19-2x^2
2LH = 38 - 4x^2
WH = 12x-2x^2
2WH = 24x-4x^2

[north wyvern]

so, how i would go about this is to start with the total area

subtract off to find the area of the inner rectangle

then the volume = that area * x

[chrome agate]

what would i subtract off?

[north wyvern]

you need to subtract off each rectangle's area except the inner one

[chrome agate]

i have the length, width, and height all solved if you want that

let me look at my notes cause we did something similar to this

[north wyvern]

so all of these guys

[chrome agate]

my notes say l•w - corners

for surface area

[north wyvern]

oh sorry i read volume and thought thats what were doing

[north wyvern]

yes do this

[chrome agate]

haha no, i figured out volume already

let me see what i get

for the final product for l•w i got -4x^2-62x+228

[north wyvern]

no

where did the -62x term come from

[chrome agate]

when i multiplied both terms

[north wyvern]

not like this

(l*h) - corners

[chrome agate]

wait i solved for all of this lol

LH is 19-2x^2

[north wyvern]

this is a really long way to go about it

whats l*h

[chrome agate]

^

since L itself is 19-2x

[north wyvern]

no, we are given l and h as numbers

[chrome agate]

and H is x

OH

[north wyvern]

yeah

[chrome agate]

228

[north wyvern]

now whats the total area of the corners

[chrome agate]

-4x^2

[north wyvern]

ill take it

[chrome agate]

but it says the width is 12

[north wyvern]

so the surface area is?

[chrome agate]

228-4x^2

oop

so does the width act as the height when it comes to outside surface area then?

[north wyvern]

i just cant believe the google ai made you do all that

[chrome agate]

cause it was l•w - corners

[chrome agate]

i know 😭

should've just followed my notes

thank you so much!

.close

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[snow hare]

What topics in unit 1 for AP Calculus AB help solve this question?

[floral arrow]

Just limits, along with basic understanding of factorization and function composition but that's not calculus

[snow hare]

Oh okay thanks

.close

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[floral arrow]

If you need help solving these correctly, you can ask

[cedar kiln]

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[soft path]

Consider a group of 21 people. A and B are considered to know each other if A is friend with B or A is friend with C1, C is friend with C2,... Cn is friend with B. Given that in any 6 people from the group, there would be 2 people who know each other. Prove that there exists 5 people in this group who pairwise know each other.

[soft path]

I translated this from another language so I'm very sorry if the language is misleading/difficult to understand

I'm thinking about a proof by contradiction, but idk how that could be done

[cedar kiln]

@soft path Has your question been resolved?

[tropic oxide]

well hm "knowing each other" is an equivalence relation isnt it

yeah, it is

if you make a graph of all their friendships, "A and B know each other" <=> "A and B are in the same connected component"

try to reason about how many components there can be

[cedar kiln]

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[rancid prairie]

Can I get some help on this derivative problem?

[frosty creek]

Find the year by year slopes first

For 2005-06, it will be like 5.2 -6.3

Approx

So slope will be -1.1

Then plot the dy/dt graph

[rancid prairie]

Ok but its asking the slope for each part?

I thought it said it was asking for the derivative between 2005 and 2001

[frosty creek]

Yeah do it separately for each year

[rancid prairie]

2011*

shi ok

[frosty creek]

Yeah so you can do directly, you have to plot the years that are present betweel 2005-11

Hope its clear

If you question has been solved then pls close the ticket

[rancid prairie]

So the dates are the x values right?

like 2005 is x=0

[frosty creek]

Yeah we want dy/dt

So x is not the thing here

We need to find delta y and plot the graph

Delta t would be 1 as its 1 year change

[grave osprey]

There is no "x" here.

[rancid prairie]

ok let me find the slopes of each then I'll come back to you

[frosty creek]

Yep

[frosty creek]

Okay

[rancid prairie]

ok I got the slopes

now what?

[rancid prairie]

wdym?

[grave osprey]

What you refer to as "x" is "t"

[rancid prairie]

Yes

[frosty creek]

Plot the graph, also do you mind sending the slope values?

[rancid prairie]

so time is the x-value here

[frosty creek]

Yeah now plot slope versus graph time

Thats what it said in the question

Graph dy/dt

[rancid prairie]

Ye but wouldn't that give me the same slopes?? thats what I'm confused about

I understand it better physics wise

like the derivative of the postion over time is the velocity

and the derivative of velocity over time is acceleration

[frosty creek]

No, the graph would be a horizontal line and then a vertical line where it changes the slope

[rancid prairie]

oh so the y value is the slope?

[frosty creek]

Yeah

Delta y

[rancid prairie]

change of y?

I still don't understand what you mean

I think I understand but also don't

Let me draw it

Is this what your saying?

@frosty creek

[frosty creek]

Yeah

And when slope changes, it’ll be a vertical line

Have you ever drawn SFD?

[rancid prairie]

Ok I got it

thank you so much

[frosty creek]

All good mate have a nice day

You may close the ticket @rancid prairie

[rancid prairie]

ok wait I got one more question

[frosty creek]

Yes?

