And how do I start this question and solve using matrices?
#help-38
sand forge
boreal dome
You have $AA^T = 225I_3$
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boreal dome
Multiply both sides by A^-1 on the left
sand forge
So A^-1AA^T=A^-1(ANS)
IA=A^-1(ANS)
A=A^-1(ANS)?
boreal dome
no
$A^T = 225A^{-1}$
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sand forge
**Ｈｅｒｅｌｓ**
wait how to proof this?
boreal dome
wait how to proof this?
thats... your result
$\begin{pmatrix} 225&0&0\0&225&0\0&0&225 \end{pmatrix} = 225I_3$
solid kilnBOT
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sand forge
oh i didnt know that
sand forge
**Ｈｅｒｅｌｓ**
then how do i get inverse mat. of A
boreal dome
jus divide both sides by 225
sand forge
So A^-1 = 1/225A^T?
boreal dome
yes
sand forge
ohhh
then once i get A^-1
how do i cont
Then after that just AA^1=I and solve?
oh wait nvm
use inverse matrix method
thanks @boreal dome
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wraith hinge
need help to find x
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noble fable
Since in the first part, you have the fraction can be rewritten as fractions over r+2, r+3 and r+4.
and importantly, the coefficients total 0
if you add together lots of these, with different values of r, there will be some cancellation.
this should be called telescoping somewhere in your textbook.
the idea is that when you do the partial sum r=1 to n, nearly everything will cancel, except some terms in the beginning, and some terms at the end
try and figure out what the terms at the end look like.
they should depend on "n"
@coarse bramble
I don't know of something called the method of differences
if you explain it, I should be able to answer
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hot fulcrum
Im confused on how 1 is not the upper limit on this problem
sudden obsidian
Use the equation you have shown to write the sum as
\begin{align*} \sum_{n=1}^\infty r^{\ln n} = \sum_{n=1}^\infty n^{\ln r} \end{align*}
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nvx
sudden obsidian
Then recall for what conditions p-series converge
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hot fulcrum
Then recall for what conditions p-series converge
um im still a little confused
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zinc ginkgo
um im still a little confused
do you know what p-series is?
hot fulcrum
well p series diverges if p is less than 1
hot fulcrum
would it not be e?
sudden obsidian
https://cdn.discordapp.com/attachments/657617458915442688/1036767569820004382/392341051794391040.png
hot fulcrum
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hot fulcrum
i feel kinda stuck on understanding how to create a general formula
not quite sure how to approach it
im also very confused on as to how part(c) isnt 1/3
hot fulcrum
ive looked in my notes
hot fulcrum
i realzie the limits of summation or whatever are different
i just dont know exactly how changing that range affects the answer
zinc ginkgo
use the definition of the sigma notation
or better just write out the first few terms of both
hot fulcrum
for the first one its 1 + 1/4 + 1/9.....= pi^2/6
i mean i did the math and starting at n=3 should give you 1/3
zinc ginkgo
i mean i did the math and starting at n=3 should give you 1/3
nope
zinc ginkgo
for the first one its 1 + 1/4 + 1/9.....= pi^2/6
what's the second one
hot fulcrum
1/9 + 1/16 + 1/25....
zinc ginkgo
1/9 + 1/16 + 1/25....
this is almost pi^2 / 6. it's missing a couple terms
specifically n=1 and n=2 if you compare the indices
hot fulcrum
okay i got the right answer
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tepid owl
Hello. Having issues with this problem. I found y' to be 2e^x+1, then set y' = 2, so 2 = 2e^x+1. Now I need to solve for x, and I got to the point where 1/2 = e^x
tepid owl
I am stuck from there, how am I supposed to solve for a variable in the exponent?
inland orbit
That's exactly what logarithms are for
In this case a natural log, since that's log_e
inland orbit
Yep 👍
Oh wait
I don't think this is the equation you should be solving for this question, though
tepid owl
?
inland orbit
Why did you set y'=2?
tepid owl
If I solve for x I should get the slope, and I have a point so I should be good from there?
