help-49

[midnight plank]

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[craggy owl]

need some help with this question

[craggy owl]

1. im confused on how fxx (-1, 2) is not negative

to me it looks like its going down?

then up at a certain point but still down first

2. could someone explain how to do f xy? its change of slope but not sure how to see that from the graph

[midnight plank]

@craggy owl Has your question been resolved?

[craggy owl]

<@&286206848099549185>

[midnight plank]

@craggy owl Has your question been resolved?

[tall lintel]

Hi! With graphing questions, it's kind of hard to tell (I will admit, this is a bad question). In any case, the point (-1, 2) looks a little further about.

[craggy owl]

yea...really dont like those graphing ones

[tall lintel]

If you look here, the point (-1, 2) seems to be on a concave up portion (i.e with respect to the x axis, since f_xx is the concavity of f by varing x).

So it seems that f_xx is more positive rather than negative.

[craggy owl]

interesting! do you have any suggestions how to do f xy of that point?

[tall lintel]

Well, fx tells me the slope along a certain fixed y. So, (fx)y tells me how that slope is changing as we move y, so I think graphically analysing that is the way to go.

I.e at (-1, 2), fx looks fairly positive, but what happens when shift y a little bit to the left or right? Does the slope increase or decrease?

[craggy owl]

sorry shift a little to the left/right in the y direction?

im a bit slow

we would be looking at the

fy (-1,2) is negative

and fx (-1, 2) also -

[tall lintel]

Welll

This is the tangent line to the surface at (-1, 2), right?

So here's the thing,

fx is the slope of the tangent line (in the x direction).

If we then take the derivative of this function (fx) with respect to y, what is this derivative doing?

[craggy owl]

how the slope in x-dir changes as we move in y?

[tall lintel]

Exactly.

So what that means is, we're still looking at the slope in the x direction, but we want to see how that slope changes.

[tall lintel]

Imagine that the blue and yellow points are just teeny tiny nudges of the red point in the corresponding y direction. What do you think happens to the slope as it moves from the red point (specifically from the red to the yellow point, since one-variable derivatives follow the positive direction)? Does it increase or does it decrease?

^ This is btw not a conceptual question, it's hard to tell from the graph.

[craggy owl]

yea...the point they chose is interesting

[tall lintel]

Ngl I think the slope ||increases. This is because the concavity (looking from the x direction) actually decreases significantly as you move in the y direction.||

[craggy owl]

https://tenor.com/view/jgmm-cat-meme-ragebait-cat-think-cat-ponder-gif-7034802297606826390

[tall lintel]

Yeah bum question. Interpreting a 3D question using a 2D frame of reference is not that great.

I'm sure there's some arguments regarding saddle points that apply here or something.

[tall lintel]

But anyway, as long as recognise some logic like this, then I'm happy.

[craggy owl]

i just hope nothing like this will be on the exam

[tall lintel]

Hopefully so

[craggy owl]

i will ask a TA about this too but its an online course so i had this to go from

[abstract perch]

oh

[craggy owl]

but thank you

i'll suffer a bit more

.close

[midnight plank]

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[tall lintel]

👍

[midnight plank]

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[twilit field]

I would like to prove this

[twilit field]

So I need to show $g \ker(\phi) g^{-1} \subseteq ker(\phi)$

[grand pond]

wai

[twilit field]

Let $x \in \ker ( \phi)$. Then \phi(g x g^{-1}) = \phi(g) e_{\overline{G}} \phi(g^{-1})$

[grand pond]

wai
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[twilit field]

okay

got it

tq

.close

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[frail carbon]

you're welcome?

[midnight plank]

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[twilit field]

Kind of confused

[pine wave]

what do you think is confusing here?

[twilit field]

so firstly, every element of the range need to have a pre-image?

[pine wave]

"onto", yeah, I believe so

wait codomain

range by definition, every element has a pre-image

[twilit field]

In that case I need to find all those combinations of elements whose product is 30

So the kernel would be 0,1, 2,3,5,15

[subtle zinc]

Z_30 is an additive group btw

[twilit field]

F

right

[subtle zinc]

try using 1st iso thm

[spiral rock]

You have a surjective homomorphism

Also in this case if it weren't surjective it would be the trivial homomorphism

[twilit field]

this?

Also, I believe the kernel has 6 elements

[spiral rock]

Yes

[subtle zinc]

nice one

take the order of both sides of the iso

[twilit field]

Lemme first justify this

Z_30 can be partitioned into 5 cosets

As for why

I think I'm forgetting some thoerm

[subtle zinc]

ok so |G/ker(phi)| = |phi(G)|

[restive sleet]

From what I understood, the subgroups of Z_30 correspond to divisors of 30.

[subtle zinc]

cuz both groups are isomorphic (hence bijective)

what's the |phi(G)|?

