So 45 + 45 = 90 but we need another 90 so thats
#help-13
frosty grail
Yes
mild narwhal
Alr ty
frosty grail
Good
sharp lotus
.close
ionic finch
Why does the definition of prime ideal requires a commutative ring ?
ionic finch
The definition i have says, Let R be a commutative ring, I is a prime ideal if ab in I, for some a,b in R then it follows either a in I or b in I.
<@&286206848099549185>
lyric narwhal
!nogpt
cedar kilnBOT
Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).
wispy talon
Thanks 👍🏼
ionic finch
Does that some connections to right and left ideals?
scarlet sand
There is a definition of prime ideal for noncommutative rings
The primary use of the notion of prime ideals are for algebraic geometry and algebraic number theory. We generally study commutative rings here
ionic finch
I mean, why to define prime ideals with in commutative rings?
what is the motivation
or what goes wrong in non commutative rings?
scarlet sand
you are asking why are prime ideals important?
oh
In noncommutative rings, you have left-ideals and right-ideals
which, pardon the pun, is not ideal
just as you have left and right cosets in groups
ionic finch
so if it’s non commutative, one of a or b may belong either left or right somehow
scarlet sand
umm
the definition of prime ideal for commutative rings is that if ab is in I, then a or b is in I
now, we have that we have to consider left ideals and right ideals I
ionic finch
when i say an ideal exists i really mean both ideals exists and equal?
scarlet sand
oof, that is not a good definition for ideals in a noncommutative ring
Do you understand the definition of ideals in a noncommutative ring?
ionic finch
I think the definition is have is more general
scarlet sand
If you force the left and right ideal from a generator to be equal, you lose a lot of ideals
ionic finch
applies to both
For a ring R, I is an ideal, if I pick any r from the ring R, operate it in the left or right their product ar or ra will belong to I.
scarlet sand
Well, the definition of your ideal is something called a two-sided ideal
but the definition of a generic ideal over a noncommutative ring is somewhat broader
In some sense, you have the left and right product to consider, so you have the capactiy for more ideals
ionic finch
For any r in R and a in A, if ra is A then it’s a left ideal
I suppose this is def im looking
scarlet sand
yes, that is the correct definition
ionic finch
so as you said, for non commutative i might have more ideals
scarlet sand
this definition is different from the previous definition "For a ring R, I is an ideal, if I pick any r from the ring R, operate it in the left or right their product ar or ra will belong to I."
ionic finch
yes, i see that
scarlet sand
yes
ionic finch
this definition is different from the previous definition "For a ring R, I is an...
how does the notion of prime ideal is tightly defined here
scarlet sand
What does tightly defined mean?
ionic finch
just defined
scarlet sand
Ah, okay. Good question.
ionic finch
maybe define, if ab in a left ideal then a or b in the left ideal then I is prime left ideal?
or equivalent for right
scarlet sand
P is prime when the following is satisfied: if the product of ideals AB is contained in P, then at least one of A and B is contained in P.
ionic finch
I see that unlike the definition of ideals as in forward direction, prime ideals comes converse direction
scarlet sand
you've defined a completely prime left ideal.
this is a notion that can also be useful in places. Once you leave commutative algebra, more weird defintions arise
i've defined a general prime ideal over noncommutative rings
ionic finch
So my definition is defined in commutative rings to avoid the ambiguity of left or right ideals?
lyric solar
I need a bit of help showing if this converges or diverges
cedar kilnBOT
I need a bit of help showing if this converges or diverges
lyric solar
I want to apply the Leibniz test but sin is not a strictly decreasing function
lyric solar
how so, I'm confused
ohhhhhhhhhh
1/n will keep having a smaller value
so sin of that will be smaller and smaller
.close
cedar kilnBOT
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young flame
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young flame
How should I go about proving this? I have no idea what to do
should I prove this by definition (epsilon delta)
rose oak
How should I go about proving this? I have no idea what to do
this kind of proof is easily solved by assuming towards contradiction. suppose that the series of a_n + b_n converges, and try to get to the (contradictory) conclusion that b_n converges :)
using other theorems you learned about convergence of series
cedar kilnBOT
@young flame Has your question been resolved?
rose oak
@young flame ATC lim (a_n + b_n) converges.
Then, as lim(a_n) and lim(-1) also converges, by limit algebra theorems we have that lim(-a_n) converges
Thus lim(a_n + b_n) + lim(-a_n) = lim(a_n + b_n - a_n) = lim(b_n) converges, in contradiction to the assumption
young flame
<@728716838363594772> ATC lim (a_n + b_n) converges.