Bc the point is 0,2
The y value is second
zinc ginkgo
Bc the point is 0,2
that's y(0) = 2, not y'(0) = 2
inland orbit
Right, that's y, not y'
tepid owl
Ah
inland orbit
You need to find the slope (y') at (0,2)
tepid owl
Hang on so what am I supposed to do after finding y' then?
inland orbit
You have y'=2e^x+1, and you want to find the slope at (0,2)
inland orbit
So what could you plug in for x in order to find y' at that point?
tepid owl
Oh, 0
tepid owl
So 3
tepid owl
And the rest should be easy
Yup, got it. Thanks. I keep making these things harder than they actually are
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barren tusk
how is this wrong
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how is this wrong
limpid sinew
can you maybe get a better picture 
barren tusk
grim saffron
Pick two dots
stoic saddle
how is this wrong
your equation states the intercept is at -7 the furthest the diagram goes is -6 should be a sign it might not be correct and the line intercepts at different point entirely ^
barren tusk
is a the 2nd x?
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Clarkie
stoic saddle
$m=\frac{y_2-y_1}{x_2-x_1}$
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Clarkie
stoic saddle
should be all you need to solve that q above
barren tusk
ty
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stable jacinth
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stable jacinth
Im totally lost. Does anyone know how to do this please?
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stable jacinth
<@&286206848099549185>
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manic root
I suspect this has something to do with some kind of force action involving some physics formula
I don't know anything about this unfortunately, but if you can convert it into a purely calculation problem, I can probably help
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sacred jackal
draw FBD (free-body diagram) first
kind sorrel
rugged mortar
go to discord.gg/physics with that one
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kindred terrace
so i learned the basic vector formulas/rules and stuff but im not sure how to begin doing these word problems; i tried setting CB to u and CA to v and then using the distance formula for magnitude to get
sqrt((||u||cos30°)^2+(||u||sin30°)^2) + sqrt((||v||cos50°)^2 + (||v||sin50°)^2) = 1000
but that doesnt look right
kindred terrace
oh it spoilered instead
its supposed to be ||
$sqrt((||u||cos30°)^2+(||u||sin30°)^2) + sqrt((||v||cos50°)^2 + (||v||sin50°)^2) = 1000$
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Autumn
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wraith hinge
I don’t know how to prove it’s congruent, I’ve been told it is but how do I prove that it is.
unreal loom
wraith hinge
No, this was using Fermats little theorem
wraith hinge
I’ll try
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wraith hinge
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dark lily
why does the square root cancel? we have sqrt of G over L times sqrt of g over L. why does the sqrt cancel and everything relse stays the same
dark lily
Is this right?
dense scroll
yh
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formal crystal
Hello I don‘t know how I can proof if a function is axial symmetry or point symmetry.
wraith hinge
if you square a negative number you will get a positive number
so if you square the absolute value of that negative number ( the positive value of that negative number ) you will get the same result
therefore you can conclude (-x)^2 = x^2
and you can extend that reasoning to (-x)^3
-x * -x * -x
the product will be negative
formal crystal
There are some tasks in the book
formal crystal
Yea
wraith hinge
well the function is even when it is symmetrical ya
ya
so to know if a function is even iirc
its f(-x) = f(x)
formal crystal
And what is a point symmetry function then
wraith hinge
if you plug in -x for all the x in the function and you get the same original function, then you can conclude that it is even
uhh i'm not entirely sure of that
it could just be that the function / graph is symmetrical about a certain point like the origin
formal crystal
No I know what it is but idk how to proof that it‘s a point symmetry
It‘s for example (5/6) and (-5/-6)
@wraith hinge what is -f(x)
wraith hinge
times the whole function by negative one
royal ridge
@zinc ginkgo #help-7｜zen1thxyz message
How do I do this without that part at the end? I cant really remove it. Did you use the hypothesis somewhere else?
royal ridge
use 2 = 1 + 1
did you pull it into the numerator afterwards?
zinc ginkgo
did you pull it into the numerator afterwards?
don't think so?
not sure what you mean though
you're pretty close. like 2 more lines
royal ridge
not sure what you mean though
i dont know what to do after using 1+1. I have two products that go to n and that part at the end
i dont know how to use that x_(n+1) to make one of the products to n+1 without having something at the end that i cant remove
zinc ginkgo
you know you want this to look something like
$\left(\prod_{i=1}^n(1+x_i)\right)(1+x_{n+1})$
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riemann
royal ridge
yeah
zinc ginkgo
so factor your product out
then use induction hypothesis 1 more time on the $x_{n+1}/\prod (...)$ term
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riemann
zinc ginkgo
might be more obvious to first re-arrange your hypothesis to be $(...) \ge 1 + \sum x_i$
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riemann
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royal ridge
so factor your product out
do i factor the product that goes to n out?