[twilit field]

6

[subtle zinc]

no 5

[twilit field]

right

[subtle zinc]

it's the image group

ok so |G/ker(phi)| = 5

and we know |G| = 30

do u know lagrange's thm

[twilit field]

yea, I forgot that the converse is true for cylic groups

The possible generators are

[subtle zinc]

ok but we just need the orignal thm

for now at least

[twilit field]

why, to determine the elements of the kernel surely we want the cosets

[safe remnant]

hello, im new here, im interested in math so i joined here

[subtle zinc]

we're trying to justify this first

[safe remnant]

ok, ill wait

[subtle zinc]

lagrange's thm says that |G/ker(phi)| = |G|/|ker(phi)|

[subtle zinc]

wasn't talking to u buddy 😂

[subtle zinc]

!help

[midnight plank]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[safe remnant]

Omg i thought you were talking to me 🫠

[twilit field]

that follows as to apply the first iso theomr we want to map 5 cosets injectvely to this other group

[subtle zinc]

ehh

u kinda need the fact that cosets all have the same size

which u get kinda used to and is easy to prove but really that's lagrange's thm

like lagrange's thm is kinda an easy thm

[twilit field]

I know

I didn't realise you were askig for that 😭

[subtle zinc]

?

ok so ker(phi) has 6 elements

and it's a subgroup of a cyclic group so is cyclic

[twilit field]

we know 0 is in the kernel

[subtle zinc]

therefore it must have a generator of order 6

[twilit field]

yea, so 6 works

[subtle zinc]

what are the order 6 elements of Z_30

[subtle zinc]

yes and 6 is the only one

[twilit field]

{0,6,12,18,24}

this is only 5 elements though

[twilit field]

we need a genertor of order 5

each subgroup need to be of order 5

[subtle zinc]

ah sorry i meant 5 is the only element of order 6

[twilit field]

yea

[subtle zinc]

no order 6

cuz ker(phi) is a cyclic group of order 6

[twilit field]

{0,5,10,15,20,25}

[twilit field]

got it

[subtle zinc]

yep

[twilit field]

Any such function is one -one and onto, there's only 1

[dreamy lichen]

why is there only 1?

[spiral rock]

There are at least 2

Trivial + identity

[twilit field]

oops

yes

[spiral rock]

Think what determines such a homomorphism uniquely

[dreamy lichen]

wdym by identity btw?

[spiral rock]

Identity map

[twilit field]

n!

my bad

[spiral rock]

Too much

[dreamy lichen]

not all bijections are homomorphisms though

[dreamy lichen]

everything to identity?

or everything to itself?

[spiral rock]

No, everything to itself

[twilit field]

Is there any efficient way to do this

[spiral rock]

Yes

[dreamy lichen]

and trivial means everything to identity?

[spiral rock]

.

[subtle zinc]

homomorphism not ismorphism

[twilit field]

I basically need to solve f(x+y)=f(x)+ f(y) modulo n

[twilit field]

right

[dreamy lichen]

yes, but idt its useful to think about it like that

[twilit field]

its kernel

[dreamy lichen]

maybe it would be best to start by constructing few explicit examples

can u find all the homomorphisms from Z4 to itself?

or at least some?

[twilit field]

the homomorphism onto 0

[dreamy lichen]

so everything to 0, yeah, that works

what else?

can u find something less trivial

[twilit field]

the identity is the other extreme

[dreamy lichen]

yeah, thats also quite trivial though

what else?

[twilit field]

oh, as Z_4 is cyclic

the set of outputs will have a genertor

Z_4 can have cosets of order 1,2, or 4

Phi(0)=0; phi(1)=2

@spiral rock , "‼️ ". Is that wrong?

[dreamy lichen]

no, u're going in the right direction now

[twilit field]

there are 4 such functions

[spiral rock]

Its ths key fact that makes this nice

[twilit field]

I can't believe I forgot that , ( I've proven this too 😭 )

[twilit field]

Okay, so the number of functions would be given by $\sum_{i=1}^{n} \frac{n}{\phi(n)}$

[grand pond]

wai

[spiral rock]

Why phi

Its cyclic groups

And remember you only need homomorphisms

[twilit field]

number of genrators

*generators

oh

[twilit field]

so what should I be doing

[twilit field]

Last attempt : Everything is determined by where 1 goes

[misty gorge]

thinking about examples of homomorphisms

[twilit field]

1 has n-1 places to go to

[misty gorge]

you should just be thinking

it's not about computing things like an integral or whatever

[twilit field]

I got it

[misty gorge]

you are in the wrong mindset

[twilit field]

I just tracked where 1 gors

This shouldn't have taken me so long 😔

[misty gorge]

why? there is no reason why it should take you any particular amount of time to complete. you just have to get out of the mindset that "X problem should take me Y minutes" because that's not really how math works

like you discovered something about groups today that allowed you to find this trick of tracking where 1 goes

and the trick is simple but the discovery process is not

[twilit field]

Tbf went through this same thing yesterday

was solving a functional equation for an assignment

took me 5 hours to realise a simple result would mean that no such function can exist

[misty gorge]

i don't understand what your point is

[twilit field]

I know that I leanr a lot by taking time is my point

[misty gorge]

ah

yeah

[twilit field]

the sideroads can be more enlightening

[misty gorge]

yeah. if you complete a problem in 30 seconds and it takes you no effort at all then that's more of a waste than your functional equation problem

[midnight plank]

@twilit field Has your question been resolved?

