help-23

[foggy salmon]

one

fat

chunk

[opaque sorrel]

[foggy salmon]

no idt it affects

o wel

[nova creek]

It's all one big scheme

[opaque sorrel]

this server has now become my resting server instead of my studying server

[toxic stratus]

its not message count

[opaque sorrel]

wat is thi

[toxic stratus]

its exp

i got before goddess

[foggy salmon]

despite

[opaque sorrel]

i have 13+ years of life experience

[toxic stratus]

w less messages

[foggy salmon]

my

[opaque sorrel]

more than u guyz

[foggy salmon]

30mill msgs

D:

[opaque sorrel]

wait

[foggy salmon]

im

bro

[opaque sorrel]

come back

hush

[foggy salmon]

ye imma get out

[opaque sorrel]

yes

little child

[toxic stratus]

hello

[opaque sorrel]

$det(A^{-1})$

[edgy ore]

👀

[opaque sorrel]

hi

[flat frigate]

HellO

[opaque sorrel]

so

this is the same as

$det(AI^{-1})$?

[flat frigate]

HellO

[toxic stratus]

uh

[opaque sorrel]

no its not

nvm

also that emoji is cool

hmm

$det(A^{-1})$

[flat frigate]

HellO

[toxic stratus]

maybe

we can write the thing we need to prove

a bit better

[opaque sorrel]

$det(A^{-1}) = det(A)^{-1}$

[flat frigate]

HellO

[opaque sorrel]

O

[toxic stratus]

P

[opaque sorrel]

why cant i recall

asdjaosdk

[toxic stratus]

what do we know about dets

and matrices

[opaque sorrel]

AH

[toxic stratus]

whats special about inverses

[opaque sorrel]

$det(A)det(A^{-1}) = I$

[toxic stratus]

:O

[flat frigate]

HellO

[toxic stratus]

no

[opaque sorrel]

heh

no wait

[toxic stratus]

that shouldnt be an I

[opaque sorrel]

$det(A)det(A^{-1}) = det(I)$

[flat frigate]

HellO

[toxic stratus]

yeah aight

[nova creek]

The product of determinants is a matrix

[opaque sorrel]

shush

okay then

[toxic stratus]

a 1D matrix

[opaque sorrel]

wait

a ha

$det(A^{-1}) = \frac{det(I)}{det(A)}$

[flat frigate]

HellO

[opaque sorrel]

$\frac{1}{det(A)}$

[flat frigate]

HellO

[opaque sorrel]

ez

i am evolving

[toxic stratus]

amazing

[opaque sorrel]

cant see me

my time is now

[toxic stratus]

👀

[opaque sorrel]

ok done it

OKAY

more qns

[toxic stratus]

more 👀

[opaque sorrel]

this

is just

finding the deerermintant of A

via the row operations to get to b

wait...

oh

i see

nvm

[opaque sorrel]

nice felex

wait can i do that

upper $\triangle$

[toxic stratus]

yeah you can

[flat frigate]

HellO

[opaque sorrel]

noob

ok recap time

[opaque sorrel]

so

[toxic stratus]

omg

[opaque sorrel]

the det of the tranpose

is itself

the scllar multiplier is

2^n where n is the mtriax size

the inverse of A^-1 is the inverse of itself

we proed this like 5 times

then det(AB) is yes

then yes

okay

[toxic stratus]

ys

[opaque sorrel]

now this is the true question

[toxic stratus]

adjoint

[opaque sorrel]

yes

[toxic stratus]

omg

this thing

holy

i remember this

[opaque sorrel]

war

giving u war flashbacks

[toxic stratus]

this is really kinda yuck

[opaque sorrel]

i know

XD

pls help

so like wat the

i donut understand

A is the cofactor

then why and how do we flip it

[foggy salmon]

so many ppl

not in help channels

are like

who is

circle

n that chinese name

[opaque sorrel]

wow

why am i not there

[foggy salmon]

maybe

cuz u arent at the top of the leaderboard

[opaque sorrel]

wow

[foggy salmon]

[opaque sorrel]

i will spam

[foggy salmon]

go on

[opaque sorrel]

where did snow go

[foggy salmon]

go on

[opaque sorrel]

i need halp

[foggy salmon]

probably

had better things to do

its ok

[opaque sorrel]