I tried to do some stuff on the side to see what you mean and while I know what you mean i dont really understand
zinc ginkgo
combine these two
you're so close!
all you need is to show this is bounded above by 1+x_{n+1}
royal ridge
combine these two
but how do I combine it if the above goes to n and the lower one goes to n+1. Do I need to pull the factor out of the bottom one?
zinc ginkgo
but how do I combine it if the above goes to n and the lower one goes to n+1. Do...
oh i missed that. use the inductive hypothesis which is just n instead of n+1
royal ridge
isnt the induction hypothesis only 2 to the n with n+1 in the denominator?
zinc ginkgo
errrrr no i lie
royal ridge
combine these two
this has 2 to the n+1 though. Do i pull the 2 out?
zinc ginkgo
multiply by n+1 and divide by 2^n in your inductive hypothesis
i might have done this wrong
royal ridge
This i started with
This is the first induction proof we received after we got the induction axiom a couple days ago
Induction schmema or so
royal ridge
all you need is to show this is bounded above by 1+x_{n+1}
i dont think its possible that this is bounded by 1+xi because even if xi=1 then 1+1+... is already bigger than 1+xi
because none of the numerators are 0
so I guess I did something wrong before that already
wraith hinge
hii
zinc ginkgo
i dont think its possible that this is bounded by 1+xi because even if xi=1 then...
i was over complicating it with the second inductive hypothesis
observe $\prod_{i=1}^n (1+x_i) \ge 2^n$
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riemann
wraith hinge
How many positive integers less than 1000000 have exactly one digit equal to 9 and have a sum of digits equal to 13?
zinc ginkgo
How many positive integers less than 1000000 have exactly one digit equal to 9 a...
this channel's occupied. read #❓how-to-get-help
wraith hinge
Any one know this problem?
royal ridge
**riemann**
use it to cancel the product in the denominator?
zinc ginkgo
**riemann**
so using this you know 2^n divided by the product is less than 1
i think the rest is algebra
oh wait there's a pesky extra factor of 2
oh there's an n+2 in the denominator i left out
zinc ginkgo
observe $\prod_{i=1}^n (1+x_i) \ge 2^n$
yea from here you can bound this from above by just x_{n+1}
royal ridge
observe $\prod_{i=1}^n (1+x_i) \ge 2^n$
wont it be above 1+xi because of the +2 in front of this?
zinc ginkgo
yea you can deal with that easily
i only selected the complicated term for simplicity
royal ridge
Which complicated term
Isnt the last line in the pic above already too big
For all x_(n+1)
zinc ginkgo
**riemann**
you didn't use this
zinc ginkgo
observe $\prod_{i=1}^n (1+x_i) \ge 2^n$
this complicated term
zinc ginkgo
Do i have to use it before? I replace the 2^n right
just try it
i took a screenshot where you should use it if you go back and read
royal ridge
observe $\prod_{i=1}^n (1+x_i) \ge 2^n$
this
Ah nvm
No not nvm
Was thinking i cant use <= but i can
Ill try again i replaced the wrong thing
No its still the same
I dont see how doing anything to 1+1+(…) will make it be bounded by 1+x_(n+1) as its already bigger for some x_(n+1)
I think i am not understanding what to do with the observation above
Alternatively I tried to replace it sooner
royal ridge
Alternatively I tried to replace it sooner
royal ridge
Not x1 but x_(n+1)
royal ridge
i took a screenshot where you should use it if you go back and read
I tried to do it at multiple points after the first ind. hypothesis and it always ends up with the product*(number bigger than 2)
think i know why. didnt do the the part before the ind. hyp. correct
anyway will go sleep now i as its 1am. will try to do it tomorrow.