[zealous schooner]

why do we not like functional equations

[misty gorge]

i didn't say or imply that

[oak nymph]

Well, atleast now you have some facts to remember that may come in handy later on: Any morphism of a cyclic group is determined by where a generator is mapped

The answer to the problem is a consequence of the above simple fact, which you should engrave into your mind

[midnight plank]

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[safe onyx]

Is my understanding of implies correct?
(Here X means don't care/irrelevant)

[safe onyx]

(1 means true 0 means false for P / Q values)

[midnight plank]

@safe onyx Has your question been resolved?

[midnight plank]

@safe onyx Has your question been resolved?

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[modern lynx]

does this function have a vertical assymptote at 0

[modern lynx]

i mean it looks like it

[dreamy lichen]

by the defn u mentioned earlier, it does

[midnight plank]

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[keen harbor]

In this passage, is there a reason the author wrote one with brackets, and the other without?

[sick fossil]

No, just to emphasize where the sides of the original equation were

[keen harbor]

ok, thanks.

.close

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[modern lynx]

should i find both the LHD and the RHD and make them equal or should i just try to find the limit definining the derivative

(i already got past the continuity part of this Q)

[sick fossil]

Just find the limit defining the derivative

[modern lynx]

nd what should i write before it

(as my reason)

[sick fossil]

"The derivative is defined as"

or something similar

"...defined as this limit"

[sick fossil]

or however your teacher usually writes it, or tells you to write it

[modern lynx]

okie

,lose

.close

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[royal dirge]

i used this fact

AB:DA = 2:3

then k:w = 4:9

well, its wrong

i think i assumed these are similar triangles

when they are actually not

but then i feel like they are similar

they have that shared vertical angle

anyways, im not really sure how else to approach this

[shell wigeon]

Yeah they aren't similar

ABCD being cyclic gives you BCD = DAE = BAF, ADC = ABF, and ABC = ADE

[royal dirge]

cyclic means what? is that a synonym for inscribed in a circle?

[shell wigeon]

Yes

[royal dirge]

interesting, i've never seen that before

[shell wigeon]

Opposite angles are supplementary (BCD+BAD = ABC+ADC = 180º)

[royal dirge]

oh yeah, that makes sense

because the circle adds to 360

[vast anchor]

what's up guys i'm new here

[fallow scarab]

General chat goes in #discussion

[royal dirge]

how do i relate those angles to area though?

[shell wigeon]

Well, there are similar triangles in the figure, find them

[royal dirge]

if i cut the quad in half it forms two triangles

but i dont think they are similar

i make this cut

but now i have no triangles that share two angles

same with this cut

[shell wigeon]

Here, this is to scale, maybe that will help you find the similarities

[royal dirge]

ohhh

CDF is similar to ABF

and CBE to ADE

[shell wigeon]

Yes

[royal dirge]

that makes more sense

ok allow me moment now

ADE has sides 15, 3, 6 (out of order)
ABF has sides 2, (8/3), 2

huh

i got nonreal finding the area of ADE

[shell wigeon]

No those side lengths are wrong

[royal dirge]

oh no

[shell wigeon]

I don't think you need them anyway

I computed them to draw the figure but I don't think the question would require that

(I haven't solved the question yet, just waiting for you)

[royal dirge]

well, before we continue, can you explain why those sides are wrong

what i did was

BC/AD = EC/ED

and so on

to find each one

does BC not correspond to AD?

[shell wigeon]

It's not EC/ED since BC and AD are not parallel

[royal dirge]

ah

[shell wigeon]

Try imagining flipping the triangle ADE about the angle bisector from E

Then D goes onto EB and A onto EC, and that new segment is parallel to BC

[royal dirge]

oh

so i can use my side length strategy, i just have the wrong sides being compared

wait, i dont think that is possible though

[shell wigeon]

Sure, but maybe thinking of areas is better

[royal dirge]

going back to this

i can relate the area of the big triangles to their smaller similar triangles

like A_CBE:A_DAE = 16:9

[shell wigeon]

Right

[royal dirge]

A_CFD:A_ABF = 25:4

now i need to relate these two pairs

well A_CBA:A_CDA = 15:8

thats not the triangles im looking at though

[shell wigeon]

You don't need to cut the quadrilateral

(that 15:8 is false anyway)

[royal dirge]

oh lol

how do i relate these two ratios then?

sorry im kinda tired of math for today lol, maybe its obvious

[shell wigeon]

Use r,g,b for the areas

[royal dirge]

r+g:g = 16:9

r+b:b= 25:4

[shell wigeon]

Right

[royal dirge]

let me write as fractions so its easier to see it

$\frac{r+g}{g} = \frac{16}{9} \ \frac{r+b}{b} = \frac{25}{4}$

[grand pond]

UCYT5040

[royal dirge]

there we go

i need to find b/g

$g=\frac{r+g}{\frac{16}{9}} \ b=\frac{r+b}{\frac{25}{4}}$

[grand pond]

UCYT5040

[shell wigeon]

That looks complicated

[royal dirge]

yeah am i overthinking it?