Then, as lim(a_n) and lim(...
oh that's smart
thanks
gentle yacht
A triangle ABC has an angle A equal to 50,2 degrees, and a side AB = 5,7 cm. Set BC = a. Which values can a be so that there can be
- one
- two
- zero triangles that fulfill the task's requirements?
gentle yacht
So far, I believe that I have calculated the values for a in which we get two triangles that can fulfill the requirement
4.4 < a < 6.8
No, I think that's also wrong...
I don't know where to begin, actually
cedar kilnBOT
@gentle yacht Has your question been resolved?
earnest hound
How would you do part a
cedar kilnBOT
How would you do part a
odd verge
How would you do part a
hi can I help?
cedar kilnBOT
@earnest hound Has your question been resolved?
prisma glacier
I’ve been stuck on this problem for quite some time and wondering if anyone could help confirm my answer below. Thank you!
west mason
prisma glacier
.close
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proven sierra
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proven sierra
I help with this one
cedar kilnBOT
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proven sierra
No like a biryani my problem don’t get solved easily
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proven sierra
I’ll close it for now
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hot cloak
scarlet sand
YAY!! group theoryyyy
can you find a way to pair natural numbers with subgroups of Z
well first, can you give me an example of a subgroup of Z
hot cloak
a subgroup of Z could be {0 , 1 , -1}
scarlet sand
thats a good thought, but Z is only a group under addition
hot cloak
a subgroup of Z could be {0 , 1 , -1}
is this not a subgroup under addition?
it has closure
an identity
scarlet sand
1+1=2, which is outside the subgroup.
scarlet sand
this is a good exercise and one that is valuable so you should continue thinking
hot cloak
i guess nZ is a valid subgroup
hot cloak
is nZ the only subgroups?
scarlet sand
if I give you any integer n, nZ is a distinct subgroup, proving the forward direction
Try to show that nZ are the only subgroups
This is also a good exercise in developing intuition of relatively prime numbers
hot cloak
i guess a proof by contradiction would make sense here
assume nZ are not the only subgroups
then there exists a subgroup where there are elements a,b such that gcd(a,b) = 1
scarlet sand
yes!! one more step
scarlet sand
It is the fact that if you have two coprime integers in the subgroup, then you can reach any integer by some linear combination of those integers
for example, if I have 5 and 7 in my subgroup
hot cloak
oh right i remember this
scarlet sand
How can I get to 1?
hot cloak
bezouts lemma right
scarlet sand
yes!
hot cloak
5x3 - 2x7
scarlet sand
exactly, conclude the proof by bezout's lemma and we're done
therefore, nZ are the only subgroups of Z
reverse direction
hot cloak
then there exists a subgroup where there are elements a,b such that gcd(a,b) = 1
how can we be certain that this holds
hot cloak
how can we be certain that this holds
in fact this is actually wrong lol
scarlet sand
right thats not necessarily true
but close enough lol
Suppose a subgroup contains a,b then gcd(a,b) must be contained in the subgroup
why?
hot cloak
because
scarlet sand
Once we answer that, we repeat the process for all the numbers and the gcd of all the numbers, denote it g, must be within the set. We can then show that g generates the whole subgroup and therefore the ideal is gZ
Maybe this proof I found online makes more sense
hot cloak
Suppose a subgroup contains a,b then gcd(a,b) must be contained in the subgroup
i think i figured this out
by euclids algorithm
hot cloak
you'll end up with
gcd(a,b) = gcd(qa+kb,1) = qa+kb where q and k are some integers
because throughout the algorithm you'll subtract combinations of a and b
oh i see or this
hot cloak
Once we answer that, we repeat the process for all the numbers and the gcd of al...
Wdym repeat the process for all numbers?
scarlet sand
we simply take the gcd of all elements in the subgroup
this is in fact finite and a number within the subgroup.
Therefore, it generates the whole subgroup
cedar kilnBOT
@hot cloak Has your question been resolved?
scarlet sand
well, heres the thing
gcd(gcd(a,b),c)= gcd(a,b,c)
so we know gcd(a,b) is in the set and c is. Therefore, gcd(gcd(a,b),c) is in the set, we can repeat this
pure flume
Mk so i have a question abt notation
cedar kilnBOT
Mk so i have a question abt notation
pure flume
this is what I was taught for the form of a transformed function
where a=vertical dilation
k= horizontal dilation
d=horizontal shift
and c = vertical shift
what's confusing me is:
say I have f(x) = (1/x) + 2
2f(x) is not only dilating the function but its also shifting the horizontal asymptote
why is this?