😦

HELP ME

goddess

[foggy salmon]

ur

questions

are funny

they smell weird

[opaque sorrel]

why

u so mean

[foggy salmon]

im

always mean

muhahahaha

[opaque sorrel]

[toxic stratus]

here

[opaque sorrel]

ill wait for snow

[toxic stratus]

i found my notes on it

[opaque sorrel]

oh

wtf

[foggy salmon]

what even

[opaque sorrel]

wait wat

is that

[toxic stratus]

thats the visual explanation

for why the cofactor stuff works

[foggy salmon]

no what even is cofactor

[worthy hemlock]

Because math works

[opaque sorrel]

im lost

[foggy salmon]

sameeeee

[toxic stratus]

basically like

along the diagonals

when you do the matrix multiplication

you end up getting the same expansion as if youd done the cofactor expansion along that row

and so you end up with the determinant

but along the non-diagonals

when you do the matrix multiplication

[opaque sorrel]

wat

[toxic stratus]

you end up doing the cofactor expansion as if two of the rows of the matrix had been duplicated

so you end up with 0

[foggy salmon]

i dont even get whats cofactor

am i supposed to have learnt

[opaque sorrel]

wait goddess

im confused

how are u doing

complex analysis

if u

havent learnt

linear algebra

[foggy salmon]

im

[opaque sorrel]

heh

[foggy salmon]

self taught

[opaque sorrel]

same

u

nob

[foggy salmon]

[opaque sorrel]

anyway i dont get thisl mao

[foggy salmon]

same

[opaque sorrel]

is there a

[foggy salmon]

lmao

[opaque sorrel]

better visual representation

cough

[worthy hemlock]

Co-factor of an element within the matrix is obtained when the minor
of the element is multiplied with (-1)^(i+j)

[foggy salmon]

HAHAHAA

whats

[opaque sorrel]

a minor

[toxic stratus]

[foggy salmon]

minor of the element

[toxic stratus]

so like

[worthy hemlock]

Do you know the concept of finding the det of a 3 x 3?

[toxic stratus]

if you calculate this entry

[foggy salmon]

yes

[toxic stratus]

the green parts are the original matrix

[opaque sorrel]

ye

[toxic stratus]

the blue parts are the corresponding entries in the adj

[worthy hemlock]

That 2 x 2 is the cofactor

https://www.cuemath.com/algebra/cofactor-matrix/

[foggy salmon]

are we multiplying the green with the blue matrix?

[toxic stratus]

yes

you end up computing exactly the cofactor expansion

of the determinant

[opaque sorrel]

wait what happens after

cofactor expansion?

WAT

how

[foggy salmon]

whats the

cofactor for

[opaque sorrel]

determinnat

[worthy hemlock]

Help find the inverse of a matrix

[opaque sorrel]

wait wat

[toxic stratus]

first of all like

[foggy salmon]

ill

just

sleep

[toxic stratus]

you need to know what the entries in the cofactor matrix are

[worthy hemlock]

You can use the cofactor matrix to find the inverse

[foggy salmon]

n do my other work after

too much pain already

[opaque sorrel]

cofactor is

[opaque sorrel]

noob

[foggy salmon]

pain

[opaque sorrel]

like

we cover up

that elements row and column

[toxic stratus]

its the minors right?

[opaque sorrel]

then we take the rest

yes

[toxic stratus]

you delete the row and column

and you calculate the det of the minor

[opaque sorrel]

and then (-1)^i+j

[toxic stratus]

• the sign correction

yeah

okay

[opaque sorrel]

ye

[worthy hemlock]

https://www.youtube.com/watch?v=c9m9PdvZZKs&ab_channel=Mathispower4u

[toxic stratus]

so in this image

this column

of blues

correspond to the row of greens

but the blues represent the cofactors

[worthy hemlock]