[shell wigeon]

You should isolate r instead, to use elimination

[royal dirge]

$r=(\frac{16}{9}-1)g \ r=(\frac{25}{4}-1)b$

[grand pond]

UCYT5040

[shell wigeon]

Right

[royal dirge]

$\frac{7}{9}g=\frac{21}{4}b$

[grand pond]

UCYT5040

[royal dirge]

now i can just undo cross multiply

hold on one sec

ok sorry im back

$\frac{7}{9}=\frac{21}{4}\frac{b}{g}$

[grand pond]

UCYT5040

[royal dirge]

i get 4:27

answer key agrees!

thank you so much for your help!

this was the last problem i got wrong. and im not doing any more math for today lol

again, thank you!

.close

[midnight plank]

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[sly cape]

What is the logic behind putting numbers in brackets and not actually an equation?

[grim ferry]

Well some countries use BODMAS, some use PEMDAS.

so you’d kind of do “Brackets” or “Parentheses” before u multiply

[sly cape]

but you cant do anything with a number in a bracket...

like (-2) is just -2

[burnt flame]

sometimes its confusing just to have a floating negative, it can get confused for a subtraction sign

[grim ferry]

well in the case of this, there isn’t really any difference

[grim ferry]

^

Especially if you’re in exams and you get something like multiple negative signs..

[sly cape]

well but i can write 3*-2 and still be correct

[grim ferry]

Why not?

[sly cape]

what about 3*(+2)

does something like this exist

[burnt flame]

you can yes,

[burnt flame]

+2 is pointless, we already have 2

-2 is negative 2

[sly cape]

kk

.close

[midnight plank]

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[sly cape]

try moosy and maddie

ty*

.reopen

[midnight plank]

✅ Original question: #help-49 message

[sly cape]

but this −1(−1)

its -1 times -1

@burnt flame

[grim ferry]

negative multiply by negative is positive

-a * -b = +ab

[sly cape]

so whenever something is near brackets, you multiply?

[subtle blaze]

Some say subtraction isn’t real

[sly cape]

ik

[grim ferry]

Follow PEMDAS or BODMAS, whatever you’re taught

[sly cape]

wdym?

[subtle blaze]

There’s only ever negative numbers

[burnt flame]

whenever something is outside of brackets, just next to them, it signifies multiplication yes

[sly cape]

WHAT IS THE DIFFERENCE?

[grim ferry]

No clue i’ve seen people arguing about it before

[subtle blaze]

-2 is one symbol, it is the symbol such that when you add it to 2 you get 0

[burnt flame]

in general when you have variables next to each other too, its multiplying them, or a variable next to a number

2x=2*x

xy=x*y

[sly cape]

okay but why for example -(-5+3) we switch them to +5-3, how does it make sense?

[burnt flame]

-(-5+3)=-1*(-5+3)

-x=-1*x

[subtle blaze]

It so happens that the -1 symbol is special and you can rewrite the -2 symbol as the multiplication 2 * -1

[burnt flame]

distributive property

[subtle blaze]

- signs can’t exist on their own, whenever they are by themselves it implicitly means there’s a 1 after it

[sly cape]

so whenever theres a - or + i can add 1?

[subtle blaze]

So here it’s (-1) * ((-5) + (3))

Brackets are there so that we can package a whole symbol into a “box”

[sly cape]

but signs arent symbols

[subtle blaze]

I did not say so for +

[burnt flame]

look, think of it more like the 'opposite of'

[subtle blaze]

• is a real thing (addition)
  - is a fake thing (subtraction)

[burnt flame]

-(-5+3) is the 'opposite' of -5+3

[subtle blaze]

“-1” is 1 symbol

[burnt flame]

when we add them together, we should get 0

[subtle blaze]

It’s just 2 disconnected parts

[sly cape]

but when you have 5-(-5+3), then you have to muplitply them by 5?

[burnt flame]

if it were -5(-5+3) we would multiply by -5

[quiet parcel]

practically the same paradox as having said that all real numbers are complex itself because they have their Im(z) = 0

[subtle blaze]

The only 2 properties you know about -a is

1. whenever you do -a + a or a + -a you get 0
2. -a = a * -1 = -1 * a

[sly cape]

but if we had +(-5+3)

[subtle blaze]

You would not write that

[sly cape]

why is 1 and 0 so special numbers in math..

[burnt flame]

but the way you have written it would be read 5 minus the quantity (-5+3)

[subtle blaze]

They are the additive and multiplicative identities

[sly cape]

why don't?

