oh yeah
#help-49
buoyant fjord
oh yeah
junior flower
that sequence converging to 0 for any (x_n) for your specific problem and what i said here are very related
buoyant fjord
wait one sec im trying something
radiant roost
why are you using two sequences?
junior flower
because the slope of that secant line is the value (f(b) - f(a))/(b-a)
junior flower
and that value converge to 0 when you take b values close to -2
buoyant fjord
an N such that for all n>N $d(x_n,-2)<d(x,-2)/2$ maybe lemme try it
grand pondBOT
Julian
junior flower
let me ask you something. if the function was instead defined by f(x) = 3(x+2)^2 for all real x, would you be able to show, using the sequence interpretation, that f is differentiable at 0?
buoyant fjord
yeah cuz
junior flower
ok now i’m going to tell you that it’s very easy to adjust that proof to apply to your problem too
because…
junior flower
when you take an irrational point, the secant line expression is 0
buoyant fjord
yeah
radiant roost
yeah it's not a product, it's cases
case 1, if it's irrational then it's 0 and it's in a certain range
case 2, if it's rational then it takes a certain value and u have to show that's in the range too
buoyant fjord
i thought the irrational case was trivial right
junior flower
yes
radiant roost
yes but you still have to state it
buoyant fjord
i am yea
radiant roost
i mean if you're doing a proof
buoyant fjord
i will include it
radiant roost
yeah but i can't understand using two sequences, i think you just need to take one arbitrary sequence and show that its limit is 0
buoyant fjord
yeah but i can't understand using two sequences, i think you just need to take o...
im not doing 2
im doing an arbitrary mixed one]
radiant roost
what is a mixed sequence?
buoyant fjord
like with irrational and rational
junior flower
let me ask you something. if the function was instead defined by f(x) = 3(x+2)^2...
btw this can be shown with a pretty simple limit computation. are you allowed to do that or do you really need to do epsilon delta stuff?
radiant roost
do you mean that a_n and b_n are subsequences?
buoyant fjord
btw this can be shown with a pretty simple limit computation. are you allowed to...
probably
junior flower
do you mean that a_n and b_n are subsequences?
no
buoyant fjord
do you mean that a_n and b_n are subsequences?
no its the fact that the difference quotient is 0
for irrationals
so thats less than whatever epsilon
buoyant fjord
btw this can be shown with a pretty simple limit computation. are you allowed to...
oh do you mean
3(x_n+2)^2's limit?
cuz the division
junior flower
do you mean that a_n and b_n are subsequences?
(x_n) is an arbitrary sequence converging to a. and (f(x_n)- f(a))/(x_n-a) can be viewed as a sequence
radiant roost
shouldn't it converge to -2?
buoyant fjord
it does but the difference quotient converges to 0 is what we wanna show
junior flower
(x_n) does by assumption
junior flower
(x_n) is an arbitrary sequence converging to a. and (f(x_n)- f(a))/(x_n-a) can b...
the long one here should converge to 0
radiant roost
you said it converges to a
buoyant fjord
well a is -2 i think
buoyant fjord
oh yeah
radiant roost
what is b?
buoyant fjord
wheres b
radiant roost
^
radiant roost
oh
buoyant fjord
oh i can prob just assume the limit cuz we had this on an older homework
junior flower
if you have that the quotient limit is 0 with rational x_n’s, it’s easy to say it’s all 0 with irrational x_n’s too
since the quotient is 0 when x_n is irrational
midnight plankBOT
junior flower
great
midnight plankBOT
Available help channel!
tidal turret
midnight plankBOT
grand pondBOT
Renato
lyric charm
where is your question
tidal turret
can someone explain what does this mean?
lyric charm
there's several different points that could demand explanation
tidal turret
i just find it confusing
lyric charm
can you at least narrow down to which of the 4 paragraphs you want explained from this picture?
that or we could do it the painful way and i could extract this info from you like up to 4 rotten teeth
@tidal turret
tidal turret
can you at least narrow down to which of the 4 *paragraphs* you want explained f...
i think second paragraph is kind of confusing
lyric charm
paragraph 2 is an intuitive explanation of WOP
ripe juniper
tidal turret
paragraph 2 is an *intuitive* explanation of WOP
it gets fuzzy when it talks about n0
lyric charm
it gets fuzzy when it talks about n0
it says:
if A is infinite, pick whatever element you want from there, and consider only the elements of A that are leq that one
tidal turret
can we narrow it down to a down to earth example test case?
lyric charm
can we narrow it down to a down to earth example test case?
sure
suppose A is the set of all primes
it's infinite, so big
tidal turret
yes
lyric charm
but. 67 is in A
so we can narrow down to the finite set {primes <= 67} and the minimum won't change
tidal turret
so we can narrow down to the finite set {primes <= 67} and the minimum won't cha...
we could have also took the first prime from the set of all primes
2
lyric charm
we could have also took the first prime from the set of all primes
sure we could
we have freedom of choice here
i am just giving an example
tidal turret
okay let's try 67 and then 2 please
tidal turret
so we can narrow down to the finite set {primes <= 67} and the minimum won't cha...
sure i think
lyric charm
you want this worked out formally?
it will be very boring and not remotely enlightening
tidal turret
you want this worked out formally?
not really no. im not in that stage of my career
lyric charm
ok then let's not
tidal turret
this is an intro to wop
tidal turret
so we can narrow down to the finite set {primes <= 67} and the minimum won't cha...
the part about the intersection it gets fuzzy
lyric charm
and consider only the elements of A that are leq that one
i translated it into plain English
tidal turret
i translated it into plain English
i see
is tricky to follow
why they say consider finite then consider infinite set
lyric charm
why they say consider finite then consider infinite set
finite sets are more straightforward
with a finite set we can think about "iterating" over all its elements and it'll come to an end
because. well that's the entire thing with finite sets isn't it
midnight plankBOT
Available help channel!
viral dagger
,rccw
grand pondBOT
viral dagger
hold on nvm
blue, yellow, and green are only adjacent for (528,529) (1495,1496)
so except them, each blue, each yellow, and each green contributes 2 pairs
so (floor(2023/23)+floor(2023/88)-floor(2023/(23×88))×2
viral dagger
blue, yellow, and green are only adjacent for (528,529) (1495,1496)
-2 for overcounting
viral dagger
yeah thats what i thought
viral dagger
so (floor(2023/23)+floor(2023/88)-floor(2023/(23×88))×2
but this gets 2(87+22-0)-2=216
anywhere i went wrong?