[rancid prairie]

This problem is asking me for the first and second derivative.

[frosty creek]

Then whats the problem?

[rancid prairie]

I'm confused about what rule to do?

Power rule?

[frosty creek]

You need to find dy/dx and d^2y/dx^2

[rancid prairie]

then product rule?

[frosty creek]

What product rule?

No, just differentiate

[rancid prairie]

let me explain my thinking

[frosty creek]

4x^3/3 would be 4x^2

-x would be -1

And 2e^x would be the same

First derivative is done

And then differentiate again

[rancid prairie]

why 4x^2?

[rancid prairie]

wouldn't that be 0?

since the derivative of e is 0

[frosty creek]

Noo

[frosty creek]

Whats the derivative of e^x

[frosty creek]

e and e^x is different

We need to find derivative of e^x

Search the derivative of e^x

[rancid prairie]

so e^x is just the same?

[frosty creek]

Yeah

[rancid prairie]

ok but why this?

[frosty creek]

And 2e^x too cuz we take 2 out and we dont differentiate 2

[rancid prairie]

so e^129830192x

[frosty creek]

x^n is nx^(n-1)

[rancid prairie]

is

e^129830192x

[rancid prairie]

ye but the denominator?

is 0

Im sorry I sound very dumb rn

just don't understand fraction of derivatives

[frosty creek]

If x is a different varible then the e term and x are seperately differentiated

[frosty creek]

You’ve been taught differentiation right?

[rancid prairie]

yes

I just forgot

do I quotent rule it?

oh wait

its 12x^2/3

12/3 = 4

but still

[frosty creek]

Yes

Yes

Yes

You got it

[rancid prairie]

why doesn't the 4 differentiate?

[frosty creek]

You just got it

[frosty creek]

Have you ever seen a constant term differentiate?

Search the derivative of 4

[rancid prairie]

its just 0

since there is no power

[frosty creek]

No bro

[rancid prairie]

???

[frosty creek]

Constant term’s derivative is 0 when its alone

But when its with a variable

The constant term remains the same

Its not differentiated

[rancid prairie]

Where is the variable???

[frosty creek]

In our question, it is

Yeah only derivative of 4 is 0

[rancid prairie]

its just 3 tho

[frosty creek]

Because there is no variable

[rancid prairie]

like

[frosty creek]

Yeah but its with the variable

[rancid prairie]

3x would be 3

[frosty creek]

Yes

[rancid prairie]

but 3 would be 0

[frosty creek]

yes

[rancid prairie]

???

its 4x^3 / 3

where is the variable in the denominator?

[frosty creek]

4x^3/3 can be written as 4/3*x^3 right?

[rancid prairie]

(4/3)x^3

[frosty creek]

4/3 is the constant with variable x^3

[rancid prairie]

?

[frosty creek]

Yeah

Correct

[rancid prairie]

Oh so x applies to both?

Wait then still

why don't we then use power rule on it

to make it 12x^2 / 9

[frosty creek]

Yeah

[rancid prairie]

or wait

[frosty creek]

Why 9?

[rancid prairie]

that would be 3/3 (12x^2/3)

[frosty creek]

We can only differentiate x^3

[rancid prairie]

Ye okok

I got it

do u recommend always rewriting it in

[frosty creek]

After differentiating we get (4/3)*3x^2

[rancid prairie]

(4/3)x^3?

[frosty creek]

Rewriting what?

[rancid prairie]

or in some cases no

[frosty creek]

If you find it difficult or think you’ll be confused in the exam then yes

But if ask me for time efficiency, dont

Cuz every second matters

[rancid prairie]

ok i got the derivative of the first

what about the second?

I just use power rule again?

[frosty creek]

Derivate the first derivative again to get the second derivative

[frosty creek]

Same thing you used for the first one

[rancid prairie]

so first derivative I got

4x^2 - 1+2e^x

[frosty creek]

Yeah now differentiate it again to get the second derivative

[rancid prairie]

second I got

8x+2e^x

[frosty creek]

Yep

Its done

Good job

[rancid prairie]

Alright bet

thanks bro

mb for being a bit dumb

[frosty creek]

No worries

Having doubts is a good thing

[rancid prairie]

I got a quiz tomorrow I just need details

because yk

in invesigation details matter

if yk yk

[frosty creek]

Glhf for the quizz

[frosty creek]

Yes

[rancid prairie]

"Good luck have fun"

[frosty creek]

Yes

[rancid prairie]

https://tenor.com/view/caseoh-waffle-house-ai-caseoh-ai-gif-13944294636252468146

[frosty creek]

I play games so thats the reason

[rancid prairie]

my reaction

[rancid prairie]

hahah

[frosty creek]

Maths quizzes are fun

You better ace that thing

You may close the ticket if your problem is solved

[rancid prairie]

I hope

bro gives us impossible questions

like

for example

lim as x approaches 0

of

[frosty creek]

It only feels impossible when you havent practiced

Practice more then you’ll reach close to your answer

[rancid prairie]

$sqrt {|x|} sin(1/x)$

[wraith dagger]

Adrift

[frosty creek]

My math is weak bro

I cant solve this

Ig it can be solved using a property tho

[rancid prairie]

ye

[frosty creek]

Good luck for your quiz tho

Have a nice day

Lmk how was your quiz

And try to drop some questions too

[cedar kiln]

@rancid prairie Has your question been resolved?