[subtle blaze]

Because the + symbol requires you to have things on both sides of it

[quiet parcel]

0 is special because it doesn't change anything qhen it comes to addition but yet it changes everything when it comes to multiplication

[subtle blaze]

$+2$ is as bad as $\frac{}{2}$

[grand pond]

frosst

[burnt flame]

-something is just a quick way of writing -1*something

[subtle blaze]

That’s technically not true but…

[sly cape]

yeah so

5-(-2) = 3

[subtle blaze]

It’s true in a field at least (ignore this if you don’t understand it)

[quiet parcel]

and it's practically just a concept, hell I even had that straight thought once when I was trying to find out the proof for why zero is an integer, and why it's not just a concept, zero has a duality of being both real and complex, so many things make it special

[burnt flame]

yes, i know, but i think overcomplicating it now is pointless

[grim ferry]

No offence to anyone but I think we’re overcomplicating this for the poor op 😭

[sly cape]

but they are classified as every other number..

[subtle blaze]

This is 5 + (-(-2))

The - is attached to the (-2) object

And the - in here is attached to the (2) object

[burnt flame]

subtraction is just adding negative numbers

[subtle blaze]

This is why I said some say subtraction isn’t real

[sly cape]

yep...

[burnt flame]

yes but i think you could've just phrased it like that initially 😭

[sly cape]

okay so - + - = -

[subtle blaze]

I wanted the intrigue and delving into that thought..

[grim ferry]

Okay, well I wrote this for someone just now

[subtle blaze]

You can’t add symbols together

[grim ferry]

Take a and b as any real number that you want. Then,

a + b = a + b. eg. 1 + 1 = 2.
a - b = a -b. eg 1-1 = 0
a + -b = a - b. eg. 1 + -1 = 1-1 =0.
a - +b = a-b. eg 1 - +1 = 0.
a * (-b) = -ab. eg 2 * (-2) = -4

[subtle blaze]

But -a + -b = -(a + b)

[sly cape]

why?

[burnt flame]

distributive property

[quiet parcel]

think of it like this, you have 5 apples, and you owe 2 so you're in debt of a 2 apples (debt is the subtraction), now if they tell you that you're no longer in that debt, you actually no longer owe those 2 so you actually gain them to yourself, and gaining is addition, so you'd have 7 now in total

[subtle blaze]

That’s just an axiom

We pick our maths so that it’s true

It turns out to be useful if we require it to be true

It somehow models something the real world does, and that’s useful so we pick it to be so

[subtle blaze]

Like this example, maths wouldnt be very useful if it didn’t work

[sly cape]

so if we remove the brackets, its -a + -b?

[subtle blaze]

You can’t “remove” the brackets

[burnt flame]

the brackets are grouping it together

[sly cape]

why? its two numbers

[subtle blaze]

But you can distribute

You shouldn’t think of it as removing the bracket

You should think of it as the distributive property

[sly cape]

so is it a single number?

[subtle blaze]

a*(b + c) = a*b + a*c

[burnt flame]

https://www.mathsisfun.com/definitions/distributive-law.html

[subtle blaze]

This we take for granted

We just assume it is true

(Maths is based on a bunch of arbitrary assumptions that turn out to be useful)

[sly cape]

but is (a+b) a single number?

[subtle blaze]

Yes

[sly cape]

but a + b arent

[subtle blaze]

Yes

Well, you could say it is too, but now we’re being pedantic

[sly cape]

-(-a-b) how is this possible?

they are negative already

how can you already make them negative again?

like twiec

[subtle blaze]

That is a shorthand for -((-a) + (-b))

We know that -a and -b are numbers, in particular they are the numbers which when you add it to a and b respectively you get 0

So (-a) + (-b) is addition between two numbers which (we assume) is again a number

This number also (by assumption) has its negative version

[sly cape]

so is the minus glued to (-a-b) in -(-a-b)?

so if i wanted to add 5 i'd have to minus it with (-a-b)?

[quiet parcel]

i once heard of this one thing from reddit or somewhere...

imagine you're standing looking at something in the right (positive, +) direction and your task is to turn around in the opposite (negative, -) direction.

if you turn around from the right direction, you'll be facing the opposite (negative) direction, if you turn around again, you'll be facing the opposite direction again but this time it'll once again be your starting right direction

every "turn around' and "opposite direction" is a -
every "right direction" is a +

[subtle blaze]

Yes

5 + (-(-a-b))

[sly cape]

kk.

[subtle blaze]

You can somehow “unstuck” the - sign if you do -a = -1 * a

[sly cape]

so this will result into 5+a+b?

[subtle blaze]

Yea

Technically it results in 5 + (a + b)

[sly cape]

so it will result into being 4+a+b in this logic?

[subtle blaze]

But for addition we also require (a + b) + c = a + (b + c) so it doesn’t matter if you have brackets or not

[subtle blaze]

No there’s a multiplication sign

It’s 5 + (-1 * (a + b))

There are a whole lot of brackets everywhere but we omit them for the convention of pedmas or whatever you learn it as

[sly cape]

kk

okay

but what is the difference between podmas and bemdas

i was taught this:

()
x^2
.* / :

• / -

[twilit jetty]

this is the usual order, these different abbreviations just use different names for each one
(), then ^ (right-to-left), then . : / (left-to-right), then + - (left-to-right)

for example 5 - 5 - 5 is seen as starting at 5, doing - 5, then doing - 5 again

5 * 5 / 5 does * 5 first then / 5

however, for 2^3^4, you start with 3^ then 2^

2^3^4 = 2^81

[sly cape]

im confused..

which does what?