scarlet sand
,w graph 1/x+2
wraith daggerBOT
scarlet sand
,w graph 2(1/x+2)
wraith daggerBOT
patent belfry
2f(x) is not only dilating the function but its also shifting the horizontal asy...
it doesnt tho?
scarlet sand
do you mean the horizontal asymptote? or the vertical asymptote?
patent belfry
oh horizontal?
pure flume
i meant the horizontal asymtptote lols
patent belfry
oh then it shifts it bc the +2 also gets doubled
scarlet sand
right it is stretched relative to the x-axis not relative to the asymptote. Since the asymptote is originally at y=2, it gets stretched to y=4
pure flume
Mk so i have a question abt notation
in relation to this
a=2
k=1
d=0
c=2
so from the parent function
patent belfry
no
its just
a=2
compared to f(x)
compared to 1/x it's a=2, c=4
and k=1 d=0 obv
cedar kilnBOT
pure flume
ty
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Available help channel!
turbid spoke
how
livid hound
angle addition / subtraction
turbid spoke
how
weak otter
what do you know about radians
turbid spoke
full circle is 2pi
turbid spoke
angle addition / subtraction
show me
sharp lotus
full circle is 2pi
so that's the red right? what about to get up to the red + green parts?
turbid spoke
3pi
sharp lotus
yeah
turbid spoke
after?
sharp lotus
and then going from - x axis to - y axis (the blue) would be another pi / 2 right? but you're pi / 6 short of that, so what would be the rest of the angle?
turbid spoke
and then going from - x axis to - y axis (the blue) would be another pi / 2 righ...
always is pi/2?
sharp lotus
always is pi/2?
well its pi to go halfway around like you said isn't it (half of 2pi) so to go a quarter of the way around is half of that isn't it?
2pi - all around
pi - halfway around
pi / 2 quarter of the way around
turbid spoke
well its pi to go halfway around like you said isn't it (half of 2pi) so to go a...
how
can show me othjer example
make a example
sharp lotus
what's unclear about this 🤔
half of a half is a quarter isn't it
i don't really know how else to break it down
livid hound
adding/subtracting known/given angles
cedar kilnBOT
@turbid spoke Has your question been resolved?
wicked plinth
cedar kilnBOT
wicked plinth
Can someone explain e to me i don’t get why he calls it divergent by comparison since it’s look like it’s less than the divergent p series
main needle
hmm it's less by that little +3 bit in the denominator, which you can deal with by like comparing to say 1/(2n^1/2) which should be less after n=3
it's essentially greater than a small constant times the divergent p series
main needle
uhh maybe there's a convention that restricts you that way but if something is divergent then so is it times a constant
wicked plinth
Oh ok but when we were doing integral apps we couldn’t do that when we were just comparing their integrals using p series
Is it because this is a series so it keeps going
main needle
oh can you be more specific
with divergent things there's a lot of ways to stay divergent
main needle
hmm
formally there's the limit comparison test which is basically the generalized version where you're allowed to compare to a constant times the series
cedar kilnBOT
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ruby cape
my friend has a question related to the spinning of a roulette wheel
suppose every time i spin a roulette wheel there is a 50% chance its black and 50% chance its red
suppose red had just spun 10 times in a row
my friend argues that its 50 50 chance that its red or black will show up next
I argue that since red just spun 10 times in a row, its more likely that the roulette wheel spins black since the chance of red hitting 11 times is 0.048828125% chance
who is right
patent belfry
my friend has a question related to the spinning of a roulette wheel
suppose ev...
Your friend
This is known as the gambler’s fallacy
It is a misunderstanding of the law of large numbers
Each individual roulette spin has 50/50 odds
@ruby cape understand?
dusk goblet
it isn’t bound to be black
ruby cape
but i dont understand
if i spin the roulette wheel twice and the first is red is it not more likely that black will come up next
cause the chance of hitting red twice is 25%
dusk goblet
each spin is independent of the previous ones
patent belfry
but i dont understand
if i spin the roulette wheel twice and the first is red is...