runic hamlet
hmm if yes I dont see where
midnight plankBOT
midnight plankBOT
Available help channel!
lusty python
Hi, I needed help with set cardinalities to solve a coding problem (specifically advent of code 2025 day 5 part 2)
midnight plankBOT
Hi, I needed help with set cardinalities to solve a coding problem (specifically...
lusty python
I needed to calculate $\abs{A_1 \cup A_2 \cup A_3 \dots \cup A_n}$ where $A_i$ is a finite interval of integers
grand pondBOT
1 divided by 0 equals Infinity
lusty python
like you want code? <:kekhands:1294941772841091112>
...
lusty python
**1 divided by 0 equals Infinity**
i wanted a general formula for this
without having to do a lot of code
gaunt jetty
well you probably gotta use a sort and merge algorithm
lusty python
?
gaunt jetty
like, sort all the intervals based on their starting integer, iterate through them and merge if they overlap or touch
lusty python
like, sort all the intervals based on their starting integer, iterate through th...
what if they are disjointed?
that's the problem
gaunt jetty
well since it is sorted if they don't overlap the next one is going to be a new separate interval
gaunt jetty
idk if thats sarcasm or not but ok
cerulean oyster
I needed to calculate $\abs{A_1 \cup A_2 \cup A_3 \dots \cup A_n}$ where $A_i$ i...
Do they share any integer?
lusty python
Do they share any integer?
can you elaborate on that please?
cerulean oyster
If you had any probability class this should sound awfully similar
but if they are independent (dont share any integer), then the cardinality of that set is the sum of the cardinalities
lusty python
they may share any integer
cerulean oyster
Okay, then you probably want a hash map here
sadly you will have to iterate through all the arrays
lusty python
sadly you will have to iterate through all the arrays
oh i tried that
maximum limit is around 15 digits
so that won't help
cerulean oyster
as to cardinality of individual sets?
lusty python
#include <iostream>
#include <cstdio>
#include <string>
#include <algorithm>
struct Range {
size_t start;
size_t end;
};
void push_digit(size_t &n, char digit) {
n *= 10;
n += digit - '0';
}
Range parse_range(std::string s) {
size_t start = 0;
size_t end = 0;
bool dash = false;
for (char &c : s) {
if (c == '-') {
dash = true;
} else if (dash) {
push_digit(end, c);
} else {
push_digit(start, c);
}
}
return {start, end};
}
int main() {
std::ios_base::sync_with_stdio(false);
std::cin.tie(nullptr);
std::cout.tie(nullptr);
std::freopen("testfile.INP", "r", stdin);
std::freopen("testfile.OUT", "w", stdout);
size_t m;
std::cin >> m;
Range r[m];
for (size_t i = 0; i < m; i++) {
std::string s;
std::cin >> s;
r[i] = parse_range(s);
}
std::sort(r, r+m, [](Range a, Range b) {
if (a.start != b.start) {
return a.start < b.start;
}
return a.end < b.end;
});
size_t res = 0;
Range current_interval = r[0];
for (size_t i = 1; i < m; i++) {
if (r[i].start <= current_interval.end) {
current_interval.end = r[i].end;
} else {
res += current_interval.end - current_interval.start + 1;
current_interval = r[i];
}
}
std::cout << res + current_interval.end - current_interval.start + 1;
return 0;
}
@gaunt jetty
wrong answer
lusty python
as to cardinality of individual sets?
as to the maximum end value
cerulean oyster
we dont really care about the values really
just about their multiplicity (how many times they appear)
cerulean oyster
**1 divided by 0 equals Infinity**
Oh goddam i didnt read this is an interval not an array
yeah no clue
like, from the theory side, you can do the sum of all the intervals - the sum of all their intersections
but thats O(n^2)
lusty python
but thats O(n^2)
Think hard
subtle zinc
isn't it an interval
cerulean oyster
not even n^2, its n!, lmao.
pine wave
Yeah you would want to be looking for an algorithm
Main idea is basically a sweep-line-like algorithm
lusty python
Main idea is basically a sweep-line-like algorithm
Alr
lusty python
Main idea is basically a sweep-line-like algorithm
How does it work?
pine wave
You can also start from how you sorted your intervals
And consider the subproblem of "how many elements in the union of first k intervals
lusty python
And consider the subproblem of "how many elements in the union of first k interv...
Dp question eh?
I was thinking of that approach
Got 2 base cases
And don't know how to move on
cerulean oyster
Main idea is basically a sweep-line-like algorithm
Someone correct me if im wrong, but on this
take, A1 and A2, find Union (U1) and Intersection (S1)
U1 and A3, find (U2) and (S2)
etc... until you covered all A
Un is just the union of all sets, sum(S1...Sn) should cover multiplicity of the intervals.
pine wave
>Someone correct me if im wrong, but on this
take, A1 and A2, find Union (U1) a...
hmm... possible, takes n log n time iirc. This is not a sweepline-like algorithm
cerulean oyster
ill draw a thing rq just to check cause im doubting myself
lusty python
ill draw a thing rq just to check cause im doubting myself
Take your time
pine wave
ill draw a thing rq just to check cause im doubting myself
I know of an approach which goes along those lines which uses an ordered-set data structure
pine wave
Dp question eh?
Not very dp, simple iteration will do from k to k+1. Use the fact that you sorted your intervals. There are two (or three) cases to consider
cerulean oyster
na i was wrong
lusty python
```cpp
#include <iostream>
#include <cstdio>
#include <string>
#include <algorit...
So is it the same as this?
pine wave
na i was wrong
A union of intervals can be written as a union of disjoint intervals
pine wave
So is it the same as this?