[wicked mantle]

$\sqrt {|x| \sin\frac{1}{x}}$

Is this what you're looking for?

[frosty creek]

I think the whole thing is rooted

[wraith dagger]

Swaamii

[cedar kiln]

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[noble granite]

i need help

[noble granite]

i want to solve a linear algebra equation Mx = b

where M is a 5x5 matrix and b is a vector

and entries are integers with magnitude around 10^5

doing this by hand is obviously (usually) infeasible

but how / is it possible to try approximating the solution?

[tropic oxide]

by hand???

[noble granite]

yes

these are the calculations (which are irrelevant but just in case anyone is interested) leading up to and the final augmented matrix

i can take big calculations, but not like super big

[tropic oxide]

dear lord

i would not expect this to be done manually

[noble granite]

just for fun :-)

im transcribing this into latex

$$\left[\begin{array}{cccccc|c}+25&-26&-124&+53&-18&-9&+1470\-26&+696&+2753&-989&+106&-4978&-7100\-124&+2753&+11180&-4052&+409&-20010&-30030\+53&-989&-4052&+1691&-220&+7424&+10170\-18&+106&+409&-220&+60&-693&-1320\-9&-4978&-20010&+7424&-693&+38405&+40950\end{array}\right]$$

bad

finally

[wraith dagger]

Number Basher

[cedar kiln]

@noble granite Has your question been resolved?

[fallen moat]

Usually for such long calculations, i will do something to "simplify" the numbers of the first column first
E.g.
R2+R1→R2
R3+5R1→R3
R4+2R2→R4
R5-2R6→R5
R1+3R6→R1

[noble granite]

thats what i did initially, but thats not sustainable

i think

[fallen moat]

Hmmm

[noble granite]

i think finding the exact value is pretty doomed because the denominator (which is like correlated to the determinant of the matrix) is like probably ~15 digits

its not impossible but not very nice; which is why i want (preferrably decimal) approximations

[noble granite]

specifically, following i think actually exactly this and some more, ive successfully eliminated the first column, but now the numbers are ~ 7 digits

[fallen moat]

I did recall theres some approximation technics(but j forgot about the names), but somehow i don't know which is exactly for this.

Did you try your luck on google about "system of linear equations approximation"?

[noble granite]

mostly computer cases

nothing useful, unfortunately

[fallen moat]

Sad

[fallen moat]

I'll coppy this down and see theres anything can be done with it

[noble granite]

ty!

[cedar kiln]

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[noble granite]

.reopen

[cedar kiln]

✅ Original question: #help-13 message

[noble granite]

Repost: I want to solve the following augmented matrix by hand, which is infeasible. I'm trying to settle on a decimal approximation. How do I do that?

$$\left[\begin{array}{cccccc|c}+25&-26&-124&+53&-18&-9&+1470\-26&+696&+2753&-989&+106&-4978&-7100\-124&+2753&+11180&-4052&+409&-20010&-30030\+53&-989&-4052&+1691&-220&+7424&+10170\-18&+106&+409&-220&+60&-693&-1320\-9&-4978&-20010&+7424&-693&+38405&+40950\end{array}\right]$$

[wraith dagger]

Number Basher

[frosty creek]

Nice chat room basher

[north wyvern]

At least the name is fitting

https://en.m.wikipedia.org/wiki/Bareiss_algorithm

Heres an algorithm for you

[noble granite]

haha

[frosty creek]

Want me to help or you’ll figure it out?

[noble granite]

i am not executing a O(n4L (log(n) + L) log(log(n) + L))) algorithm where n=6 and operations are like 10 digit multiplications which take 1.5 minutes

[noble granite]

help me please

pleeeease

[frosty creek]

😔

[frosty creek]

Try gaussian elimination

[noble granite]

i am like 99% certain that any method producing the exact result is bad

[frosty creek]

The only way to be 100% sure is to try

[noble granite]

i eliminate one column, numbers become like 7 digits long. if i eliminate another, i think it becomes >10 digits, and another becomes >17 digits, etc...

ive eliminated one column and gotten something terrible(y big)

[frosty creek]

Can you share?

[noble granite]

i dont have it near me

[noble granite]

but i essentially followed this

[frosty creek]

Do you know partial pivoting?

[noble granite]

no — lemme search

[frosty creek]

Swapping row with the largest absolute pivot

Swapping and dividing by pivots might help

Tho i havent tried it myself

[north wyvern]

Youre like super limited in how you can solve these algorithmically

[frosty creek]

By hand

[north wyvern]

O(n^3) is as good as it gets

[noble granite]

interesting, possibly

[noble granite]

yah, which is why i just want approxs

[frosty creek]

I think it can make the matrix easier to solve

[north wyvern]

Chop off you your approx first

Maybe rational reconstruction

[frosty creek]

Even if its decimal approximation, its not 100 % sure that the approximation will be even a bit near to the exact

Ykwim

[noble granite]

thats not recoverable

unless i find like a 300 digit prime or something

[noble granite]

yah

[north wyvern]

I meant partial elimination + rational recon

[noble granite]

hmm, lemme think

[north wyvern]

or just chop them off completely

[noble granite]

well, im imagining the entries of the solution will have 40-digit denominators or something?