No, each spin is independant
dusk goblet
independent events
patent belfry
OVER TIME the number of spins TEND to a perfect split
ruby cape
okay each spin is independent
but as a collection is it more likely that black is next?
dusk goblet
no
patent belfry
No
ruby cape
OVER TIME the number of spins TEND to a perfect split
does that have to do with normal distribution or whatever its called
dusk goblet
without knowing that 10 times it was red
and you spun it
would it be more likely to be black
even if you didn’t know
patent belfry
does that have to do with normal distribution or whatever its called
It has to do with central limit theorem probably
Idk
But its more about law of large numbers
ruby cape
even if you didn’t know
if you didn't know then yes i agree that its 50 50
lyric leaf
these kind of questions confuse me because isnt it always just semantics/language
patent belfry
if you didn't know then yes i agree that its 50 50
So does knowing change the outcome?
lyric leaf
people say 'the chance of landing on black right now' but they mean P(A|B) dont they
isnt it just that
dusk goblet
if you didn't know then yes i agree that its 50 50
so why would you knowing make any difference
dusk goblet
yea we’re assuming that red and black have equal chance
it’s highly unlikely to get black 10 times in a row
and you might be inclined to think it’s rigged because of that
or red right
ruby cape
yeah
sharp lotus
the unlikely event already happened (same 10 times in a row), now it's 50/50 whether it's gonna be 11 of the same in a row or change
ruby cape
alright i understand thanks everyone
cedar kilnBOT
@ruby cape Has your question been resolved?
rose cliff
cedar kilnBOT
rose cliff
My proof is just to say that since they are 3 consecutive numbers, at least one is a multiple of 2 and at least one is a multiple of 3, therefore the product of the 3 numbers will be of form 6N
It feels sketchy for some reason though anyone have a better proof
rose cliff
Theres like not enough math and all the proof working outs ive seen all use algebra or smth to prove
Heres the worked solution
what do they mean by "integer since all binom coeff are integers"
sharp lotus
that feels like such an overworked proof, i like yours better
rose cliff
Yeah most of the suggested solutions are all like this
Do you know what they mean by the binom coeff
sharp lotus
rose cliff
They are doing n+2 choose 3 but where does the coefficients come in
sharp lotus
they're using the fact that they are rewriting the expression in that form on the right, which is known to always be integers
but all this seems unnecessarily complicated lol, like you say three consecutive factors are going to contribute both a factor of 3 and one of 2, it's that simple...
lyric narwhal
The worked solution is more rigorous though
crimson sedge
My proof is just to say that since they are 3 consecutive numbers, at least one ...
you can do this if you mention 2 and 3 are coprime
but when you generalise it, product of 100 consecutive integers, divisible by 100!
it's not as clear
because for example 9 is divisible by 3 and 9
so you can't say at least one is a multiple of 2,3, etc 100
rose cliff
Wdym I dont follow
sharp lotus
The worked solution is more rigorous though
there's nothing unrigorous about the simple argument imo
you can say that if your numbers are n, n+1 and n+2
n can be written as one of the following form n = 3m or n = 3m+1 or n = 3m+2
if the first one then n is divisble by 3, if the first second one then (n+2) is divisble by 3, if the third one then (n+1) is divisible by 3. In all cases (n)(n+1)(n+2) is divisible by 3
Similar argument can be made for why n(n+1)(n+2) is divisible by 2
if it is divisble by both 2 and 3, then it is divisble by 6
crimson sedge
in this case, yes, because its such a simple example
if you generalise to product of n consecutive integers is divisible by n!
you must use the binomial coefficient method
rose cliff
so you can't say at least one is a multiple of 2,3, etc 100
Wdym by this
sharp lotus
if you generalise to product of n consecutive integers is divisible by n!
ok but the question isn't about generalizing a method its about a specific problem...
crimson sedge
it isn't
thats why i said in this case is fine
but mention 2 and 3 are coprime to each other
otherwise for example 9 is divisible by 3 and 9, doesn't mean its divisible by 3x9
crimson sedge
by induction
rose cliff
Oh true
For this one i let p = 2m + 1
square would be 4m^2 + 4m + 1 = 4m(m+1) + 1
since m and m+1 are consecutive one is a multiple of 2
8N + 1
How would i get a 3
cedar kilnBOT
@rose cliff Has your question been resolved?
rose cliff
.close
little hull
Is Out of Bag (OOB) an error test or training error, and why? (In terms of random forest)
OOB is a test error since it is an estimate of the model's performance on unseen data ("out of bag") and is used as an approximation of the testing error.
It this the correct way to think about this question?
cedar kilnBOT
Is Out of Bag (OOB) an error test or training error, and why? (In terms of rando...