[twilit jetty]

you tell me, you typed it like that

try reading it slower

[sly cape]

but in maths you do everything from left to right..

you don't do specifically one thing...

[twilit jetty]

if you do 1 + 4^2,

do you do the ^ or the + first?

[sly cape]

differently

[sly cape]

sure because you use pedmas

[twilit jetty]

lets ignore pemdas for now

Im talking about you bro

if YOU do 1 + 4^2, do you do the ^ or the + first

[sly cape]

^

[twilit jetty]

so much for left-to-right

[twilit jetty]

whats the difference then

[sly cape]

We're talking about this

5 + 9 - 4

[twilit jetty]

yes, you view this as 5, then + 9, then - 4

each operation you can view is something that takes an input number, does something to it, then leaves an output number

commonly this is read as starting at 5, then doing + 9 to it to get 14

this is technically different than just viewing + and - as operations that take two numbers together

but its more in line with the usual left-to-right that people actually do

[twilit jetty]

if you do it like this, the operations themselves wont tell you what order to do them in, you have to decide on an order

[sly cape]

but what if this

5 * 9 : 2

[twilit jetty]

5, then * 9, then : 2

[sly cape]

,calc 45/2

[grand pond]

Result:

22.5

[sly cape]

,calc 4.5*9

[grand pond]

Result:

40.5

[sly cape]

not the same result

[twilit jetty]

4.5 * 5, no

be consistent with your inconsistency

also, this is an example where either way is correct

[sly cape]

well in bodmas you do division first and PEMDAS you do multiplication first

[twilit jetty]

(5 * 9) / 2 = 5 * (9 / 2)

buddy pal you did it wrong

,calc 5 * (9 / 2)

[grand pond]

Result:

22.5

[twilit jetty]

find a better example

for example, 6 / 2 * 3

[sly cape]

i think you're starting to get aggressive.

[twilit jetty]

I was just trying to mirror yours

[twilit jetty]

like look at this

and you presented the example without even testing it first

Im trying to match your energy you know

as I was saying, theres a reason why PEMDAS or BODMAS isnt actually airtight

thats because of a common rule that usually people use to read the operations instead

the strongest operations go first,

[lusty python]

well if you remove the brackets, it would be 3 - 2 and that can be mistaken as a minus sign

[twilit jetty]

and with operations of same strongness, they are read left-to-right if they do actions

[sly cape]

but divison and multiplication are equal

[twilit jetty]

they are, yes

as I was saying, same strongness

[twilit jetty]

listen, please

[sly cape]

see how division and multiplication is different

different order

[lusty python]

no

[twilit jetty]

the word "or" is in there

that implies you can do one or the other

[lusty python]

yes

[twilit jetty]

you can also see theyre of the same color here

compare that witht he brackets, order, and addition/subtraction

[sly cape]

no you can't do any order..

[lusty python]

wdym by that?

[twilit jetty]

thats not what the "or" means

the "or" means neither of them are stronger than the other

the "or" means we have to use a different rule other than "multiplication first" or "division first"

the "or" is equivalent to using the same color background

it does not imply you can do them in any order

do not misunderstand the image to get 'extra points'

[twilit jetty]

no one image alone can explain how we do things, ok?

Im right here you know, let me tell you what the reason is

[lusty python]

if according to your logic, why does according to PEMDAS, you have to perform multiplication first?

that is what your logic is here

[sly cape]

6 / 2 * 3

but here.. doing in any order will get you different results

[twilit jetty]

yes, so heres the idea that we have

[lusty python]

well

[lusty python]

perform from left to right

[twilit jetty]

the idea is that we stick to one way of viewing things, and we stick to it hard for consistency

this way isnt for "correctness" as much as it is for "consistency"

this is a key idea in math that often gets overlooked

[sly cape]

why? How is that mathematically proven

[twilit jetty]

consistency is more important than correctness

[lusty python]

it's a rule we have to follow

we can't directly 'prove' it

[lusty python]

well then prove for me that 1 = 1

[twilit jetty]

look at it this way

[sly cape]

so why don't bodmas and pemdas have the same order?

[twilit jetty]

oval,

are you aware different countries do things different ways?

you know theres a U in colour in british english?

theres no "special reason" the u has to be there

well for the U in there, there actually is a reason, but thats besides the point

[sly cape]

okay so i can decide if im going to do division first or multiplication

[twilit jetty]

more often or not, changes happen and theres no good intentional reason for them

PEMDAS and BODMAS both communicate the same idea

[sly cape]

it does, accent.