[frosty creek]

Just try what i said and even if the ticket is closed, you can dm me for help

[noble granite]

based on my understanding of how it works

[noble granite]

yep ok

do you have time to roughly teach me how to execute partial (or even complete) pivoting

[north wyvern]

Its always a tradeoff between speed and accuracy

[frosty creek]

I’ll go for now, have a nice day folks @noble granite @north wyvern

[noble granite]

well, rat recons seems either 100% accurate or 0% accurate

[frosty creek]

Idk any fix time but whenever i get time, i’ll text you

[noble granite]

ok!

[north wyvern]

Rat recon accounts for error from partial elimination

[noble granite]

good! my dms are open and im not in a rush or anything, ill bump u as well

[frosty creek]

You can YouTube gaussian elimination with partial pivoting

[north wyvern]

But yeah, partials are as good as it gets

[noble granite]

ok, will try

[north wyvern]

Only way to go faster is to cut off numbers

[frosty creek]

Also i have never really practice partial pivoting that much, so will have to brush myself up too🤣

[north wyvern]

Or invent a cool new algorithm which will probably make you rich

[noble granite]

its fine

[noble granite]

haha

[frosty creek]

Its gonna be fun

[noble granite]

ok ill try to study & apply partial / complete pivoting

im going to https://www.youtube.com/watch?v=3IyumVeeUgo and https://www.youtube.com/watch?v=tpRBmGnXoyM and https://www.youtube.com/watch?v=RnCx8xq5NO4

[cedar kiln]

@noble granite Has your question been resolved?

[noble granite]

ill say no

[north wyvern]

How are you multiplying the ints?

Long mults are O(L^2) i think

[noble granite]

by hand? how specific are you asking

[north wyvern]

What algorithm

[noble granite]

yes (if you do them in any normal way)

[north wyvern]

Toom cook will get you O(L^1.5) iirc

For L digit lengths

[noble granite]

https://worldmentalcalculation.com/how-to-multiply-large-numbers-in-your-head-cross-multiplication/

[noble granite]

it gets you to O(L^(1+epsilon)), but its rarely better than O(L^2) in practice

[north wyvern]

Yeah i suppose you need to be pretty large for a big difference between O(n^2) and O(n^1.5)

Only thing better than toom is FFTs and those are for millions of digits

[noble granite]

yes

ffts are nice

[north wyvern]

But not helpful whatsoever here :P

[noble granite]

well, i think there are fft optimizations

[fallen moat]

Hey hey, just now i was calculating, and if this is [A|b]
i think you forgot to tell us A is symmetrical

[cedar kiln]

@noble granite Has your question been resolved?

[cedar kiln]

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[clear basin]

Could anyone help me with holes and vertical asymptotes, we can start on 1b). I just need help doing one of each so i can get an idea for the rest

[cedar kiln]

@clear basin Has your question been resolved?

[last apex]

holes will be when fraction is undefined but there is a finite limit at that point

vertical asymptotes happen when the two-sided limits to some point a are infinite

for 1b you should start by simplifying the fraction

[cedar kiln]

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[mental trail]

$P(X\in A)$ means $P(X^{-1}(A))$. For example in particular $P(X=x) = P(X\in {x})$

[wraith dagger]

Raphaelisius Maximus MMIII

[last apex]

well usually P is defined on subsets of our probability space

the example you gave doesn't demonstrate that

[mental trail]

"such that"

same as | yes

[cedar kiln]

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[mental trail]

different mathematicians

and also different topics, a same notation can mean very different things in different domains

[cedar kiln]

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[faint bison]

I know R\ {1} in in the topology as 1 is finite

and the same applies to -1

[frosty creek]

So it does not converge to 1 and -1 right?

[faint bison]

right

[faint bison]

and we're only considering 1 and -1 because of the sequence as we lnow it

[frosty creek]

So now we’ve to only prove that yn does not converge to any other point

[faint bison]

right

[frosty creek]

L belongs to R \ {-1,1}

[faint bison]

why? because of thr sequence, right?

[frosty creek]

Yes

Now consider an open set

A = R\ {-1,1}

Now why an open set you may ask

Cuz its complement, {-1,1} is finite

Which we know

[faint bison]

right

[faint bison]

and this proves that there can't be another limit?

[frosty creek]

Yes

[faint bison]

okay, again, because of the sequence itself

right?