This video might be helpful

[toxic stratus]

say you take the first blue circle

that comes from

deleting the first row and column of the original matrix

then calculating the determinant of the resulting minor

the second blue circle is

deleting the first row and second column of the original matrix

then calculating the det of the resulting minor

• -1

etc

but if you multiply the green circles with the blue circles

you get exactly the cofactor expansion

for the det of the matrix

so its equal to the det

[opaque sorrel]

maybe its easier to exxplain

with this

[opaque sorrel]

i legitamately dont understand this sorri

[toxic stratus]

like

theres so many entries

that are being multiplied

[opaque sorrel]

wait is this correct

[toxic stratus]

like yeah thats

the cofactor matrix

[opaque sorrel]

thats not the adj matrix?

so theyre different

HEH

oh wait

ofc they are

[toxic stratus]

same thing

[opaque sorrel]

o nvm

[toxic stratus]

well

theres like transpose shit going on

[opaque sorrel]

wait the cofactor matrix is like

lets say we have a matrix

cofactor matrix of a_11 is

[somber hamlet]

Why is it telling me to come here

[opaque sorrel]

idk

then the cofactor matrix of a_21 would be 0

right?

[toxic stratus]

yeah that

[opaque sorrel]

ok

then

the cofactor matrix

of the WHOLE THING

is

[toxic stratus]

here is wikipedia

its the whole thing

[opaque sorrel]

so its just

cofactors of each element

inside the matrix

replacing their original spot

[toxic stratus]

and det is you take the sum of the row * cofactors

[opaque sorrel]

ye

ok wait solike

the cofactor/adj matrix for this

would be

[toxic stratus]

well

[opaque sorrel]

wait

me typing

[toxic stratus]

you're gonna have to calculate like

9 dets

its

[opaque sorrel]

o

[toxic stratus]

kinda not good for your health

[opaque sorrel]

ill do for first two

[toxic stratus]

doing the calculation isnt too helpful

doesnt actually teach you anything

[opaque sorrel]

like this?

[opaque sorrel]

oh

[toxic stratus]

yeah

[opaque sorrel]

ahhhhh

so i dont care about the original one

the "0"

[toxic stratus]

no

[toxic stratus]

thats the adjugate

[opaque sorrel]

adjugate/cofactor matrix right?

[toxic stratus]

adjugate is cofactor transposed

this is cofactor

[opaque sorrel]

ooh

OH

OHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

[toxic stratus]

but like

okay so

[toxic stratus]

take note of the way the cofactors are indexed here

because we're gonna multiply it with A

[toxic stratus]

so if you take the product

[toxic stratus]

we get this whole thing

[opaque sorrel]

yes

[toxic stratus]

just look at what multiplying the first row and column gives

its gonna be

[opaque sorrel]

wait so were multiplying the a_ij element twice

to get the adj matrix

[opaque sorrel]

wait nvm

wait

wait

[toxic stratus]

this happens for everything on the diagonal of the resulting product

[opaque sorrel]

ok to recap

A is the original matrix

[toxic stratus]

yes

[opaque sorrel]

C is the cofactor matrix

where each element is

a_ij * det of the minor

[toxic stratus]

yes

no

no

[opaque sorrel]

and (-1)^i+j

[toxic stratus]

no

theres no a_ij

[opaque sorrel]

ah WHAT

okay

wait

oh wait

OH YEAH

THATS FOR FINDING DEETERMINANAINTS

[opaque sorrel]

okay

understoodeded

[toxic stratus]

its just the det of the minor

[opaque sorrel]

yes

[toxic stratus]

• sign correction

so thats why

[opaque sorrel]

okay then

[toxic stratus]

when you do the product

and you multiply with the a_ijs

you get the det

[opaque sorrel]

OH

[toxic stratus]

this

[opaque sorrel]

lemme just put it here

okay then C^T is the ajacent matrix

so the adjacent matrix is basically the cofactor matrix but transposed

then after that if i multiply the original matrix and the adjacent matrix

ill get the determinant

because its bascially the cofactor expansion

but in matrices form

[toxic stratus]

rip lol

[opaque sorrel]

ha

[opaque sorrel]

is this correct

[toxic stratus]

yes

thats what happens

but only on the diagonals

[opaque sorrel]

wat

oh

[toxic stratus]

because the indices dont match properly

on the off diagonals

[opaque sorrel]

whats that

[toxic stratus]