Same way of sorting, different processing afterwards
lusty python
Same way of sorting, different processing afterwards
Kk
cerulean oyster
yeah, union of previous intervals is just a loss of information
Sorting them out and using the end-start values is prob best
midnight plankBOT
cerulean oyster
gl
midnight plankBOT
Available help channel!
hybrid sage
i need help in understanding the answer of this question
midnight plankBOT
i need help in understanding the answer of this question
hybrid sage
placid abyss
Do you remember what a closed circle (filled in) vs open circle represents)
hybrid sage
Do you remember what a closed circle (filled in) vs open circle represents)
yes a closed circle is equal to f(x), and the open circle is excluded from the function
placid abyss
Right
If we start with the 2nd question which is simpler
What does the question ask for?
hybrid sage
What does the question ask for?
f of -2
placid abyss
Yeah
hybrid sage
but idk what answer to put
placid abyss
Is that what you circled?
placid abyss
Are you sure? Look at the x-value
hybrid sage
its the other one right
placid abyss
Yeah,
hybrid sage
so how do i look for the answer?
placid abyss
So think about what a fuller circle means vs a non filled circle
hybrid sage
At x=-2 theres two circles
i put the closed circle? correct
placid abyss
Yeah exactly
hybrid sage
and what value is the closed circle? it doesn't indicate in the image
placid abyss
It does say
You can count the y-direction
The vertical axis
In the same way you counted x=-2 is two blocks to the left
Apply the same concept to the y axis
placid abyss
Did you get an answer?
placid abyss
Not exactly. count again
And you should start from the point where the two axes intersect
hybrid sage
is it a negative on the y axis?
hybrid sage
And you should start from the point where the two axes intersect
ohh
placid abyss
It is negative!
hybrid sage
-2?
placid abyss
Exactly
hybrid sage
Do you know what this limit means?
it's looking for the value or limit if x is -2 to the left
am i right?
placid abyss
Yeah, so as x approaches -2 from LEFT to RIGHT, what does the value of x reach?
In the same way, if it said -2^(+) it would mean the value as x approaches -2 from RIGHT to LEFT
hybrid sage
wouldn't - mean to the left? or is it the other way around?
placid abyss
Other way around
hybrid sage
ohh so it would be to the right?
placid abyss
I like to think about it as: if we have a negative we start from the negative side and move in the opposite direction (so to the positive)
hybrid sage
I like to think about it as: if we have a negative we start from the negative si...
ohhh
makes sense
placid abyss
And if we have a positive sign, we start from the positive and move in the opposite direction so towards the negatives
This is a good thumb rule to remember
Alright so if you start from the very left side of the graph (negative side) and move tot he positive side, and stop at x=-2 what does the graph read
placid abyss
Yup
hybrid sage
ohh nice
placid abyss
One important thing: make sure you follow the graphs line
hybrid sage
Alright so if you start from the very left side of the graph (negative side) and...
how would i know
placid abyss
Look at the graph
placid abyss
You’re going the wrong way now
You want to start in the bottom left corner
And follow the graph to the right
hybrid sage
until we reach -2?
placid abyss
Yeah
hybrid sage
okay so we would stop here?
hybrid sage
and then we get the value of the closed circle?
placid abyss
Exactly
placid abyss
Yup
jade magnet
hey i dont think i understand the difference between prims and nearest neighbour
midnight plankBOT
hey i dont think i understand the difference between prims and nearest neighbour
jade magnet
or even between prims and kruskals tbh
whenever i want to find a minimum spanning tree i just add the node with the smallest weighted edge that's not included in my graph if that makes sense
but i dont even know which algorithm that is
i dont understand the differences can somebody help
nvm
my teacher explained it
.solved
midnight plankBOT
midnight plankBOT
Available help channel!
slate crater
Let 35 people seated in a circular table, in how many ways can we pick 6 out of 35 people given no any 2 of them sitting next to each other.
midnight plankBOT
Let 35 people seated in a circular table, in how many ways can we pick 6 out of ...
gaunt jetty
Let 35 people seated in a circular table, in how many ways can we pick 6 out of ...
What have you tried
slate crater
The answer is $\frac{\binom{28}{5}}{\binom{35}{6}} \cdot \frac{35}{6}$
grand pondBOT
Poly(^///^)
slate crater
But I don't get where the 35/6 come from
Oh
Mb
Probability
Sorry bruh the question was asked yesterday so I didn't remember it correctly
But everything else is right
junior flower
so the question is what’s the probability 6 people picked at random are all not sitting next to each other?
slate crater
Are there a different between choosing 6 people from 35 people in a circular table and just choosing 6 from 35?
slate crater
so the question is what’s the probability 6 people picked at random are all not ...
Yes
junior flower
what would “just choosing 6 from 35” mean?
it would be different if they were e.g. in a line
slate crater
$\binom{35}{6}$ by that I mean is the the sample space?
grand pondBOT
Poly(^///^)
junior flower
because the ones on the end wouldn’t be next to each as they would be if they wrapped around in a circle
you can still view the sample space as something with 35 choose 6 elements
slate crater
Yeah that's why I asked and that what I did
Then fix one from 6 people, then choose 5 others people from the rest give binom{28}{5} and there're 6 ways of choosing the fixed person so divide it by 6
But where the 35 come from
I guessed the sample space is binom{35}{6}/35 so it satisfies the answer but I couldn't wrap my head around why it's the sample space
junior flower
there are 35 seats you can choose to fix the first person at
slate crater
So the sample space is binom{35}{6}/35?
junior flower
it depends you how do the problem really
slate crater
Are there a different between choosing 6 people from 35 people in a circular tab...
So like, is that the fixed person make the difference ? that's why I asked this question
slate crater
it depends you how do the problem really
hmm
junior flower
So the sample space is binom{35}{6}/35?
but yes this is a fair thing to say
there are 35 choose 6 ways to put the people around the table, but if you take a particular way and rotate it, each of the 35 rotations are kinda the same thing
you can treat them as the same or different and still do the problem
slate crater
you can treat them as the same or different and still do the problem
How can I do the problem with the sample space binom{35}{6}?
Like what's the thought process for it
junior flower
there are 35 seats you can choose to fix the first person at
if you imagine the tables as not being the same under rotations, so that there are 35 choose 6 tables, then you can get the “good ones” by first choosing one of the 6 people and of the 35 seats to assign them
if you consider tables the same under rotations, there are (35 choose 6)/35 of them and that’s where the 35 comes from under that interpretation
junior flower
if you consider tables the same under rotations, there are (35 choose 6)/35 of t...
with this interpretation there isn’t really a “choosing a first seat”
do you know what i mean by “same under rotations”?
slate crater
do you know what i mean by “same under rotations”?