[frosty creek]

Because for all yn, n will be greater than the integer

And for all n, yn is either 1 or -1

[faint bison]

right

[frosty creek]

Neither of which is in A

Which is our open ser

Set

[faint bison]

exactly

[frosty creek]

Therefoe yn does not converges

[faint bison]

okay got it

[frosty creek]

My math is improving with y’all

Thanks guys

[faint bison]

haha

lets go to b)

[frosty creek]

Try b) alone

It’ll be fun

I mean more fun when we solve together

But still give it a shot

Lmk if you get stuck at any point @faint bison

[faint bison]

okay will do

thank yu

[frosty creek]

No need

If you find the answer for b) explain it to me aswell

[cedar kiln]

@faint bison Has your question been resolved?

[cedar kiln]

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[weak granite]

how does this even make sense? adding up two irrational values will also result in an irrational value... so both are irrational should be a solution

(thats my only concern, their question is 100% way beyond me as iam yet to enter collage or learn about... proofs. but this lowkey doesnt make sense to me)

[weak granite]

this is a question sent by jaineel in channel number 18

x and r can be e and pi so- yeah thats one solution where r+x is irrational when x and r are irrational

unless, I misunderstood something?

[sudden cairn]

First understand that the converse is not always true

Consider x=sqrt(2)-1, y=sqrt(2)+1

Their sum is rational

[north wyvern]

Yeah this is only true if r is rational

[weak granite]

OH

okay...

[sudden cairn]

Wait not the converse

Sorry

[weak granite]

idk what that word means but i understand your counter example

[north wyvern]

Rational numbers are fractions of counting numbers

Irrationals are then not fractions of counting numbers

[sudden cairn]

Hang on

[north wyvern]

Perhaps sqrt(2) and 1-sqrt(2) is a better example

[sudden cairn]

How does the question define r and x?

[north wyvern]

But the difference here is rational

[sudden cairn]

Because that statement is only true if r is an integer

[sudden cairn]

Yeah my bad

[north wyvern]

Iff r is rational, specifically

[sudden cairn]

Yeah

[weak granite]

oh

ok that solves it

yeah x + r can only be irrational when x is irrational if r is always rational

[north wyvern]

Right

[weak granite]

i had a stroke writing that

[north wyvern]

Can you prove it tho?

[weak granite]

yeah i better not study thisone day

im-

im like- a highschooler

can i?

[north wyvern]

You can

[weak granite]

ok i can try

[north wyvern]

Do you want to learn how to?

[weak granite]

hell yeah

this seems cool to think about

ill try to think of it first

[sudden cairn]

I can give you a hint

[weak granite]

is there a way we could denote a number by only its- whole part? like 2.5 --> 2
pi --> 3
e --> 2
is there a way we coulddenote this

[sudden cairn]

Why?

[north wyvern]

Floor, also you dont need floor here

[weak granite]

I'm thinking of substraction

oh okay

what's the hint then

[sudden cairn]

Do you know what the contrapositive is?

[north wyvern]

Sum of rationals is rational is your hint

[weak granite]

hell no

[sudden cairn]

Basically

Here's the statement you want to prove:

If r is rational and r + x is irrational, then x is irrational.

Here's the contrapositive:

If r is rational and x is rational, then r + x is rational.

You swap the conditions and negate them

[weak granite]

rational + rational = rational
r is 100% rational
x + r is a sum of two values one of which is rational
irrational + rational = irrational
x is irrational?????????? i feel like i just did the most redundant proof

[north wyvern]

Its okay new math is hard

[weak granite]

yeah okay

[north wyvern]

So sum of rationals is rational, i would start with this, we can prove it later if need be

There are actually two directions were interested in here

[sudden cairn]

Now you can consider yourself a contrapositive expert

[weak granite]

rational + rational = rational
irration + rational = irrational
if r is rational
x+r will always be rational if x is rational
x+r will alwaysbe irrational if x is irrational

so x has to be irrational for x+r to be irrational

[weak granite]

yay.

[sudden cairn]

This is regarding the 'if and only if' part

[weak granite]

i mean- we know r is rational so- we['re just trying to prove that x should be irrational for x+r

[north wyvern]

So consider
Let x be irrational, and r be rational, prove that x+r is irrational

I say, let x be irrational and r be rational, assume x+r is rational, then show this is absurd

[sudden cairn]

Proving the contrapositive is easier

[weak granite]

oh

[sudden cairn]

That works too

[weak granite]

rational - rational cannot be irrational

[north wyvern]

This is kind of the same thing as a contrapositive, but you seem like a highschooler, so i think dropping contrapositive arguments might be very unintuitive for you

[north wyvern]

Exactly

[weak granite]

x (irrational) + r (rational) = z (rational)
x (irrational) = z(rational) - r(rational)
and thatcannot be so z should be irrational

[north wyvern]

Well lets fix this argument really quick

[weak granite]

please do

[north wyvern]

Irrational = rational - rational
Consider the negative number
Irrational = rational + rational
So a number is both rational and irrational, a contradiction

[sudden cairn]

That doesn't make sense

[weak granite]

i dont understand?

[north wyvern]

Because a rational is the sum of rationals

[sudden cairn]

That's what we want to prove

[north wyvern]

Nono

We are proving that a rational plus irrational is irrational here

[sudden cairn]

They are the same statement

[north wyvern]

They arent the same
Tbey are logically equivalent but thats by a law

[sudden cairn]

Yes

[sudden cairn]

This is the way

[weak granite]

I get how this is a contradiction, but i dont get how it relates to what you said?

rational + rational / rational - rational shouldboth give us rational. where did you conclude that it could be irrational?