[twilit jetty]

which is that multiplication and division arent done first

theyre just done left-to-right

[twilit jetty]

by "special reason" I mean an intentional reason to insert the U in there

again, please dont misconstrue my words

accents are not an intentional choice

[lusty python]

i get what you are saying

[twilit jetty]

you seem to attribute some holier-than-thou status to the exact order of BODMAS and PEMDAS

[sly cape]

oh... but so its american's choice to not speak british

[twilit jetty]

that is not supposed to communicate to you the correct order

[lusty python]

to tell you in simple terms

BODMAS

and PEMDAS

they all mean the same thing

the thing is that people call it differently

depending on their country

[sly cape]

but mtt said it was different in the beginning

[twilit jetty]

let me look at my exact wording

[twilit jetty]

hmmm doesnt seem to place any one of them first

[lusty python]

where is that message may i ask?

[lusty python]

i think you misunderstood his words oval

[sly cape]

then why did you say to start with the right side and then do the left side?

[twilit jetty]

this is just telling you that its possible to get different answers if you do things the wrong way

[lusty python]

he was saying what i said right here

[sly cape]

first that he sent

[lusty python]

well i can't just scroll up

[twilit jetty]

oval, so far youve misconstrued two messages so far, can you actually reply to the exact message that gives you this idea

common misconception: people cant read your mind, you have to tell us in more detail what the confusion is

[sly cape]

also frosst said that "whatever you learn it as" so i supposed that these are 2 different things.

[twilit jetty]

oval thats not how you read that

the difference here could just be in how the same information is presented in two different ways

[sly cape]

are you a mathematician or a philosopher?

[sly cape]

yeah presented the way that the things had multiplication and division in different order

[twilit jetty]

for example, color and colour are spelled differently, but they both mean the same idea

are you sure?

[lusty python]

well, you can

[twilit jetty]

this picture sure is saying the order doesnt matter

[lusty python]

BODMAS

or PEMDAS

[twilit jetty]

its not muliplication or division first thats the rule

[lusty python]

they said that parentheses should be calculated first

[twilit jetty]

its multiplication and division, left to right, which is the rule

parentheses and brackets are the same word in american and british english

[lusty python]

and since in the brackets, there was only a -2, so it means -2

[twilit jetty]

consider for example -2^2
we usually do the ^ first, so -4

if on the other hand, you want to enforce the -2 is the number being squared, so -2 * -2,
you place parentheses around the -2
so (-2)^2 is 4

[lusty python]

and as they said, if you remove your parentheses then it can be mistaken as a minus sign

[sly cape]

ok but why cant it be bomdas and pemdas?

so the order is actually 'the same'?

[twilit jetty]

no reason, its just that BODMAS is a popular way of viewing it, and BOMDAS is less popular

you know "order" isnt common in america right

we call it exponentiation

we dont call them indices

you may as well ask why neighbor has a silent "gh"

its not math, its history

[lusty python]

or listen

has a silent sound t

[sly cape]

how does that make sense?

[twilit jetty]

lets go by the usual rules

there are two operations here, ^ and -

[sly cape]

but not list

[lusty python]

well, according to PEMDAS, or BODMAS, we perform exponetitation first

[twilit jetty]

now -x has a similar operation that is short for "0 - x"

[lusty python]

it's not my rule

[twilit jetty]

the rules arent very consistent, listen is just a counterexample

[sly cape]

well but -x is a single number , not an equation

[twilit jetty]

theres plenty of these around

[twilit jetty]

and unfortunately for you, the - is a lot more separated from the number than you think

it is common to view -x^2 as -x * x instead of -x * -x

[sly cape]

which is correct?

[twilit jetty]

bold to ask

[sly cape]

okay can you explain with the brackets and exponents

[twilit jetty]

I just told you its common to view -x^2 as one instead of the other

shouldnt that be reason enough

-x^2 is more commonly viewed as -x * x

therefore -x^2 is consistently -x * x

this is a matter of convention, not correctness

do you know what convention means here? you seem to want to look for a right answer where there is only a consensus answer

if you want -x * -x, you do (-x)^2 to signify that the - sign goes first (called "negation" when used on a single number)

[sly cape]

but ^2 is just a number being duplicated...

[twilit jetty]

not true

^2 specifically means multiplying with itself

and in -x^2, whats being duplicated is x, since the negation operation - hasnt been done yet

-x^2

[sly cape]

so (-x) (-x)?

[twilit jetty]

(-x)(-x) and -x * -x are the same

[lusty python]

yes

[twilit jetty]

and they both mean x^2

[sly cape]

but why is -x^2 not -x * -x

[twilit jetty]

to remain consistent with the existing rules we already have

consider 0 - x^2 for example

[subtle blaze]

If you look at 2x-4^2(3-(x+y))

[twilit jetty]

would this really be (0 - x)^2?

what about 1 - x^2?

we view that -x should be the same as 0 - x

instead of being "the minus sign"

[sly cape]

1 + x * x

[twilit jetty]

this is so common that even if its attached to the number, we still put in ()s around it

for example, if I want to square -4 with itself, Id write (-4)(-4) or (-4)^2

I cant write -4^2

you can see here an extra () has been used, to denote that the - cannot be done last

operations have context

you cannot "regular expression" your way out of this

you cannot "string replacement" your way out of this
when you see a -2, it does not need to mean -2, it only means 0 - 2

and so -2^2 is 0 - 2^2, which is 0 - 4 or -4

the alternative is messy

[subtle blaze]

This is really (2*x) + (-(4*4))*(3 + (-(x+y))))

[twilit jetty]

lets say we view -2^2 as -2 * -2

[subtle blaze]

But why have so many brackets when you can have a convention

[sly cape]

but why cant you do -2 * -2?