the non-diagonals

the entries

that arent on the diagonals

dont get their indices matching

[opaque sorrel]

wait so

thats if

i wanna do cofactor expansion

on the diagonals

is hat right

[toxic stratus]

kinda hard to see the colours

but

only when you match the column and the row correctly

do you get the cofactor expansion

for the original matrix

[opaque sorrel]

is that a question

[toxic stratus]

no

[opaque sorrel]

or

[toxic stratus]

its a statement

[opaque sorrel]

oh

lmao

okay

i understandeded

u r the bsest

[toxic stratus]

that only happens on the diagonals of A C^T

but thats what we want

cuz we want it to be 1 eventually

we're dividing by det(A) remember

[opaque sorrel]

so basically

the colour matching

ok gimme 10 mins

i brb

[toxic stratus]

LOL

okay

[opaque sorrel]

hi @toxic stratus

ur child has returned

[toxic stratus]

child D:

[opaque sorrel]

yes

u owe me life

anyway

let us cotiue

[toxic stratus]

where were we

[opaque sorrel]

wait no

this first

so here they wanna show what u just proved

[toxic stratus]

yeah

good

[opaque sorrel]

so um

why i = j

[toxic stratus]

thats saying

you're on the diagonal

i = row, j = column

[opaque sorrel]

o

so like u said just now

wait

diagonal

is it

east and south

[toxic stratus]

its

[opaque sorrel]

or southeast

[toxic stratus]

\
 \
  \

going that way

[opaque sorrel]

wait why are we multiplying that

[toxic stratus]

wait

what do you mean

i = j as in

the entry

[opaque sorrel]

i thought we are multiplying

|
|```

[toxic stratus]

on the ith row

and jth column

of A C^T

you get that by multiplying the ith row of A with the jth column of C^T

which is the ith row of A with the jth row of C

[opaque sorrel]

ohhh

that makes sense now

wait then

whats this part

[toxic stratus]

thats the off-diagonals

wait

that might not be the right term

[opaque sorrel]

O.O

[toxic stratus]

the entries not on the diagonal

when i != j

its

when you arent on the diagonal

[opaque sorrel]

wat

wait

how does diagonal

even tie into this

[toxic stratus]

because the

[opaque sorrel]

isnt it just rows and columns

[toxic stratus]

identity matrix

has 1s on the diagonal

so we need to make sure

that A C^T

has det(A) on the diagonal

and 0 everywhere else

[opaque sorrel]

o

hmm

hmmmmmmm

okay

anyway

[opaque sorrel]

then this

is trying to say

we swap rows?

[toxic stratus]

its trying to say that

we we aren't on the diagonal

we essentially end up calculating the cofactor expansion

for a matrix

that is the same as A

except

one of the rows is duplicated

and whenever you have a matrix with two rows that are the same

the det is 0

[opaque sorrel]

OK I THINK I KINDA SEE IT

OK

NOWE

ok last part

CRAMERS RULE

last one

[toxic stratus]

oh lord

cramer

[opaque sorrel]

yes

[toxic stratus]

thats like

[opaque sorrel]

HELP

[toxic stratus]

i dont even remember how it works

[opaque sorrel]

ah

[toxic stratus]

the algebraic proof for it is terrible

just like every other algebraic proof in linalg

[opaque sorrel]

agree

but what is it even

[toxic stratus]

cramers rule lets you solve a linear system

without calculating the inverse

[opaque sorrel]

O>O

[toxic stratus]

you can calculate the components x

in the equation Ax = b

without inverting A first

and calculating x = A^-1 b

[opaque sorrel]

how

[toxic stratus]

its just all determinants

you just calculate a bunch of determinants

the jth component of x is given by the formula in blue at the bottom

oh wait

okay its pretty simple how it works given the cofactor stuff

but

still yucky

[opaque sorrel]

simple but im lost

[toxic stratus]

so

we know that

if A is invertible

then

[opaque sorrel]

yrd

yes

[toxic stratus]

right

and then we also know

[toxic stratus]

where C^T is the adj matrix

[opaque sorrel]

wait

er

[opaque sorrel]

oh

yes

[toxic stratus]

this stuff

[opaque sorrel]

ye

[toxic stratus]