[sudden cairn]

Can I try and clear things up? This chat is getting congested

[north wyvern]

Its how contradictions works, we are always free to assume x is irrational and r is rational
So the assumption that their sum is rational is the contradiction

So their sum must be irrational by contradiction

[sudden cairn]

Let me present @north wyvern's argument in a clearer way

Let r be rational and x be irrational. Show that r + x cannot be rational.

You do this by first assuming that r + x is rational

Then showing that this assumption leads to a contradiction

[weak granite]

which is

[sudden cairn]

Right

Now present it mathematically

[north wyvern]

Yes but you have to make the contradiction immediately clear

[weak granite]

that wasnt mathematical 😭 ?

[sudden cairn]

Remember the definition of a rational number?

[weak granite]

can be expressed as a/b

wAIt

ratIonal

can

[sudden cairn]

Yes

Show that z - r is also rational

In other words, it is a fraction of integers

[weak granite]

and how do we do that 😭

[north wyvern]

-r is also rational

[sudden cairn]

Let r=a/b and z=p/q

What is r - z?

Or z - r

Doesn't matter really

[north wyvern]

In other words, can we put that as a single fraction?

||(Cross multiply)||

[weak granite]

ive been tryingto do that with my eyes. i failed. i needpaper 😭

(bp-aq)/bq

this

[north wyvern]

Yeah

This is z-r but yeah

Same diff

[sudden cairn]

So we have x = z - r = (bp - aq)/bq

A rational number

But we assumed that x was irrational

Hence a contradiction

[weak granite]

oooooh

okaay

yeah that makessense

[sudden cairn]

Now keep in mind that this is just the forward direction of the proof

You also need to prove the reverse direction

[weak granite]

i wasgonna ask how do we know for sure that this is rational but a, b, q , p are rational so this shouldbe rational too

[sudden cairn]

Because of that if and only if part

[north wyvern]

Great question actually

Because a rational is a fraction of counting numbers (may be negative)

[sudden cairn]

a, b, p, q are integers

[north wyvern]

Since b,p,a,q were all counting numbers, then it follows

[weak granite]

i seeee

[north wyvern]

A proof of this part isnt so straightforward because wed need to construct numbers from here

But we do generally assume it

[weak granite]

this means assume z is irrational and both r and x are rational?

[sudden cairn]

You just swap the conditions

If r is rational and x is irrational, r + x is irrational

By proving the forward and reverse directions, you prove that the conditions imply each other

In other words, they are logically equivalent

[weak granite]

so we would start by assuming r+x (z) is rational with x being irrational and r being rational then we'd write

x (irrational) = fraction so r +x had to be irrational.

or does that imply the same thing

[sudden cairn]

We can do a proof by contradiction again

[weak granite]

thats what ive been trying to do but somehow

[sudden cairn]

This time assuming that r + x is rational

[weak granite]

i use the same arguments over and over again

[sudden cairn]

Same process

[weak granite]

but! thats what i did

[sudden cairn]

r = a/b, r+x = p/q

You want to find x in terms of a, b, p, q

[weak granite]

them

[sudden cairn]

Yes

You will get a rational number

[weak granite]

which is a contradiction

[sudden cairn]

Thus proving the statement

You're getting the hang of this!

[weak granite]

i believe this is an *easier * proof?

i can see how other proofs would be complex as hell

eh what majors study this anyways?

[sudden cairn]

Every field involves proof

Proof is the backbone of mathematics

[weak granite]

oh i mean isnt this just real analysis

yeah its everywhere

[sudden cairn]

So basically any pure mathematics major

[sudden cairn]

I don't believe this particular problem falls into a specific field

We're just dealing with rational and irrational numbers

But it is real analysis ig

[weak granite]

ig soo

alrighty

ty @sudden cairn and @north wyvern

[sudden cairn]

Your welcome

[weak granite]

.close

[cedar kiln]

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[abstract anvil]

Hi, I'm bad at math, and this step could be very easy, but I don't know how to do it.
My question is, how do I move the square root from the numerator to the denominator?

[ancient lodge]

$\frac{\frac{1}{a}}{b}$. Now, multiply the numerator and deominator by $a$:
$$\frac{\frac{1}{a} \cdot a}{a \cdot b}=\frac{1}{ab}$$
Here, $a=\sqrt{x^2+1}$ and $b=x^2+1$.

[wraith dagger]

Civil Service Pigeon

[abstract anvil]

thank u very much

[ancient lodge]

!done

[cedar kiln]

If you are done with this channel, please mark your problem as solved by typing .close

[abstract anvil]

.close

[cedar kiln]

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[warped owl]

i have k=3

which is correct

[queen stirrup]

so how would you approach the second part

[warped owl]

well

one of the roots is four timers the other

so 4a or 4/a

i think

then idk

[queen stirrup]

yes

[warped owl]

oh

Ok

So

[queen stirrup]

so $\alpha = 4 \cdot \frac{1}{\alpha}$

[wraith dagger]

CherryMan

[queen stirrup]

how would you solve that

[warped owl]

common denominator

[queen stirrup]

continue

[warped owl]

4a/a

[queen stirrup]

why?