I'm trying to wrap my head around it, It's just that I can't imagine the difference between linear and circular case
gusty falcon
junior flower
the difference is not between linear and circular here
junior flower
<:kongouderp:483988392099446794>
slate crater
the difference is not between linear and circular here
doesn't the rotation make them different?
junior flower
imagine that you are going to sit in a circle with 3 of your friends and you wonder to yourself how many different ways you can sit down. anyway, you guys sit down some way in 4 seats. if you all move over to the right one seat, should that be considered a different circle? it’s kinda the same circle, right? but it’s also different in the sense that you are at different seats. well they could be counted as different or not, it’s really up to you
there are (35 choose 6)/35 tables if you consider rotations of tables to be the same and 35 choose 6 if you consider them to be different
also generally in school it’s expected for you to treat tables that are the same up to rotation as the same
junior flower
you are not
slate crater
okay changed my mind set a bit, it makes sense now
One question
How can we do the problem with the sample space 35 choose 6?
slate crater
if you imagine the tables as not being the same under rotations, so that there a...
ngl I'm not sure what you said here
junior flower
How can we do the problem with the sample space 35 choose 6?
it will be the exact same except there are 35 ways to pick which seat to put the first person at, so that’s where the *35 will come in
slate crater
Okay hang on, why does the step of choosing the first seat not matter here? When in sample space we're not choosing the first seat
junior flower
do you mean in the tables are the same under rotations interpretation?
junior flower
because no matter where you put the first person, the table looks the same
when they are the same under rotations
so there’s only one table
not 35 different ones you could make with one person
slate crater
Okay tbh I'm not sure what you're taking about really
junior flower
😭
junior flower
imagine that you are going to sit in a circle with 3 of your friends and you won...
did you understand this?
slate crater
No I meant
slate crater
because no matter where you put the first person, the table looks the same
these
I get that
slate crater
did you understand this?
that*
junior flower
let’s say we consider tables that are the same under rotation to be the same. no matter where i put the first person, i’m forming the same table right?
slate crater
yeah
junior flower
now let’s say we consider them different
i can put the first person at any of the 35 seats and those tables will all be distinct
slate crater
okay
junior flower
so that’s why the placement of the first person matters in one case but not the other
slate crater
but shouldn't they be different? I meant the assumptions are different
junior flower
what’s they
slate crater
let’s say we consider tables that are the same under rotation to be the same. no...
"consider tables that are the same under rotation the same " and "consider them different"
junior flower
i’m not sure if this is what you’re asking, but at the end of the day the probability that the 6 people are not sitting next to each other is the same regardless of which of the two interpretations you pick. the interpretations just affect things like “the total number of tables” for calculation purposes
slate crater
Okay so in 35*(28 choose 5)/6
junior flower
if i roll some different colored 6 sided dice and ask what’s the probability they sum to 15, it doesn’t matter if i consider them all the same or distinguish them by color when looking at all the possible rolls. the probability they sum to 15 is unaffected
slate crater
(28 choose 5)/6 is the way that we count if the rotation doesn't matter, and we multiply it by 35 so it match with the sample space where we take 35 choose 6 where the rotation does matter
slate crater
if i roll some different colored 6 sided dice and ask what’s the probability the...
No I meant if we're counting different for sample space so do the event
junior flower
No I meant if we're counting different for sample space so do the event
yes that is right
slate crater
i get that...
junior flower
that’s why the 35 gets corrected for both in the total and in the good events
slate crater
But I've thought you said smth different 😭
Okay clearly I'm wasting your time bruh
Thanks for the help
slate crater
It does for me haha
Well it used to
Okay Imma close the channel
Have a nice day!!
.close
midnight plankBOT
junior flower
cya
midnight plankBOT
Available help channel!
lethal aurora
Hi, Im trying to understand Cramers rule better,
so essentially we plug in the output vector in a row i followed by calculating the determinant, this gives us the signed value of b
relative to our basis consisting of the column vectors. considering each coordinate get scaled by a value det(A)
we get det(A)*v = (len(b1) len(b2) ... len(bn) )
and thus vi = signed volume(bi)/det(A)
is my reasoning correct?
midnight plankBOT
Hi, Im trying to understand Cramers rule better,
so essentially we plug in the ...
@lethal aurora Has your question been resolved?
midnight plankBOT
@lethal aurora Has your question been resolved?
blissful trench
A curve on a highway has a 1000 foot radus with lanes that are 14 feet wide. If the curve makes a 180 degree turn how much longer is the outside edge of the far lane than the inside edje of the close lane? This is a 3 lane highway and assume the 1000 radius is to the inside edge of the cloest lane.
midnight plankBOT
A curve on a highway has a 1000 foot radus with lanes that are 14 feet wide. If ...
blissful trench
dunno where to start
spiral rock
its the ratio of the radii
because of the formula pi * r for the perimeter of a half circle
blissful trench
ik r_1 is 1000 but idk r_2
spiral rock
3 lanes 14 feet
blissful trench
so add them?
spiral rock
multiply
blissful trench
,calc 1000143
grand pondBOT
Result:
42000
like this? 
shell wigeon
Add the lanes, multiply with the ratio
spiral rock
no like 1000 + 3 * 14
midnight plankBOT
midnight plankBOT
Available help channel!
fossil ember
Okay, I have P+(O-P)t+(X-O)x+(Y-O)y=Q+sum((V_i)s_i) where all upper case are known and points except V_i which is known and a vector and all other things are scalar and unknown and for other things, t>=1 and I have ranges of what s_i can be like 2>=s_1>=1 and a minimum and maximum like that per s_i, what would be most efficient to solve this in 4d+ and 3d? I can make it like per dimension subtract Q and other than one V_i and s_i so that leaves that over V_i for one dimension. This is way overdetermined and all things are >=. What are some recommendations to study? X E.
midnight plankBOT
Okay, I have P+(O-P)t+(X-O)x+(Y-O)y=Q+sum((V_i)s_i) where all upper case are kno...
fossil ember
We can also have (V_i s_i)>=V_i bound. X E.
There are things to do this but this is a lot. X E.
The end result should be x, y, and t for where the outer edges of the shape are, maybe even eliminate some back faces. X E.
Just looking for things to look up. X E.
Honestly, this is just ideas. X E.
runic hamlet
linear programming
fossil ember
linear programming
Any more wisdom?
Well I know linear programming can but efficient algorithms?
midnight plankBOT
@fossil ember Has your question been resolved?
fossil ember
Trying for O(n), not O(n^(n^2)). X E.
runic hamlet
n being what? dimension?
not even solving linear systems is O(n)
and thats already basically the easiest thing ever
linear programming is polynomial time
fossil ember
n being what? dimension?
Maybe, or number of equations. X E.
Okay, n^2?
I don't want the second option I mentioned. X E.
runic hamlet
linear programming is polynomial time
.
fossil ember
polynomial time is short time
So O(?)?
Estimate?
Well, any good books on linear programming?
fallow scarab
fossil ember
Well, there is one valid link in the whole years long discussion and I am unsure if it is relevant. X E.