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compact saddle
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compact saddle
how do i answer c? (im assuming its simplifying)
i keep getting this
but its incorrect
<@&286206848099549185> (posted this q a while ago but thread got shutdown because i kept trying)
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compact saddle
bump
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errant tiger
I would recommend to use lhopitals rule and i can show a pocture how soon
errant tiger
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eager juniper
Just divide the numerator and the denominator by e^-5x like you do when indeterminate forms involve polynomials
raw gate
I guess this is what it looks like?
glossy marsh
Yes it is
raw gate
is there anything wrong here?
raw gate
no
raw gate
Do you have any values for fs ?
how would I solve for A if static friction is not given
glossy marsh
I'd say its 0 because it rolls
glossy marsh
No when it rolls there is nothing but the normal force
raw gate
huh
if it's just the normal force then the disk would float
would be launched or something
glossy marsh
No there is weight
raw gate
static friction cannot be zero
glossy marsh
Then idk
Sry
This is a common situation i think you can get something on youtube about that
Or google
raw gate
I haven't found any
raw gate
You have to do the same thing for the y axis
do what
raw gate
I already did that wdym
raw gate
?????
glossy marsh
Oh yes u did
Then you have n
And n is linked with fs
By some formula
Yes fs = n * us
raw gate
but that's the maximum static friction
raw gate
hmm I think there should be another way
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raw gate
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raw gate
you would find better help in discord.gg/physics
I would but no one really answers there
raw gate
you would find better help in discord.gg/physics
what's different in an annular disk compared to a normal disk when solving problems like these?
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slim knot
wouldn't the integrated function of $2/3 x^2$ be $8/3 x^4$
solid kilnBOT
yomiko
slim knot
why is it $1/6 x^4 ?$
edgy cedar
No
solid kilnBOT
yomiko
edgy cedar
No
slim knot
how
edgy cedar
2/3 * 1/4 = 1/6
wraith hinge
yeah the fractions at the denominator do throw off a lot of people
edgy cedar
Just write fractions separately
wide cargo
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stupid bmp
Hi, so if I have dx/dt being x with dot
what dould d/d(x with dot) of x be
I need this for setting up lagrange equations of second kind for a dynamics problem
in the original problem it's a partial diferential though, if that alters it
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idle canopy
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idle canopy
any help?
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idle canopy
Let m be Mahlelet's age
idk
i got 22 as my answer
3x4=12
12x4=48
48+3=51
73-51=22
@silent canyon
anyone?
lethal anvil
whoah I'm not sure what you're doing there.
If Mahelet's age is m, and Mr. Jones' age is 3 more than Mahelet's age, then what is Mr. Jones' age?
aka if I have m apples and I add three more, how many apples do I have?
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marble dawn
what would be the answer to this?
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what would be the answer to this?
marble dawn
so answer to 2 dp would be 0.99 ?
hushed ferry
i dont know
marble dawn
and how many solutions would there be?
sorry, largest solution would be cos(5) ?
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frosty frigate
hey so 😭, im genuinely bottom in math and have no clue at all how to work out angles / even what to search to get answers lol can someone help work out the angle of O
modest ice
A circle has 360 angle. You have some values for making a circle . 86+67+122+ O(the letter in this picture) = 360
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marble dawn
how would i solve this?
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rugged hollow
I have a question:
If out of a group of 20 people, three people win a prize. and Jane and her three housemates are in this group ( they are four of those 20 people.) What's the probability that they win all the 3 prizes?
And what's the probability that exactly two of them win the prizes?
Can I use binomial in this case?
can someone help? this question is super urgent. Thank you!!
split sundial
I have a question:
If out of a group of 20 people, three people win a prize. and...
the chance that they win all 3 prizes is
4/20 * 3/19 * 2/18 assuming no replacement
rugged hollow
4/20 * 3/19 * 2/18 assuming no replacement
that's what i was confused about. in the case like this, is it with or without replacement?
rugged hollow
no
split sundial
4/20 * 3/19 * 2/18 assuming no replacement
so its this
i dont actually know this math yet im kinda just using logic
i have no idea how to do the second question tho
fossil finch
Anyone that can help me?
rugged hollow
lilac wind
Anyone that can help me?
and yes what Vc05 is saying is correct as long as no 1 person can win multiple times
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wraith hinge
Is this correct
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Is this correct
hallow spruce
how did you get that?
wraith hinge
14 divided by 21
hallow spruce
why did you do that?
wraith hinge
My teacher told me its how you get experimental probability
seems like she was wrong
hallow spruce
she's not wrong at all
wraith hinge
oh
hallow spruce
if 14 people won a prize and 21 didn't, how many people total participated?
hallow spruce
that's better
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fickle flume
how do i find order and degree of equation y = Ax +A^2
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icy nova
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heady summit
you know the second derivative test?