[warped owl]

ill write 4 as 4/1

multiply 4/1 by a/a to get common denominator
now both denominators are the same so multiply stuff on top
a x 4 is 4a x 1 is 4a

im prob wrong idk

[queen stirrup]

well no

we use common denominator when we add fractions

[queen stirrup]

what happens in this equation when we multiply both sides by alpha

[warped owl]

4a x 1

so 4a

[queen stirrup]

like $\alpha \cdot \alpha = \frac{4}{\alpha} \cdot \alpha$

[wraith dagger]

CherryMan

[queen stirrup]

what does this give

[warped owl]

a^2 = 4a^2

[queen stirrup]

yeah one side it is a^2

but why is the right hand side 4a^2?

shouldnt the a in the numerator and the a in the denominator cancel out?

[warped owl]

i dont know what you mean

what a in the denominator

[queen stirrup]

see the right hand side

$\alpha ^2 = \frac{4\alpha}{\alpha}$

do you agree

[wraith dagger]

CherryMan

[warped owl]

uh

yes

[queen stirrup]

why

its cus we multiply right

[warped owl]

yes

[queen stirrup]

now what happens to the right hand side

[warped owl]

multiply by a

to get rid of fraction

[queen stirrup]

ok let me ask this

what is 2 divided by 2

[warped owl]

1

[queen stirrup]

what is 4 times 2 divided by 2

[warped owl]

4

[queen stirrup]

yeah so 4 times a divided by a = 4

[queen stirrup]

So $\alpha ^2 = 4$

[wraith dagger]

CherryMan

[queen stirrup]

can you solve this

[warped owl]

2

the a(s) just cancel

?

[queen stirrup]

exactly

but there is one more solution than 2

[warped owl]

im overthinking Lol

[queen stirrup]

-2 works as well right?

[warped owl]

-2

[queen stirrup]

yess

[warped owl]

ya i was about to say

[queen stirrup]

going back to this

what is the coefficient of x in terms of the roots

[warped owl]

uhm

idk whgat u mean

[queen stirrup]

you have a formula written at the bottom of the image

x^2 - (sum of roots)x + (product of roots) = 0

[warped owl]

yes

[scenic palm]

interesting

[queen stirrup]

so the coefficient of x is the sum of roots divided by the leading coefficient right

[queen stirrup]

loll

[scenic palm]

negative of that

[warped owl]

yes

[queen stirrup]

yeah negative of that sorry

[queen stirrup]

right so $-\frac{(\alpha + \frac{1}{\alpha})}{3k} = -18t$

[wraith dagger]

CherryMan

[queen stirrup]

where a + 1/a is the sum of roots

3k is the leading coefficient

and -18t is the coefficient of x

you agree?

[warped owl]

i resend this 1 sec

to help

me

[queen stirrup]

yeah

[warped owl]

why is it over 3k

[queen stirrup]

see

in the formula we have x^2 - (sum)x + (prod) = 0

notice how the coefficient of x^2 is 1

[warped owl]

oh so we need it to be 1

[queen stirrup]

yeah so we divide everything with 3k

[queen stirrup]

but here

we know the value of k

we know the value(s) of alpha

can you solve the equation

[warped owl]

ok

[queen stirrup]

try it then

[warped owl]

i have rn x^2 - 4/9=-54

[languid charm]

Wait shouldnt it be times 3k since we're making the coefficient of x² in both equations equal

[warped owl]

@queen stirrupidk what im doing

[languid charm]

X²-(sum)x +(prod) =0
<=>3kx² -3k(sum)x +3k(prod)=0
This means -3k(sum)=-18t
Right?

[queen stirrup]

theres no x^2

but you're close

[warped owl]

why

[queen stirrup]

its not involved

[warped owl]

ok

[queen stirrup]

im makign stupid mistakews

[queen stirrup]

yes this is wrong $-(\alpha + \frac{1}{\alpha})= -\frac{18t}{3k}$

[warped owl]

i multiply -54 by 9?

ts is so confusing brah

[languid charm]

Uh rewrite the equation its supposed to be -3k(sum)=-18t

[wraith dagger]

CherryMan

[warped owl]

im rewrite everything

what was the thing with the 4 times the roots again

i times a by 4 and 1/a by 4

[queen stirrup]

a = 4/a => a^2 = 4

=> a=2 or -2

[warped owl]

ok

now

idk

[queen stirrup]

then we have that the -(sum of roots) = coefficient of x / leading coefficient

[warped owl]

can you just quickly clarify how we got to a=4

again

[queen stirrup]

so we have the two roots as a and 1/a

we know that one root is 4x the other

so say a = 4/a

or 1/a = 4a

its the same thing

we multiply both sides with a

a^2=4

so a = sqrt(4) = 2,-2

[warped owl]

ok

i understand

[warped owl]

i just dont understand this

why leave out x^2

[queen stirrup]

why would we have x^2

[warped owl]

idk

ok i get sum of roots

[queen stirrup]

ok ill tell you how we get everything

we have $3kx^2 -18tx + (2k+3)=0$

[wraith dagger]