except your slides are using A

for some reason

A_1, 2 etc

oh well

anyway

ignore the division by det(A) for now

[opaque sorrel]

o

[toxic stratus]

we'll just calculate the multiplication with b

so

[toxic stratus]

from here

[toxic stratus]

so

[toxic stratus]

and then like

we do the same stuff

[opaque sorrel]

oh

to get x

[opaque sorrel]

ye

[toxic stratus]

thats the same however

as if you had taken the original matrix A

[opaque sorrel]

caant we just do dis

[toxic stratus]

and replaced the first column with b

and calculated the determinant instead

[opaque sorrel]

ok i kinda get the idea

but its still so confusing

do i really need to get this

[toxic stratus]

well

the algebra is

really quite

horrendous

disagreeable

[opaque sorrel]

im so confused man

[toxic stratus]

same tbh

[opaque sorrel]

ok ill just understand thru practicing

[opaque sorrel]

WAT

[toxic stratus]

like

i dont remember this stuff at all

[opaque sorrel]

but u can still exlapain

eee

[toxic stratus]

i understand how it works intuitively

but i dont remember any of it

i had to look up the algebra just then

[opaque sorrel]

flex

[toxic stratus]

to explain it

[opaque sorrel]

means ur goated

cuz u dont even remember

[toxic stratus]

no but

[opaque sorrel]

and could still explain

[toxic stratus]

the reason why this even works in the first place

and how the algebra is done

is so disconnected

[opaque sorrel]

ikr

wait

lemme look thru again

im so confused

breh

[toxic stratus]

bruh

like

its probably okay

if you dont remember all the algebra

this is just complicated af

just sort of generally get the idea

and itll be g

[opaque sorrel]

i just remember how to use it?

[toxic stratus]

you'll almost never need it

[opaque sorrel]

o wut

[toxic stratus]

you'll almost never need to use it

unless like

your profs are evil

but

ive never

had to use this formula

[opaque sorrel]

o

wit

wait

uve never used cramers rule

[toxic stratus]

never had to use it for anything serious

if you want to solve a linear system by hand

just do row reduction

[opaque sorrel]

so this

[toxic stratus]

yes

that

[opaque sorrel]

wtf

[toxic stratus]

ive

[opaque sorrel]

IT LOOKS OS SIMPLE

[toxic stratus]

never had to use it

[opaque sorrel]

WHY DID THEY MAKE IT SO COMPLICATED

[toxic stratus]

its like

a lot worse

computationally

[opaque sorrel]

wtff

[toxic stratus]

except in the case of 2x2

[opaque sorrel]

scam

[toxic stratus]

anything larger than 2x2 is just

[opaque sorrel]

i ave been scammed

[toxic stratus]

absolutely horrendous to calculate by hand

so

[opaque sorrel]

wait so thi sis

[toxic stratus]

basically you never want to use cramers rule

[opaque sorrel]

lmao

okay so just to know what its about

[toxic stratus]

yes

[opaque sorrel]

i put the first row down

wait

[toxic stratus]

column

[opaque sorrel]

no

i have an equation

[toxic stratus]

you replace columns

[opaque sorrel]

Ax = b

i multiply the right

with the det of A

then the right becomes 60

[toxic stratus]

no no

LOL

[opaque sorrel]

wait am i making sense

[toxic stratus]

this is a full stop

[opaque sorrel]

OH

LMAO

ANYAY

[toxic stratus]

its saying the det of A is 60

LOL

[opaque sorrel]

ohhh

oops

cough

ANYWAY

then to find x

i put the determinnat

on top

and replace the first row

with 0 4 3

aka the result

[toxic stratus]

column

[opaque sorrel]

then ill get x y z

[toxic stratus]

you replace each column

[opaque sorrel]

yes column

so ill just not gaf about the proof

cuz i seriously dont understand

[toxic stratus]

yeah its probably okay

[opaque sorrel]

wait then for the adj matrices

[toxic stratus]

you'll never need to calculate anything

[opaque sorrel]

can i just know that its the transpose of the cofactor matrix

nothing else

[toxic stratus]

like

you'll never need to use that either

because you can also just calculate the inverse of a matrix by row reduction

and its so much faster

than calculating a million determinants

determinants are just bad

like

you never want to calculate determinants

[opaque sorrel]

thats true

det sucks

ok

i think im good

[toxic stratus]

good

[opaque sorrel]

thanks madam/sur

[toxic stratus]

cuz i have stuff to do lol

[opaque sorrel]