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real geyser
How can I solve the area for this? I've tried a trapezoid but i don't have the h of the triangle, I'm stuck.
primal walrus
Uh Heron's formula for the triangle??
primal walrus
Divide the figure into triangle and a rectangle
real geyser
but the rectangle is not even
young swift
Divide into a trapazoid and triangle
primal walrus
I didnt read it carefully
real geyser
Divide into a trapazoid and triangle
like in the middle?
lilac wind
make the bottom a rectangle and the top a triangle, then use pythagorean to find the height of the triangle and use triangle area equation
young swift
Nah, like the bottom and top
lilac wind
oh wait nvm, the sides arent equal
lilac wind
yeah that would work
real geyser
Are these angles 90 deg
yes
real geyser
yeah that would work
problem is that i dont know the distance
primal walrus
what about like this?
triangle, rectangle, then right triangle at the bottom
solve for the area of each then add
primal walrus
[from top to bottom] Heron's formula, length x width, and 1/2 bh
edgy cedar
Herons
real geyser
this makes sense
lilac wind
not sure how you get the height there but maybe im just dumb. was trying to find the height for my own by cant
edgy cedar
Which height?
lilac wind
of the top triangle
lilac wind
oh wait, ok yeah then i see. didnt know the name of that formula lol
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rapid kiln
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rapid kiln
for this question do they want the third row third column of inverse of A right
if so why ask to compute A and b
zinc ginkgo
if so why ask to compute A and b
if you multiply the two, then you'll see how A^{-1} relates to B
rapid kiln
alri so the inverse of a is how i get the answer
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zinc ginkgo
alri so the inverse of a is how i get the answer
Do you notice any special structure of the product AB?
undone knoll
Let A, B be sets. Show the following:
(a) From A' ∩ B = ∅ follows B ⊂ A,
(b) A ⊂ B exactly if A' ⊃ B'
hello i am confused what exactly is supposed to be done here :d
hardy zodiac
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fallow spruce
The set of invertible elements of the monoid $\left( \mathbb{Z}[i],\cdot \right)$
solid kilnBOT
『Marius』
dusty hound
Notice that all nonzero elements have norm greater than 1
So yes, you need a^2 + b^2 = 1
fallow spruce
but bi/1 its still complex if b is not 0[
dusty hound
Notice that all nonzero elements have norm greater than 1
@zinc ginkgo why that pikachu face ?
zinc ginkgo
<@261933205387477002> why that pikachu face ?
i didn't realize this 
fallow spruce
but bi/1 its still complex if b is not 0[
?
dusty hound
I don't get what you're trying to say.
At least not how it relates to the invertibility of anything
random citrus
Integers
fallow spruce
but if 1 a^2+b^2 is 1 wouldn't I just end up with a - bi is this number a integer
dusty hound
if a^2 + b^2 = 1, then one of a and b is 0 and the other is +- 1
That gives 4 cases which you can solve separately to find all the invertibles of Z[i]
Unless you want to do it in one go using complex exponentials
wraith hinge
(overkill :p - by dirichlet's unit theorem, the units form a ring of rank 0, so the only invertible elements are the roots of unity)
dusty hound
,w rank of a ring
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dusty hound
Now I have to go look this up. Dammit
wraith hinge
(in case I wasnt clear, I was overkilling using dirichlet)
dusty hound
Still curious
wraith hinge
(rank is the power of Z when viewed as an isomorphism Z^r x u, where Z is infinite cyclic, u is finite)
dusty hound
Is that actually 21th century math ?
random citrus
Doesn't this defintion depend on u
dusty hound
random citrus
Rank is an adjective here?
wraith hinge
Is that actually 21th century math ?
wraith hinge
Doesn't this defintion depend on u
u is all elements with finite order
fallow spruce
so this would be the elements ?