@little hull Has your question been resolved?
cedar kilnBOT
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cosmic relic
cedar kilnBOT
cosmic relic
having trouble getting rid of the square root
cosmic relic
yeah exactly
sage field
are you sure the previous working is correct?
cedar kilnBOT
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slow otter
Question 25
cedar kilnBOT
Question 25
pastel vault
Question 25
,rotate
wraith daggerBOT
slow otter
Where do i start
pastel vault
Where do i start
Can you find the points where they intersect?
slow otter
You can also check using Desmos etc
I havent learnt that yet
slow otter
wraith daggerBOT
south
pastel vault
So $x^2 - 2x + 2 = 5$
wraith daggerBOT
south
slow otter
Oh
pastel vault
Yep don't always expect whole numbers all the time
But yeah when you get answers with radicals then usually you did something wrong
slow otter
Ok now i can get AB
pastel vault
Yeah and you should be good from here
slow otter
Now how do i find P
pastel vault
Ah so if AP/PB = 3/1, that means AP/(AP + PB) = AP/AB = 3/(1 + 3) = 3/4
So that means the ratio of both the x-coordinates and y-coordinates must also be that
slow otter
So i find length of AB
pastel vault
(Long as in the x-direction and the y-direction)
pastel vault
√((x2 – x1)² + (y2 – y1)²)
No
Not for this question
But that formula is correct btw
slow otter
So what do i do
pastel vault
So what do i do
Okay if I have the points (1, 2) and (3, 4)
What is the distance that separates them in the x-direction?
Draw a picture of the points if you're stuck
slow otter
pastel vault
Yeah so what's the distance separating them in the x-direction
slow otter
2 and 4
pastel vault
2 and 4
Well, that's in the y-direction
Yeah so the distance in the y-direction is 4 - 2 = 2
pastel vault
Im soo confused rn
Yeah, so basically we have similar triangles
slow otter
So i do the same for the question
pastel vault
We know the length ratio of small triangle / big triangle = 3/4 already
So we know the x-distance from point P to point A (-1, -4)
is 3/4 that of the x-distance from point B to point A
And the y-distance also has the same ratio, 3/4
slow otter
Brain is not braining😭
cedar kilnBOT
@slow otter Has your question been resolved?
raven shard
is this an indeterminate form? if so, which?
wraith daggerBOT
guinearW
white latch
Well I just thought about it intuitively. If I expanded it, I'd have x^(2020) - x^2020sin(2020x) and that would equal to 0
pastel vault
And 0 * (1 + 0)
white latch
if you think about it intuitively just subbing in 0 as south said
yeah.
my exact point.
Wdym it has a fraction?
pastel vault
Yeah also intuitively x^(2020) approaches 0 really quickly as soon as |x| < 1
There's no fraction here
white latch
Yeah.
wraith daggerBOT
guinearW
white latch
You can if you want to, but its good to know multiple ways of finding the limit of something.
Well you can expand it.
Right.
What confuses you?
Why is it weird? lim x --> 0 (x^(2020) + x^2020sin^2(2020x) = 0^2020 + 0^2020(sin^2(0) = 0 + 0(0). = 0 + 0 = 0
pastel vault
Maybe the 2020s are freaking you out
white latch
Yep.
pastel vault
But yeah don't be intimidated
white latch
Pandemic, we feel you man.
pastel vault
If you can substitute it works
If you can't, well try thinking of something else from the techniques you've seen
You mean .close
cedar kilnBOT
pastel vault
Nwnw
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Available help channel!
left cove
can somebody solve e^y+1=1+6/(e^2y+1)
cedar kilnBOT
can somebody solve e^y+1=1+6/(e^2y+1)
pastel vault
can somebody solve e^y+1=1+6/(e^2y+1)
Yes, it's actually a quadratic in disguise
Can you guess what you could substitute to make it a quadratic?
Also I think you need brackets
Do you mean $e^y + 1 = 1 + \frac{6}{e^{2y} + 1}$
wraith daggerBOT
south
pastel vault
Or e^(y + 1)
left cove
Or e^(y + 1)
oops yea this one
pastel vault
Wait forget what I just said
It's technically solvable using the cubic formula
If you sub in u = e^y
But it's messy
pastel vault
This is more of an intelligence test and not maths, haha
It's (bottom left - top right) * (top left - bottom right)
sage field
i wouldnt say this is math
pastel vault
There you go, I know I shouldn't give out the answer but doing this will make you learn nothing (other than making you doubt your ability ig)
sage field
its only recognising pattern problem
pastel vault
Yes I can't say the word "eye queue" cause of a filter on the server
rare ridge
So harsh, was urgent and my brain isn t working after sessions(I am at architecture and we get this type of questions, some are mathematical some arent so mathematical) Sorry i tought is math…
pastel vault
But this is literally what you do in that kind of test
sage field
Yes I can't say the word "eye queue" cause of a filter on the server
what filter
remote salmon
why tho
pastel vault
‘We do not allow discussion of [] on this server, due to the topic frequently starting heated, unproductive, and elitist arguments.’