OH

wat time

would u be free again

[toxic stratus]

now

oh

never

[opaque sorrel]

heheh

oh

WAT

[toxic stratus]

LOL

[opaque sorrel]

when

WILL U BE FREE

[toxic stratus]

idk

maybe much later

[opaque sorrel]

ill just tag u and pray

[toxic stratus]

busy today

[opaque sorrel]

wow

okay

i go eat lunch

bye boss

[toxic stratus]

lunch 🤔

.close

[safe radish]

Channel closed

Closed by @toxic stratus

Use .reopen if this was a mistake.

[opaque sorrel]

wait

[toxic stratus]

LOL

[opaque sorrel]

u can close my channel

wtf

[toxic stratus]

.reopen

[safe radish]

✅

[toxic stratus]

yeah i have

[opaque sorrel]

SCAM

THIS IS SCAM

.close

[safe radish]

Channel closed

Closed by @opaque sorrel

Use .reopen if this was a mistake.

[opaque sorrel]

nub

[toxic stratus]

D:

[safe radish]

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.
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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.

[slow fern]

sub t=log x

[safe radish]

@lean otter Has your question been resolved?

Channel closed

Closed by @terse prairie

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.

[lean otter]

f = 2x² - 1
g = x-2

[lean otter]

arw they right?

[idle parrot]

,rotate

[idle parrot]

uhh not quite correct

[lean otter]

which part

[idle parrot]

both

[lean otter]

oh damn

[idle parrot]

(f+g)(x)= f(x)+ g(x)

and you have f(x) = 2x² - 1 and g(x) = x-2

(f+g)(x)
= f(x)+ g(x)
= 2x² - 1 + x-2

and -1-2=-3

so you get

= 2x² + x-3

[lean otter]

hold on shouldnt it be -1+-2 ?

[lean otter]

.

[idle parrot]

yes it is (-1)+(-2)=-1-2=-3

[lean otter]

yuh

so

the next one

how abt it

[idle parrot]

do same analogy with (f-g)(x)= f(x)-g(x)

[lean otter]

where am i wrong

at

[lean otter]

2x² - 3 ? or is it wrong

[idle parrot]

nope check again

[lean otter]

does it affect the x²?

because

[idle parrot]

= 2x² - 1 -(x-2)

[idle parrot]

no

[lean otter]

doesnt the sign change? thus 2x² - 1 - x+2

[idle parrot]

yes

[lean otter]

2x²-x = 2x²
-1-2 = -3

which part am i wrong 😭

@idle parrot

[idle parrot]

wait 2x²-x = 2x²?

also you have -1+2= 2-1=1

[lean otter]

i thought it doesnt affect the x²?

[idle parrot]

but you can’t magically erase x term

thats not how it works

you keep it as it is

2x²-x -1+2= 2x²-x +1

[lean otter]

OHHH

[idle parrot]

uhh do you get?

[lean otter]

so i keep the -x?

yup yup

[idle parrot]

yea

[lean otter]

ty!

[idle parrot]

,tex \np

[lean otter]

.close

[safe radish]

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[silk bough]

Help

[silk bough]

4b

I got it p^2 (1/10)

What do i do now

[safe radish]

@silk bough Has your question been resolved?

[safe radish]

@silk bough Has your question been resolved?

[safe radish]

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[smoky vortex]

is this right?

[halcyon carbon]

Is the question asking you to find f(x)?

[smoky vortex]

yes

@halcyon carbon

[halcyon carbon]

If f(x) = -1/-x-2 then fg(x) is -1/-(3-2x^2) -2= -1/2x^2-3-2=-1/2x^2-5

So I don’t think that’s right

[safe radish]

@smoky vortex Has your question been resolved?