Is linear programming linear algebra?
fossil ember
Is linear programming linear algebra?
No. X E.
There is officially no recommended book yet. X E.
How is this the bottom help already?
Am I doomed?
fallow scarab
linear programming is usually beyond lower division undergrad math knowledge
you can also try #optimization
fossil ember
Well I do Calculus before done with Precalculus in school so why not?
fallow scarab
good for you
fossil ember
Yeah, I am kind of advanced for being uneducated, or kind of uneducated for being advanced. X E.
Why is there no linear programming channel?
And why not a 4d math channel?
I joke. X E.
Anyway, any specific theories I should study?
I know cosine is trash in 4d+. X E.
carmine sigil
I'm not sure I understand your meaning here? How is it trash?
fossil ember
Is this #optimization or #game-theory ?
fossil ember
I'm not sure I understand your meaning here? How is it trash?
It returns values not useful for finding one way from another, it does not work for practically anything etc. X E.
fossil ember
I have tried the lin alg version and distances. X E.
carmine sigil
You already need two values to specify a direction in 3d, you just need 3 to specify a direction in 4d.
fossil ember
Yeah, using that, cos(ABC)•distance(AC) does not return anything usable for anything. X E.
In 3d that would be perpendicular projection. X E.
carmine sigil
You can use it to build Euler angles in any number of dimensions
Combined with projections
carmine sigil
We use 2d angles in 3d all of the time, why do you think they wouldn't be useful for 4d as well?
Like I really don't understand why you think this
carmine sigil
So, you decided that, because you couldn't figure out how to use them immediately, they must be useless?
fossil ember
No. X E.
carmine sigil
Ok, that tracks actually.
fossil ember
Anyway, how do you suggest?
Like find a moving dimension, use it with another, depending which comes out not zero, relate it to others and continue so that like origin is 1, other is unit and compare the totals?
carmine sigil
Depends on what you are trying to do, which isn't super clear to me. It sounds almost as if you are attempting to find a direction in 4d space, which would be an element of S^3, which can be specified in a number of different ways, but one way is by using a 4d analogue of Euler angles, and to determine these you can use projections and dot products against basis vectors to find the angles
And a big part of that is the identity I mentioned earlier.
fossil ember
The identity you used earlier does not work unless I used it wrong. X E.
carmine sigil
Then you used it wrong? What can I say, the identity is a proven mathematical theorem
fossil ember
In 3d?
carmine sigil
In any finite d
fossil ember
Okay, then using it wrong. X E.
Can you state what cosine equals again so I can try again?
scarlet solstice
Well I do Calculus before done with Precalculus in school so why not?
I don't think you should be so proud of that tbh, because it seems you are still lacking a lot of the basics. There's a reason we usually learn things in order
fossil ember
I don't think you should be so proud of that tbh, because it seems you are still...
Yes. X E.
fossil ember
In any finite d
Alright, what was that cosine again?
How is this wrong, it is obviously wrong. X E.
Javascript. X E.
The vector cosine definition is not working. X E.
Maybe cosine is not enough?
Or perhaps it gets overloaded?
Or perhaps I am just using it wrong. X E.
Anyway regardless that cosine is not useful to me. X E.
Should I resort to my cosine and sine thing to get relative movement to one way?
I am dumb right?
carmine sigil
Alright, what was that cosine again?
a.b = |a| |b| cos θ (assuming the L2 norm)
carmine sigil
@fossil ember
"I have these two vectors, A, B, and I'm attempting to find the cosine of the angle between them. I am getting X, but I expected Y."
Can you tell me what A, B, X, and Y are here?
fossil ember
function cosine(a,b,c) {
//X E
return (dot(minus(a,b),minus(c,b)))/(distance(b,c)*distance(b,a));
//X E
}
//X E
a-b is A, b-c is B, expected something relating to the projection in the program represented by s, but nothing gets it. X E.
-156=x, 1.something=y. X E.
The program above is enough. X E.
Any more you want?
Probably a normal and perpendicular issue. X E.
Thanks for talking anyway. X E.
Well, I guess this is going nowhere. X E.
Did not even get much of an answer what to study. X E.
scarlet solstice
function cosine(a,b,c) {
//X E
return (dot(minus(a,b),minus(c,b)))/(dist...
I think Omni wanted an example with actual numbers, not just more variables. So something like, I ran cosine(1, 2, 3) and expected 0.222, but got 0.555
fossil ember
I think Omni wanted an example with actual numbers, not just more variables. So ...
The number set is random and 10d or more, I could give a program though. X E.
Thanks. X E.
carmine sigil
function cosine(a,b,c) {
//X E
return (dot(minus(a,b),minus(c,b)))/(dist...
Is a, b, and c here vectors of numbers, if so, is b the center of the angle, so you are finding angle abc considered as points?
fossil ember
Is a, b, and c here vectors of numbers, if so, is b the center of the angle, so ...
I am using points yes. X E.
Getting vectors from them. X E.
carmine sigil
Have you tested the minus and distance functions separately to ensure they are giving the results as you expect?
As well as dot
carmine sigil
Why do you have the distance function counting down instead of up?
fossil ember
Does not matter order, -squared=+square. X E.
carmine sigil
The iteration direction.
You have it starting from the end of the vector and walking back to the beginning, which is opposite of the way it is both normally done and the way it was done in the other functions
fossil ember
-98.60061513425832=X 24.008384767096388=Y. X E.
sum of all of them like that does not matter. X E.
carmine sigil
The cosine of any angle should be between -1 and 1.
So if you're expecting 24 then why?
fossil ember
Yeah, multiplying by a distance. X E.
Trying to get a total. X E.
Probably me being dumb?
console.log(point,x,proj,cosine(point,x,proj)*distance(point,proj),setx); X E.
Thanks for scrutinizing. X E.
carmine sigil
I cannot test your code, because I am currently at a Christmas party. I will test it when I get home
Can you test the following just for sanity:
distance([0,0,0,0], [1,1,1,1])
fossil ember
Merry Christmas buddy. X E.
carmine sigil
Should be 2.
fossil ember
Yes, 2. X E.
carmine sigil
So if you do cosine([1, 0, 0, 0], [0,0,0,0], [1, 1, 0, 0]) do you get sqrt(2)/2 ?