sage field
So harsh, was urgent and my brain isn t working after sessions(I am at architect...
its ok man
remote salmon
alright
cedar kilnBOT
Available help channel!
lavish pilot
I got the answer for cartesian equation which is 16/x^2 but didn't understand how the domain would be 0<x<8 as I only got that x>0, I substituted the range for t in 8cost to get the domain of x but the answer says it is 0<x<8
sage field
I got the answer for cartesian equation which is 16/x^2 but didn't understand ho...
see
you have t in the interval -pi/2 to pi/2
now plot the graph for that with x = 8cost
and you will get the range
lavish pilot
ah ok but how do you get 8 then from that?
because pi/2 is 0 for x is what I understand?
sage field
ah ok but how do you get 8 then from that?
range of cos x is [0,1] in between -pi/2 and pi/2 interval
so range of 8cosx is [0,8] in between that interval
basically multiplied [0,1] with 8
lavish pilot
ohh that makes sense then, thanks
pastel vault
Yeah basically -pi/2 < t < pi/2 only gives you the top half, the positive half of cos x
No worries
lavish pilot
.close
cedar kilnBOT
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marsh lake
are both series ok to approximate ln(1+x)?
cedar kilnBOT
are both series ok to approximate ln(1+x)?
marsh lake
i got the red line one from wikipedia
black one by myself
they have the same values from -1 to 1
cedar kilnBOT
@marsh lake Has your question been resolved?
cedar kilnBOT
@marsh lake Has your question been resolved?
warm crescent
aren't both intrinsically the same
marsh lake
aren't both intrinsically the same
really? how
cedar kilnBOT
elder sparrow
calculator
young shuttle
without a calculator?
elder sparrow
approximation?
crimson sedge
Anyone wanna help me with my math assignment
crimson sedge
how tho? 😭
patent belfry
about 5.65
elder sparrow
wait r u 13?
patent belfry
do you know prime factorization
elder sparrow
💀
patent belfry
dont bother him on it
he just wants math help
okay so
the easiest way
is to just try numbers
and get close
so you can see that 5*5<32
so try 5.5*5.5
i recommend you learn prime factorization
thats more important than finding square roots
elder sparrow
the most efficient algorithm be like
patent belfry
splitting a number into it's prime factors
all numbers can be uniquely factored like this
you know what a prime number is and what a factor is right
so lets take an example
18
18 can be written as 2*3*3
and those are all prime
thats the prime factorization of 18
8
genes
idfk
anyways
"i dont fucking know"
patent belfry
anyways, so 18
18 is 2*3*3
so imagine we want to find the square root of 18
yes
so you know how $\sqrt{ab} = \sqrt{a} * \sqrt{b}$ right
wraith daggerBOT
hired
patent belfry
well, we can prove it easily. The definition of square root of ab is that if you square it, you get ab. Try squaring sqrt(a)*sqrt(b) and see what you get
what
how do you know square root
patent belfry
...
elder sparrow
how do u compute the area of a square?
patent belfry
ok
thats an area for rectangle
but for a square all sides are same
what are length and width
add what
AREA
of a square
patent belfry
arent you violating discord tos then
im 14
yes
so the area of a square is (side) squared
patent belfry
which is side*side
quiet plover
square root of any number you mean?
patent belfry
you joined 3 years ago blud
oh yeah before i was breaking tos
quiet plover
lol
quiet plover
square root of any number you mean?
yes there is a method
patent belfry
ok so do you get squaring
patent belfry
first lets take care of squaring
patent belfry
imagine a square with side lengths A
what is the area
of the square
a^2
yes
a squared
ok
so now you get squaring
patent belfry
now a square root is like "imagine you are given a square of area A. what is the sides of the square"
quiet plover
ok just give a number
patent belfry
like the actual value
he didnt know what squaring was
quiet plover
now a square root is like "imagine you are given a square of area A. what is the...
sqrt(A) 😐
it just stops there
patent belfry
square root of A
quiet plover
oh lol
patent belfry
ok first lets prove a property of square roots
patent belfry
that $\sqrt{AB}=\sqrt{A}\sqrt{B}$
quiet plover
amazingg
wraith daggerBOT
hired
patent belfry
lets prove this
quiet plover
no im confused
like what is the question
do you want help on understanding square roots
patent belfry
so we know that $\sqrt{AB}\sqrt{AB}=AB$
wraith daggerBOT
hired
patent belfry
do you get this
quiet plover
oh i thought he was asking on how to get square roots of a non-square number
ok cool
hmm do you have a grasp of what kind of questions they can ask?