CherryMan

[queen stirrup]

that means that $x^2 - \frac{18t}{3k}x + \frac{2k+3}{3k} = 0$

[wraith dagger]

CherryMan

[warped owl]

isnt the x meant to be with 18t

[queen stirrup]

yeah i wrote -18t

i just divided everything with 3k after

[warped owl]

yes

[queen stirrup]

but we know that x^2 - (sum of roots)x + (product of roots) = 0

[warped owl]

Oh

its like algebraic identiies

[queen stirrup]

comparing, we get that $\frac{18t}{3k} =$ (sum of roots) =$\alpha + \frac{1}{\alpha}$

[wraith dagger]

CherryMan

[queen stirrup]

i mean not quite

[warped owl]

ok

[queen stirrup]

did you get this step

[warped owl]

yes

1 sec

+-5/2

is the sum of the roots

fraction is +- not 5

[queen stirrup]

yeah the sum of roots is $\pm \frac{5}{2}$

[wraith dagger]

CherryMan

[queen stirrup]

also we know the value of k right

[warped owl]

ye

[queen stirrup]

why dont we put these two values into the equation

[warped owl]

ok so

is it +- 5/2 = 2t

[queen stirrup]

yes

so t =?

[warped owl]

uhm

[queen stirrup]

man just divide by 2

t = +- 5/4

[warped owl]

is that it

[queen stirrup]

yup

k=3, t= +-5/4

[warped owl]

thank you cherryman sorry for being stupiod

[queen stirrup]

nah you were trying your best

[warped owl]

ye idk bout that

alriught thanks

[queen stirrup]

bye!

[warped owl]

.close

[cedar kiln]

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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.

[past wave]

What have you tried?

[nocturne spear]

so on a date which contains two squared numbers, the clock displays three sets of four different non-zero square numbers within an hour

[past wave]

Which are?

[nocturne spear]

so this will happen on 01/09, 04/09, 09/09, 16/09, 25/09

all hours of the day

[past wave]

?

[nocturne spear]

which seems a bit too much for the options

[past wave]

Wdym "all" hours?

[nocturne spear]

idk i thought this meant it's an hourly occurence

nvm, he says this after 4pm

[past wave]

Hmm

[nocturne spear]

so how would i find how many times this happens in a day

[past wave]

That won't work as all 4 numbers have to be different

[nocturne spear]

hm

so do i have to manually count

[past wave]

Yes pretty much

[swift moth]

But luckily i think we dont have to brute force this question since it is asking for whole september then?

[nocturne spear]

i don't get why you have to square 6

[past wave]

Makes no sense to me too

[languid charm]

Ye idk why either
Also it doesnt even take long for all possibilities

[nocturne spear]

the limit for a question is 1 min 30 sec

[languid charm]

Is that like on average or sth

[nocturne spear]

yes

[languid charm]

I mean its still somewhat doable in that amount of time, only problem is you can make mistakes

[sudden cairn]

The solution is so weird though

They didn't even count the numbers correctly

7 numbers not 6

Like, what?

[past wave]

Lmaoo

[languid charm]

Im guessing they dont count 9 since its already the month? (Doesnt make sense why they wouldnt say that tho)

[nocturne spear]

yeah i think easiest way is to list out all the solutions

ima close this channel and post again because i'm reviewing all the questions i got wrong

.close

[cedar kiln]

Channel closed

Closed by @nocturne spear

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• 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.

[glossy fractal]

Hey this is my proof for this theorem, can yall check if it's valid?

[cedar kiln]

@glossy fractal Has your question been resolved?

[glossy fractal]

<@&286206848099549185>

[fringe scroll]

ok you need to stop using the conclusion

[royal finch]

They aren't really using the conclusion. This is sloppy writing, but as a chain of if and only ifs that is better read starting from the last line backwards, that part isn't a problem.

[north wyvern]

this works, but i will say you make a few steps that arent axiomatic

[glossy fractal]

like what?

[north wyvern]

like $$P(A\cap B^c)+P(A\cap B)=P(A)$$

[wraith dagger]

Cycadellic

[north wyvern]

this is a consequence of the axioms, sure, but its not an axiom

[fringe scroll]

@glossy fractal please confirm, does your uni give clarity marks?

[glossy fractal]

so i have to prove this

[north wyvern]

yeah

[glossy fractal]

i dont know, this is just a practice question

[fringe scroll]

if so, then going backwards will absolutely get you marked down

okay

[glossy fractal]

ill work on formatting

[north wyvern]

i dont have a problem with direction as long as you only use equivalence rules

[glossy fractal]

i see

[north wyvern]

it will hold, if you really want to be pendantic about it, i suppose you could always rewrite it bottom to top

[fringe scroll]

UK unis are very particular about these kinds of things

[glossy fractal]

what's the reason for that

i dont get it

no i dont i go to a UK one

[north wyvern]

<==> is symmetric