About 0.707
fossil ember
Yes. X E.
carmine sigil
If you run cosine([2,3,4,5], [0,0,0,0], [3,4,5,6]) do you get 34/(3 sqrt(129))≈0.997844 ?
fossil ember
Yes. X E.
carmine sigil
If you run cosine([5,3,5,7], [3,0,1,2], [6, 4, 6, 8]) do you get the same answer?
fossil ember
Yes. X E.
carmine sigil
If you run cosine([5,3,5,7], [3,0,1,2], [6, -4, 6, 8]) do you get 22/(3 sqrt(129))≈0.645664?
fossil ember
Yes. X E.
carmine sigil
If yes, then your function is working near as I can tell, and I'm not entirely sure what has led you to believe it is not
fossil ember
It has no relation to the s, setx, or sety seemingly. X E.
fossil ember
setx and sety are x and y on plane and s is scaling out from a point to the end point. X E.
Clear enough yet?
carmine sigil
Not at all
fossil ember
Okay, so basically on plane x and y, then s is multiply a line from P to that by it to get end. X E.
Still unclear?
carmine sigil
Define "plane x", "plane y", "s", "P", "that", "it", and "end", please.
I'm guessing s is the name of a function
fossil ember
Plane x as in from plane center, multiply an x vector from that and add it, same for plane y, s is like 4d z, P as in the original equation pinned, "it" as the point on plane from x and y defined, and end being like right side, the end point. X E.
carmine sigil
Two points is not enough to define a plane uniquely.
carmine sigil
Center point meaning the origin?
fossil ember
Just make up a point. X E.
add 1 in a dimension and 1 in another dimension, boom, x and y vectors. X E.
carmine sigil
That makes 3 buddy. X E.
So x and y are defined as vectors from some point. So if this point is [1,2,3,4] and x is [0,1,0,1] then from the origin the head of x is at [1,3,3,5]?
carmine sigil
So you are attempting to find the angle between x and y?
fossil ember
No. X E.
Finding what originally the x,y, and s were. X E.
Or in the original equation, t. X E.
carmine sigil
s being the "center" point of this plane?
fossil ember
s being the "center" point of this plane?
No, on a line from a P point to the plane, make its magnitude times s. X E.
carmine sigil
I'm sorry I don't really understand what it is you are attempting to accomplish.
fossil ember
4d+ projection and viewing. X E.
Also simplifying the equation some. X E.
shrimplify. X E.
carmine sigil
Ok, I'm reading your OP, and I don't understand what that equation is meant to represent or how you arrived at this, but given that you are clearly attempting to create a visualization of 4d space, have you looked at prior art?
fossil ember
Yes. X E.
carmine sigil
For instance, code golf did an entire series on his 4d engine for 4d golf, and it's open source.
carmine sigil
It can't do 6d. X E.
Sorry code parade
fossil ember
Not far off, 4d golf. X E.
carmine sigil
And there is a difference between it doesn't support 6d and it can't be extended to 6d
fossil ember
And there is a difference between it doesn't support 6d and it can't be extended...
True but may as well go native. X E.
carmine sigil
I'm not sure what you mean here by "native."
fossil ember
If it gets too late for you feel free to go. X E.
fossil ember
I'm not sure what you mean here by "native."
His is for Unity, I am using Godot and Rust. X E.
And native to more dimensions, not necessarily same math in 5d as 6d. X E.
carmine sigil
The entire discipline of linear algebra was developed to handle arbitrary dimensions, it is the same math
fossil ember
We shall see what works and what doesn't. X E.
carmine sigil
Well. Best of luck.
fossil ember
What would you suggest asking tomorrow?
carmine sigil
This might help https://github.com/godot-dimensions/godot-4d
fossil ember
From what I heard CodeParade uses matrices too. X E.
fossil ember
This might help https://github.com/godot-dimensions/godot-4d
Different kind of 4d+. X E.
carmine sigil
???
fossil ember
Thanks, I know the maker. X E.
fossil ember
???
Even the creator acknowledges orthogonal vs normal projection. X E.
Check out Godot ND by same person. X E.
carmine sigil
Instead, I'm going to go afk. Have a good evening.
fossil ember
Okay, so other people, how to solve the original and what is x and y and t?
midnight plankBOT
@fossil ember Has your question been resolved?
fossil ember
I just found out 4d golf is also a different type of 4d+ maybe. X E.
<@&286206848099549185>, I am doomed right?
obtuse mason
Yep
fossil ember
Well, how should I solve?
fossil ember
Also, if I can solve, what would be the way to define boundaries of the solution?
lament fox
midnight plankBOT
latent wadi
What have you tried
lament fox
What have you tried
part a is straitghforward
i think the closed sets are sets that contain all the non-open subsets of Q and ofc emptyset
so its not hard to see that the space is conneccted because a set cant be open and closed other than empty and entire space
but im not sure how to do part iii
fallow palm
what is this X E. thing?
lament fox
what is this X E. thing?
what X E
fallow palm
I just found out 4d golf is also a different type of 4d+ maybe. X E.
here
woeful jetty
hi does anybody understand U substitution bc i CANNOT understand ts 😭
lyric charm
scroll up a bit, some guy was adding X E at the end of all of his messages
that's just a him thing
midnight plankBOT
@lament fox Has your question been resolved?
midnight plankBOT
midnight plankBOT
Available help channel!
vapid jacinth
How would I answer this question without manually calculating for 90, 91, 92, ..., 100 and adding them all up? Calculators are allowed but not those features where you just plug in the parameters.
midnight plankBOT
How would I answer this question without manually calculating for 90, 91, 92, .....
vapid jacinth
Is there some sort of formula for this?
vapid jacinth
How do I do that?
cerulean oyster
Do you already know what normal distribution is?
topaz epoch
How do I do that?
a convenient normal distribution calculator I use is desmos
vapid jacinth
Do you already know what normal distribution is?
No I don't.
vapid jacinth
a convenient normal distribution calculator I use is desmos
There's specific guidelines for the type of calculator you can bring in. I got a Casio FX-85 CW.