quiet plover
like square roots of actual numbers or like just problems involving square roots?
patent belfry
is that if you multiply sqrt(a)sqrt(a)=a
patent belfry
like square roots of actual numbers or like just problems involving square roots...
he wants to understand square roots
ok so then by definition
sqrt(ab)sqrt(ab)=ab
quiet plover
he wants to understand square roots
yeah but there has to be a place where he found square roots and got stumped for that
and im asking where that first happened
patent belfry
ok so
quiet plover
ok bruh 😭
patent belfry
now try finding $\sqrt{a}\sqrt{b}\sqrt{a}\sqrt{b}$
wraith daggerBOT
hired
wraith daggerBOT
hired
patent belfry
how did u get that though
patent belfry
**hired**
ok so just multiply this out
plus?
when did we get addition
wait
where did i put addition
did i accidently put it somewhere?
i didnt
yes, but do you get WHY
patent belfry
**hired**
do you get WHY this is ab
its sqrt(a)*sqrt(a)*sqrt(b)*sqrt(b)
what is sqrt(a)*sqrt(a)
why maybe
quiet plover
is this understandable enough
patent belfry
ok first we need to solve this property
understand it
that sqrt(AB) = sqrt(A)*sqrt(B)
its not
patent belfry
no
patent belfry
sqrt(a)*sqrt(b)=sqrt(ab)
thats what we want you to understand
so we do this by squaring both sides
quiet plover
is this understandable enough
this
sage field
read it again
sage field
so what is sqrt(ab)*sqrt(ab)
read this again
quiet plover
$\sqrt{x}\sqrt{y} = \sqrt{xy}$\\
so, $\sqrt{ab}\sqrt{ab} = \sqrt{(ab)^2}=ab$
wraith daggerBOT
sage field
no
quiet plover
ok 👍
sage field
Wdym by leave square roots
patent belfry
ok sure
sage field
it's just definition
sage field
you cant just say no what 😭
I mean don't leave square roots
quiet plover
ok ok
sage field
470?
patent belfry
ok so
quiet plover
so in one second it increases by how many metres
sage field
Then why are you asking
patent belfry
lets put this in an equation
patent belfry
$470m = 200m + 3m/s$
wraith daggerBOT
hired
patent belfry
so $470=200+3s$
wraith daggerBOT
hired
patent belfry
s being seconds
quiet plover
ok wait
patent belfry
a miner is mining
patent belfry
he's 20 meters deep
he mines at a rate of 1 meter per hour
how long until he's 100 meters deep
assuming he mines at a constant rate
exactly
quiet plover
yep
patent belfry
so you are good at this
quiet plover
yup
quiet plover
ok then what about this
a room is uniformly cooled down for an experiment. the initial temperature of the room is 27 degrees celsius and it uniformly decreases by 5 degrees every hour. if the cooling starts at 8 am, then at what time will the temperature be -8 degrees celsius?
what
that IS english
patent belfry
thats
quiet plover
k
a room is being cooled. the starting temperature is 27 degrees celsius and it decreases by 5 degrees every hour. if the cooling starts at 8 am, then when will the temperature be -8 degrees celsius?
patent belfry
already in english
quiet plover
yes
amazing
you are good at this
starting temperature is 27
reduces by 5 per hour
so 27 - 5h
easy
and the result is -8
so 27 - 5h = -8
same thing
patent belfry
you drop a ball from 7 meters up and it drops at 3m/s (we are ignoring realistic gravity here). When will it reach 5 meters
quiet plover
you drop a ball from 7 meters up and it drops at 3m/s (we are ignoring realistic...
bruh
crimson sedge
you drop a ball from 7 meters up and it drops at 3m/s (we are ignoring realistic...
am i allowed to asnwer
answer
quiet plover
thats 9th grade isnt it
patent belfry
am i allowed to asnwer
no
patent belfry
thats 9th grade isnt it
no?
patent belfry
its division
patent belfry
well, it does
quiet plover
it can, but the time is in decimals
patent belfry
it eventually falls to the ground
quiet plover
first try writing the equation
patent belfry
so at some point it reaches 5 meters
quiet plover
starting position is 7
decreases by 3 m per second
7 - 3s
ok 👍
correct
ok then that wasnt that bad
quiet plover
but for the grade in which they introduce square roots i think they would start with finding the square roots of actual numbers and not getting equations with square roots
but eh
it doesnt hurt to learn
quiet plover
hm
patent belfry
ok first we need to prove one property of square roots
thats really important
the property that $\sqrt{A*B}=\sqrt{A}*\sqrt{B}$
quiet plover
Find the hypotenuse of a right triangle ABC (right-angled at C), when AC = 12 m and BC = 5 m.