wraith hinge
yup
dusty hound
https://en.wikipedia.org/wiki/Dirichlet%27s_unit_theorem
I didn't study enough Galois theory for this
I just had the smallest intro to GT in a small end of semester class
wraith hinge
you probably dont need too much gt for (basic) algebraic number theory
dusty hound
I studied basic abstract algebra (like euclidian => principal => factorial, ring extensions, minimal polynomials, etc) but that's about it
wraith hinge
I reckon thats enough to get started with main results and calculations (ie omitting proofs)
dusty hound
Meh
Got more important things to do than study an entire branch of a field that won't be useful to me, honestly
Need to practice endomorphism reduction atm lol
wraith hinge
haha gl
wraith hinge
oh jeez, thats a lot of exercises
dusty hound
Not like we need to do all of them thankfully
That's for like 1 month, probably a little less
wraith hinge
have fun
if you need help theres a discord server that provides maths help: .gg/math
zinc ginkgo
#groups-rings-fields exists for you pros
lusty raptor
hi
lusty raptor
is there such a thing as inversion by parts
of a function
if not then the second best thing that would help me solve my problem would be if there was something like sin(x)^2+cos(x)^2=1 for arcsin & arccos
zinc ginkgo
nope
rugged raven
whats an easy way to rearange this
trim joltBOT
whats an easy way to rearange this
rugged raven
so i can graph it
dusty hound
- isolate y so you can write it in the form y = f(x)
- use desmos as it accepts almost any x-y relationship
dusty hound
Yes
dusty hound
2x - y = 5
2x = 5 + y
2x - 5 = y
y = 2x - 5 is the form you want
dusty hound
No
Also use / rather than "over" (and use parentheses if necessary to reduce ambiguity)
rugged raven
ima keep trying this 
although its been 2 hours lol
my problems they give me are harder
than youtube ones
idk why they give me -y
but they give him 5y
its not fair
dusty hound
He had -5y
rugged raven
is mine diff than his?
how can I get a question like his mine has no number
his does
dusty hound
Also it's totally fair. You should be able to do this for any numbers
dusty hound
Yes
dusty hound
If you prefer
rugged raven
it dosent change the question right?
ima show u my rough work
tell me where to go next
brb
dusty hound
Something = 1 * somethingv= something * 1 is true for any value of something
rugged raven
im visual so i need to show u
this is whati have so far
see? my problem is the -y
🙏🏿
dusty hound
Multiply by -1
Because (-1)(-1) = 1.
Also known as - - = + (minus minus equals plus)
dusty hound
2x - y = 5
2x = 5 + y
2x - 5 = y
y = 2x - 5 is the form you want
No, you only multiplied by -1 on one side there.
I already have you the answer for you to check against it
dusty hound
You really had to hit me with a blue background huh
dusty hound
The advantage of black on white is contrast
dusty hound
-y = -2x + 5
Multiply both sides by -1
y = 2x - 5
dusty hound
Try to plot it
rugged raven
alright
need to youtube that lol
gimme like 10mins
what numbers can I use for my X
does it matter?
how do you know which numbers to use?
dusty hound
You should recognize that this is the equation of a line
dusty hound
whats an easy way to rearange this
It looks like you have a button to draw a line
dusty hound
You're given a part of the x-y plane. Where would you want to put a table of values ?
Just graph it. That is to say, figure out what line it is and plot it
dusty hound
I'll be honest:
My first thought "how on Earth did he get a vertical line ?"
Second thought "actually I'm better off not asking"
dusty hound
https://www.youtube.com/watch?v=-Eze0i92WMY
You may notice that it is called slope intercept form
dusty hound
Because the first number is the slope of the line and the second one is the y intercept
dusty hound
i must of missed a step
It's not a step. It's a entire chapter at least
dusty hound
Because the first number is the slope of the line and the second one is the y in...
So it's a line of slope 2 that includes the point (0,-5)
rugged raven
ok so i needed a chart
thats the step I missed
got it
but that was hard asf tho
.close
trim joltBOT
trim joltBOT
Available help channel!
gaunt echo
Prove that every prime factorization is unique.
trim joltBOT
> Prove that every prime factorization is unique.
shrewd ridge
gaunt echo
Proof. Let us suppose that $$p_1^{e_1}\cdots p_n^{e_n} = p_1^{e_1'}\cdots p_n^{e_n’}.$$ If the exponents are the same ($e_i = e_i’$ for all $i$), then we are done. Otherwise, suppose there exist two exponents $e_i \neq e_i’$. Without loss of generality, let $e_i > e_i’$. Then we may divide both sides by $p_i^{e_i’}$ to obtain $$p_1^{e_1}\cdots p_i^{(e_i - e_i’)}\cdots p_n^{e_n} = p_1^{e_1’}\cdots p_i^0\cdots p_n^{e_n’}$$
Thus $p_i$ divides the left number but not the right, so the left and right sides cannot be equal. Therefore the hypothesis is false; it’s impossible for the exponents to differ, so every prime factorization is unique.