I think only the 83 and 85 are allowed.
cerulean oyster
The fact that it says "poor performance" makes me suspect that you are not really supposed to pull off all the calculations as Bin.dist
vapid jacinth
Ah so something like that most likely won't come up in the exam.
cerulean oyster
If it happens and you still didnt learn about bin -> normal approximation
vapid jacinth
Because I recall the lecturer didn't really calculate it, he just kinda went over it.
cerulean oyster
then you will then just have to crunch up a lot of numbers
midnight plankBOT
midnight plankBOT
Available help channel!
jade magnet
wait so this is about circle theorems but
midnight plankBOT
wait so this is about circle theorems but
jade magnet
how do i prove that an angle BAC never changes
umbral timber
!xy
midnight plankBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
jade magnet
no matter where A is placed on the circle
latent wadi
no matter where A is placed on the circle
Do you know the inscribed angle theorem?
jade magnet
how do i prove that an angle BAC never changes
i know this is true but i want to know why
lyric charm
no matter where A is placed on the circle
you sure about this
jade magnet
i heard of it but might have forgot it
jade magnet
you sure about this
oh yeah mb
latent wadi
no matter where A is placed on the circle
Yeah this isnt true
lyric charm
cause rn even if we fix B and C then the size of angle A isn't independent of its placement on the circle
!xy
midnight plankBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
jade magnet
surely the size wont change as long as a doesnt cross the chord BC
if that makes sense
latent wadi
Yeah something like that
latent wadi
You might wanna search up "inscribed angle theorem"
And also search up its derivation, which is pretty basic
Or if you really want to, you can wait for a helper to come here and explain it to you
jade magnet
You might wanna search up "inscribed angle theorem"
ok so an inscribed angle is any angle that lies on the circumference of a circle?
or at least it looks something like that
hold on im trying to understand
latent wadi
ok so an inscribed angle is any angle that lies on the circumference of a circle...
It's an angle with 2 points on the circumference of the circle and the angle at the center of the circle
jade magnet
wait so is only the red part inscribed or are the blue parts inscribed too
not "parts" sorry
angles but angle when i say angle
surely it's dependant on the point they lie in?
idk no lie in isnt the right wor
word
latent wadi
What website are you reading
jade magnet
how do i say an angle is in specific points
latent wadi
What website are you reading
Like where are you searching this
latent wadi
I think you should just search up "what is the inscribed angle theorem"
latent wadi
https://www.youtube.com/watch?v=q9bhk8tyqiY
Instead of watching this kind of video, which shows how to apply the theorem.
jade magnet
I think you should just search up "what is the inscribed angle theorem"
brev i dont even know if i know the correct definition for an inscribed angle
timber sequoia
take 2 points on the circumference of a circle
connect those to the center of the circle to make a triangle
jade magnet
oh ok
timber sequoia
now connect the same two points to the other side on the circumference of the circle
jade magnet
wait so is only the red part inscribed or are the blue parts inscribed too
i just checked and basically an inscribed angle is any angle who's point lies on the circumference
timber sequoia
the inscribed angle theorem states that the angle made by the triangle at the center is double of the angle made at the circumference.
jade magnet
the inscribed angle theorem states that the angle made by the triangle at the ce...
yeah
i remember that
ok so the central angle is double the size of the inscribed angle
am i right to say this
jade magnet
any ideas to solve the question then?
well if it's a triangle surely if we added an extra point called d between the chord b and c then
am i correct to say that BDC is the central angle?
and if it's bdc would that imply the angle is 0 or 180
timber sequoia
is BC a chord or the diameter?
jade magnet
is BC a chord or the diameter?
how would that be the diameter
wait
is it supposed to be the diameter
is that a requirement
timber sequoia
you haven't given the question so I was asking
it's not a requirement
well if you want BDC to be the central angle then yes
jade magnet
for the inscribed angle theorem or the inscribed angle A being the same
idk what theorem you call the right one
but yeah
jade magnet
well if you want BDC to be the central angle then yes
ok
well if the central angle doesnt change
does this imply the angle BAC is always equal to 180/2
which is 90 degrees
but wait
surely in my example
timber sequoia
if BC is the diameter and D is the midpoint yes.
jade magnet
it isn't 90 degrees
jade magnet
if BC is the diameter and D is the midpoint yes.
ohh ok
what if bc isnt the diameter
how do i still prove it
timber sequoia
it's doesn't change much
given BC is a chord, how would you form a central angle?
jade magnet
given BC is a chord, how would you form a central angle?
ohh yeah of course you can just create a uhh what do you call it
congruent triangle?
similar triangle?
dont remember what exactly you call it but something like that
jade magnet
it would be isosceles, though, if you're concerned
why does it have to be isosceles
if we can move point A wherever so long as
timber sequoia
it doesn't have to be
jade magnet
its not below the chord BC
jade magnet
oh ok its fine
jade magnet
it'll just be a triangle
but still if it works for another triangle
why would it work for the triangle we're looking at
we proved it for a "different case" in my eyes
which doesnt feel right
timber sequoia
can you elaborate?
jade magnet
can you elaborate?
what i mean is to form a central angle- oh wait
i get it now
you add a point to the centre of the circle
lets call it D this time
and then we know from the inscribed angle theorem that if BC was the diameter aka the vectors BD and BC has the same direction
then BAC must always be 90 degrees on that case
timber sequoia
what you have here is a non-diameter chord
jade magnet
what you have here is a non-diameter chord
i know
but what i am saying is if its true for the diameter
it must be true for a non-diameter chord as well
timber sequoia
you could take it that way sure
jade magnet
you could take it that way sure
but i guess
so long as the angle BDC doesnt change
then BAC cant change
oh ok
i just overcomplicated it
basically yeah
"if BDC doesnt change BAC also cant change"
i understand now
but then the next thing to ask would be
how do we prove the inscribed angle theorem
because we did this all "assuming" the inscribed angle theorem is true
timber sequoia
there's a simple proof you can get just by searching but I can try to explain if you wish
here's a nice one
jade magnet
there's a simple proof you can get just by searching but I can try to explain if...
ok let me check
jade magnet
wait a second
midnight plankBOT
wait a second
jade magnet
but what if a was on the other side
what if you had this
surely you cant use the inscribed angle theorem this time
oh wait
is the inscribed angle theorem still true
is the reflex angle BOC still double BAC
nvm
its fine
for a second i felt it wouldnt be true
but it still would be
.solved
thorny folio
Hello great math fellows, here I am with another question(sorry).
So it's regarding inner product spaces.
Let say we have two inner product space <,>, and [,] on a linear space V. Assuming both satisfy all required axioms to be an inner product space of course.
I'm asking if is it possible that <,> inner product space has some extra porperties which [,] lacks? or viceversa? I mean some dedicated properties regarding an inner product space.
And if yes, giving some examples would be apprectiated.
midnight plankBOT
Hello great math fellows, here I am with another question(sorry).
So it's regard...