[smoky vortex]

wait i havent actually made it clear

does it make the entire fraction negative?

thought the negatives cancel

[safe radish]

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[smoky vortex]

.reopen

[safe radish]

✅

[halcyon carbon]

But you’d get 2x^2 -5 for the last line

[smoky vortex]

i pasted the wrong thing 😭

this is the one that i think is rigbt

[halcyon carbon]

Yes that works

Btw don’t you find it difficult to read with black background and red text.

Idk lol with all the grids as well

[smoky vortex]

normally i write in white

no clue why i did it in red

thanks!

would the domain be x<=1?

[safe radish]

@smoky vortex Has your question been resolved?

[safe radish]

@smoky vortex Has your question been resolved?

[safe radish]

@smoky vortex Has your question been resolved?

[hasty wagon]

the domain can be all real numbers

[safe radish]

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[tardy copper]

any ideas

I tried to express the form

as

wait

and tried to play with this but nothing

[edgy ore]

wait, let's see if rationalization helps.

[tardy copper]

yeah, i tried to rationalize

[edgy ore]

what do you get?

[tardy copper]

just jumble

nothing nice

[hasty wagon]

do you know the answer? would it be easier to derive the working from an answer?

[tardy copper]

dont know it

[hasty wagon]

oh ok, i do have something in my mind right now, but not sure if it works

[hasty wagon]

firstly, this helps.

[hasty wagon]

do you have the jumble that we can take a look into?

[halcyon carbon]

,w sum from n=1 to 2019 of 1/(sqrt(n+sqrt(n^2 -1))

[flat frigate]

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Requested by Pure#9389

[halcyon carbon]

Nvm then

[tardy copper]

what, is it actually that number

i thought it would be smthn nice

[halcyon carbon]

It’s probably is something novice

Nice

[hasty wagon]

,calc ((2020)^(1/2)+(2019)^(1/2)-2^(1/2)-1)*(1/2)^(1/2)

[halcyon carbon]

But woldramalpha just use brute force it

[hasty wagon]

opps

forgot 1/2

[tardy copper]

,w sum from n=1 to 2019 of 1/(n*sqrt(n+1)+(n+1)(sqrt(n))

[flat frigate]

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Requested by epifor#4424

[tardy copper]

,w sum from n=1 to 99 of 1/(n*sqrt(n+1)+(n+1)(sqrt(n))

[flat frigate]

Results provided by WolframAlpha

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Requested by epifor#4424

Result:

61.846020113589

[tardy copper]

this should be like 2 or smthn

not 0.9

[hasty wagon]

there we go

[tardy copper]

this is a diff task, just testing wolfram

[hasty wagon]

thank you pure

you've helped checked my answer is right

[hasty wagon]

anyways, can you show the jumble you're into?

[halcyon carbon]

You did all the hard work lol

[hasty wagon]

because i got a decent expression after rationalizing it

[tardy copper]

can you send it

[hasty wagon]

which can make a telecope
concluding
||(√2020+√2019-1)/√2||

i just did it mentally lol, i can type in the expression if you want

[tardy copper]

wat

[hasty wagon]

well, after rationalizing, i got

$\sqrt{n-\sqrt{n^2-1}}$

[flat frigate]

OldBiscuit

[tardy copper]

what how

[hasty wagon]

pretty nice right?

[tardy copper]

i mean still dont see what to do with it

but how did you rationalize it into that

[hasty wagon]

by multiplying both numerators and denominator by this expression

[tardy copper]

you get
$\frac{sqrt{n-\sqrt{n^2-1}}}{sqrt{n-\sqrt{n^2-1}}*sqrt{n+\sqrt{n^2-1}}}$

[flat frigate]

epifor

[hasty wagon]

yea

[tardy copper]

?

[hasty wagon]

which the denominator is in the form of (a+b)(a-b)

gotta finish my shower,brb

[tardy copper]

oh you mean when we put it in the same root?

,w sum from n=1 to 2019 of (sqrt(n-sqrt(n^2 -1)

[flat frigate]

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Requested by epifor#4424

[tardy copper]

hm

i dont exactly understand how you got to the last equality

[toxic stratus]

this can be rewritten

[tardy copper]

i get it

its a square of sum

(a+b)^2

[toxic stratus]

yes

[safe radish]

@tardy copper Has your question been resolved?

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