shrewd ridge
the first sentence is the assumption
it's not clear what part is lemma 1
but the proof is talking about the "hence" part as far as i can tell
shrewd ridge
but that's like two statements then
gaunt echo
wdym
shrewd ridge
you claim the first sentence, and you claim that "hence" follows
so it's two lemmas
gaunt echo
well the first sentence is kind of obvious right
shrewd ridge
and your proof is talking about the second lemma
gaunt echo
oh
shrewd ridge
and the first one is not proven
solid kilnBOT
a.b.s._.0.
shrewd ridge
the trick in wikipedia ||is that you consider the smallest number that has two factorizations||
gaunt echo
ok wait, i don't see where that fits in exactly
gaunt echo
(is this valid?)
shrewd ridge
i'm slowly thinking
gaunt echo
I can't see anything wrong with it
shrewd ridge
**a.b.s.\_.0.**
here you're claiming that right number is not divisible by p_i
i don't think that's fair
gaunt echo
hm
shrewd ridge
you removed the entire p_i ^ something, the unproven assumption is it stops being divisible by p_i because of that
gaunt echo
I removed the entire pi^ei' ?
shrewd ridge
yeah from the right part, you divided by all instances of p_i
shrewd ridge
there's no more explicit p_i but why can't the rest of it be divisible by p_i
gaunt echo
like, I'm assuming that since it doesn't appear in the product, it can't divide the number. Which implicitly assumes that every product is unique
god that's so annoying
gaunt echo
this is like
so elementary that you have to be very very careful with assumptions
it's hard
oh I get it
Euclid's lemma, yeah
since p_i obviously divides the left side, it must divide the right side
which means it must divide one of the p_j's which are not p_i
gaunt echo
wait could you check this real quick
shrewd ridge
uh yeah, that seems correct
shrewd ridge
like I don't get it, but it's completely the same as wikipedia
"first non-prime-factorable number is composite, we can write it as two smaller numbers multiplied, which is a prime factorization lmao"
gaunt echo
oh lmfao
that's so like
stupid easy
induction is overpowered
that's a better proof really
shrewd ridge
it's the same
shrewd ridge
it's just shorter
trim joltBOT
@gaunt echo Has your question been resolved?
gaunt echo
i kinda doubt you'll prove euclid's lemma
Yeah just glanced at a proof and it’s wild
trim joltBOT
Yeah just glanced at a proof and it’s wild
trim joltBOT
trim joltBOT
Available help channel!
trim joltBOT
marble wharf
reef nimbus
How did you obtain this number?
trim joltBOT
@wraith hinge Has your question been resolved?
velvet bolt
it's irrational only if it can't be written as a fraction
yes
if 0.3 repeating irrational?
yes but is 0.3333333333333333333333333333333333333333333
rational?
3 going on forever
hm
yes
but how do you prove it
inland orbit
If the 3s go on forever, there is no last 3, where are you gonna add the 0s?
But yes it is rational
velvet bolt
but how do you prove it
?
forget the add 0 method
it's confusing
remember what i sayd?
said*
velvet bolt
it's irrational only if it can't be written as a fraction
.
if you said it's rational
it means you can write it as a fraction
so whats the fraction?
wdym
you can
look above
someone wrote it for you
shrewd ridge
it's always rational
shrewd ridge
if you can't write it as a fraction you can't write it period
there's no notation for it, in terms of this "number dot something"
solid kilnBOT
Master Oogway
solid kilnBOT
Result:
1.4142135623731
shrewd ridge
yes
inland orbit
That decimal is just an approximation for sqrt(2). It's not equal to sqrt(2)
inland orbit
Any number that can be written as a terminating or repeating decimal is rational
yes
shrewd ridge
any number with something.something is rational
velvet bolt
listen
can you write 6.210735
as a fraction?
yes or no
good
then its rational
can you write 0.(3) as a fraction?
yes or no
ok
whats the fraction
tell me the fraction
thats not a fractyion
give me numbers
,calc 3/10
solid kilnBOT
Result:
0.3