@thorny folio Has your question been resolved?
midnight plankBOT
@thorny folio Has your question been resolved?
fossil ember
Just to be clear, no need to be sorry about having questions, you learn, I am happy for you and others. X E.
thorny folio
Just to be clear, no need to be sorry about having questions, you learn, I am ha...
Much appreciated
fossil ember
Good luck solving. X E.
midnight plankBOT
@thorny folio Has your question been resolved?
thorny folio
So it seems it's not really possible. IDK. I tried some trial and error and playing around if I'm able to find anything. But seems the axioms are so strict that it's almost restricted us completely. IDK.
Math bless us all
.close
midnight plankBOT
midnight plankBOT
Available help channel!
buoyant fjord
midnight plankBOT
buoyant fjord
Let $e \in E$. Let $D={d(e,x) \mid x\in E, x \neq e}$. Since $D \subset \mathbb{R}$ and $0$ is a lower bound of $D$, $\inf(D)$ exists. Assume, for sake of contradiction, $\inf(D)=0$.
oh
grand pondBOT
Julian
buoyant fjord
Assuming $\inf(D)=0$ leads to us being able to show $\lim_n p_n=e$ possibly?
grand pondBOT
Julian
brave gorge
Assuming $\inf(D)=0$ leads to us being able to show $\lim_n p_n=e$ possibly?
What’s e?
buoyant fjord
**Julian**
right here
brave gorge
Did you try it
buoyant fjord
i was trying to figure it out
i think
at the very least we can get a subsequence
that converges to e
which already provides the contradiction because we cant have subsequences of $p_n$ converging to $e$ since $l \notin E$
grand pondBOT
Julian
lavish venture
Let $e \in E$. Let $D=\{d(e,x) \mid x\in E, x \neq e\}$. Since $D \subset \mathb...
this doesn’t make any sense
how does this in any way answer the question?
are you saying D is an open cover of E?
not sure what you’re trying to get at here
buoyant fjord
no
lavish venture
what are you doing then?
is there another question you’re answering or something?
buoyant fjord
if the infimum isnt 0
that means we can do the union of the sets of the infimum(D)/2 neighborhood of every point in E
neighborhoods are all disjoint then
@lavish venture
infimum /2
lavish venture
ok and why doesn’t that have a finite subcover?
brave gorge
I don’t really understand the construction at all to be honest
buoyant fjord
because E is infinite so we can construct a bijection
lavish venture
can’t it be that all the other points are within inf(D) of each other but farther than inf(D) from e?
like you fixed e
and this inf distance only applies for distances of points relative to e
lavish venture
like let’s say we’re on the real line and i have a cluster of points around 1 and let’s take e to be 2 or something and for the sake of argument say the closest point is x = 1.01
buoyant fjord
oh yeah that makes no sense
oh wait
we need a different radius for each neighborhood
sorry im like brain farting
uh
we can do inf of
for each point
brave gorge
Assuming ftsoc that inf(D) is zero wont really do anything
buoyant fjord
Assuming ftsoc that inf(D) is zero wont really do anything
what i meant is for every e in E we take $inf({d(e, x) |x \neq e})$
grand pondBOT
Julian
buoyant fjord
and then
we can use inf/2 for the neighborhood of that e
and thats how we get the cover
lavish venture
yes that makes more sense
lavish venture
and it’s important that l isn’t in the sequence or else the inf would be zero
and you wouldn’t cover that point
buoyant fjord
yeah cuz
lavish venture
the inf can’t be zero because the limit is unique and l isn’t in E
brave gorge
Yup
lavish venture
only one cluster point
lavish venture
it’s important you don’t use one particular inf uniformly though
as you just fixed
buoyant fjord
yeah
brave gorge
The broader point is that any covering such that each open set in the cover only captures a finite number of points will work
buoyant fjord
for some reason i like disjoint ones
lavish venture
easier to construct
buoyant fjord
easier to construct
well i dont have to do much at the end except show the disjointness
lavish venture
since p_n is arbitrary
lavish venture
well i dont have to do much at the end except show the disjointness
this is fairly straightforward
midnight plankBOT
midnight plankBOT
Available help channel!
lusty python
How do I solve for $(m, n) \in \mathbb R^2$?
$8n^2 + m^2 + 8mn - 24n + 4 = 0$
grand pondBOT
1 divided by 0 equals Infinity
lusty python
original question
find $(m, n) \in \mathbb R^2$ such that the equation above has roots $x_1 = x_2$
grand pondBOT
1 divided by 0 equals Infinity
lusty python
I started out by working out $\Delta$ for this, which is $8n^2 + m^2 + 8mn - 24n + 4$
grand pondBOT
1 divided by 0 equals Infinity
next bobcat
**1 divided by 0 equals Infinity**
that’s not the discriminant btw
@lusty python you here?
lusty python
im here
next bobcat
so first you must find $b^2 - 4ac$
grand pondBOT
BKB
next bobcat
did you do that
lusty python
**1 divided by 0 equals Infinity**
main problem is how do i solve for $m, n$ 😭
grand pondBOT
1 divided by 0 equals Infinity
lusty python
**1 divided by 0 equals Infinity**
it's this boy
next bobcat
you sure
lusty python
that’s not the discriminant btw
$\Delta = b^2 - 4ac$
grand pondBOT
1 divided by 0 equals Infinity
next bobcat
oh ok
just retry it because methinks it’s a different value
and then try to equate it to 0
that’s pretty much it
vivid yoke
,w simplify 4(n+1)^2 -4(2n(2-m)-m^2-n^2)
twilit jetty
How do I solve for $(m, n) \in \mathbb R^2$?
$8n^2 + m^2 + 8mn - 24n + 4 = 0$
your discriminant is incorrect, it should be m^2 + 2n^2 + 2mn - 2n + 1 = 0
no spoilers
lusty python
different value
vivid yoke
no spoilers
Okay...
lusty python
different value
can you check the discriminant?
twilit jetty
this is correct
now you need to find out when this = 0, to do that you can first divide both sides by 4 for extra convenience
next bobcat
your discriminant is incorrect, it should be m^2 + 2n^2 + 2mn - 2n + 1 = 0
That’s what i thought
twilit jetty
then from there, theres a few ways you can go about this
think about trying to find or form perfect squares
twilit jetty
if you dont know where to go from here, I can tell you what is essentially the answer to what the perfect squares are
np
twilit jetty
alr, which ones
lusty python
$(m + n)^2 + (n - 1)^2 = 0$
grand pondBOT
1 divided by 0 equals Infinity