huh?
patent belfry
Find the hypotenuse of a right triangle ABC (right-angled at C), when AC = 12 m ...
i dont think he knows the pythagorean theorem
wraith daggerBOT
hired
quiet plover
yippee
patent belfry
can we prove this first
quiet plover
yeah
quiet plover
can we prove this first
huh whats there to prove here
patent belfry
huh whats there to prove here
he doesnt really understand it that good
so i wanna give him an understanding
quiet plover
doesnt mean it has a proof?
quiet plover
whad do you mean
patent belfry
doesnt mean it has a proof?
it does have a proof
quiet plover
goofy abbreviations
quiet plover
it is one of the symptoms for social media addiction lmao
patent belfry
imagine we square both sides
of that equation
we have $\sqrt{AB}^2 = (\sqrt{A}\sqrt{B})^2$
wraith daggerBOT
hired
patent belfry
what is $\sqrt{AB}^2$
wraith daggerBOT
hired
patent belfry
use the definition of square roots
no
like
use the definition of square roots
we have a square who's area is AB, and we take the side lengths of that square and find the area of the square made by them
i mean the geometric definition
of squaring and square roots
i told you them
squaring is the area made by the square with side lengths A
A^2 is that area
exponent
$A^2$
wraith daggerBOT
hired
patent belfry
so a squared is the area of a square with side lengths a
sqrt(a) is like saying "this square has an area of a. what are the side lengths"
yes
patent belfry
**hired**
so what does this mean geometrically
draw a picture
so theres a square right
and we know the area inside is AB
sqrt(AB) is the side lengths of that square
no the AREA of the square is AB
the side lengths are sqrt(AB)
what sqrt(AB)^2 means is, imagine you take these sides and made a square with them
what would the area be
exactly
so the left hand side of the equation is AB
now we need to prove $(\sqrt{A}\sqrt{B})^2 = AB$
wraith daggerBOT
hired
patent belfry
here it's easier to do it algebraicly
rather than geometrically
so we know that $(\sqrt{A}\sqrt{B})^2$ is just $\sqrt{A}\sqrt{B}\sqrt{A}\sqrt{B}$
wraith daggerBOT
hired
patent belfry
right
and we know that we can rearrange multiplication
so lets rearrange it to get $\sqrt{A}\sqrt{A}\sqrt{B}\sqrt{B}$
wraith daggerBOT
hired
patent belfry
and put brackets to get $(\sqrt{A}\sqrt{A})(\sqrt{B}\sqrt{B})$
wraith daggerBOT
hired
patent belfry
so we know what sqrt(A)*sqrt(A) is
it's sqrt(A)^2
and we know that that is just A
right
and we know that sqrt(B)*sqrt(B) is sqrt(B)^2
and so we know that that's B
so finally, we are left with
$(A)(B)$
wraith daggerBOT
hired
wraith daggerBOT
hired
patent belfry
so we have proved that $\sqrt{AB}^2 = (\sqrt{A}\sqrt{B})^2$
wraith daggerBOT
hired
patent belfry
which means that we have proved $\sqrt{AB} = \sqrt{A}\sqrt{B}$
wraith daggerBOT
hired
patent belfry
do you get it
ok so
we can go back to our original problem
of finding square roots
of any number
so lets take 18
we know 18=2*3*3
so $\sqrt{18} = \sqrt{2}\sqrt{3}\sqrt{3}$
wraith daggerBOT
hired
patent belfry
which is $3\sqrt{2}$
wraith daggerBOT
hired
patent belfry
sadly some square roots you just need to memorize
like square root of 2
which is about 1.414
,w sqrt(2)
wraith daggerBOT
patent belfry
thats a better approximation
so we know that sqrt(18) = 3sqrt(2)
so ,w 3*1.414
,w 3*1.414
wraith daggerBOT
patent belfry
,w 4.242^2
wraith daggerBOT
patent belfry
its about 18
so thats approximatly sqrt(18)
so now
ok try to find the square root of 32
to do that
you do $\sqrt{num}$
wraith daggerBOT
hired
wraith daggerBOT
wraith daggerBOT
patent belfry
so thats how you find square roots
quiet plover
and also
you need to know square root values of some numbers
sqrt(2) = 1.414
sqrt(3) = 1.732
wraith daggerBOT
wraith daggerBOT
patent belfry
i meant like