[twilit jetty]

now compare -x^2 with -(-x)^2

oval, look at what I am talking about right now

are you listening?

[sly cape]

multiplication first than minusing with 0

[twilit jetty]

youve just asked a question I am in the process of answering

you do realize that, right

[twilit jetty]

as I was saying

lets compare -x^2 with -(-x)^2

show me an example of -2^2 and -(-2)^2

how do you calculate the both of these?

[sly cape]

-2 * -2 and -(-2)(-2)

[twilit jetty]

so theyre different, youre saying?

lets go with that

[sly cape]

sure, because one is with a bracket

[twilit jetty]

now lets suppose x itself is -2

x^2 is?

[sly cape]

-2 * -2

[twilit jetty]

yep, thats correct

x is -2, x * x is 4

[lusty python]

now then what

is -x^2

[twilit jetty]

however, see what happens if we test that here

[twilit jetty]

here, I mean

x^2 is 4, so -x^2 is?

[sly cape]

-x * -x

[twilit jetty]

write it out

what is the value of -x^2?

[lusty python]

for x = -2

[sly cape]

wdym?

oh

[lusty python]

you just calculated that x^2 = 4 for x = -2

[sly cape]

-2 * -2

[lusty python]

so what is -x^2

[sly cape]

the same?

[lusty python]

wdym the same?

[twilit jetty]

now consider this

[sly cape]

because two - is the same

[twilit jetty]

what is x^2 - x^2?

[sly cape]

you cant put like multiple signs like 5+++++++5

[sly cape]

0

[lusty python]

we actually can

[twilit jetty]

well yes you can

[lusty python]

but it's redundant

[twilit jetty]

but we've just agreed that x^2 and -x^2 are both 4, didnt we

4 + 4 cant be 0

[sly cape]

you'd even get an error in calculator

sure buddy

[twilit jetty]

you wouldnt get an error if you used the negation symbol thats next to the enter key on the calculator

calculators only view a limited part of how we communicate math

[twilit jetty]

go show me what happens if x is 2, and calculate each step out

[sly cape]

x * x - x * x

[twilit jetty]

thats different

in this case, why would -x^2 not be -x * -x?

[sly cape]

isn't it?

[twilit jetty]

you wrote -x * x

not -x * -x

those are different

-2 * 2 is -4,
-2 * -2 is 4

[sly cape]

x * x - x * - x

[twilit jetty]

then, if x is 2?

[sly cape]

2 * 2 - 2 * -2

16

[twilit jetty]

that would be 8, not 16

try that again

unless

unless youre viewing it as (2 * 2)(-2 * -2)

what youve done here, by accident, is inserted an extra * in the middle of things

[sly cape]

yes

that

[twilit jetty]

when previously the expression was x^2 - x^2

this is the main reason why we dont view -x^2 as -x * -x

[burnt flame]

2 * 2 * -2 * -2

[sly cape]

im lost..

[twilit jetty]

you tell me why you put that * in there

it wasnt there before

[sly cape]

x^2 - x^2

x^2 = x * x
-x^2 = -x * -x

[sly cape]

where?

[twilit jetty]

here

4 * 4 is 16, 4 + 4 is 8

the x * x would just be 4
the -x * -x would be 4 again
if it was x * -x, thatd be -4

the other *s only get you 4s

so wheres the 16?

[burnt flame]

-x^2 is -1 * x^2, while -x * -x = -1 * x * -1 * x=1*x^2=x^2

[sly cape]

2 * 2 (4) + (4)

[twilit jetty]

moosey we're currently trying out an idea where the negator has greater precedence than the exponentiation operator, which has greater precedence than the subtraction operator

[burnt flame]

o

[sly cape]

okay so can we get to the point?

[twilit jetty]

ok youre gonna have to show me where thats from

how did -2 * -2 turn to 4 + 4

[sly cape]

cause now we're making some extraordinary cofusing equations

[sly cape]

.- and - = +

[lusty python]

I think I have to step in onto this one

Let me cook guys

[sly cape]

ok

[twilit jetty]

-2 * -2 would be 4, not 4 + 4

[sly cape]

yeah?

[lusty python]

So expressions like -2 or -(4 + 3). Let's call it a unary expression

[twilit jetty]

whyd you write 4 + 4 then

[sly cape]

no you misunnderstood

[lusty python]

It has a plus or minus as an operator

[twilit jetty]

oh thats disgusting, I see

well dont invent new ways to write math in the middle of me trying out your idea

[lusty python]

And to the left of it, it has no number, or at most operators

[sly cape]

-2 * -2 is FOUR
2 * 2 is FOUR

[lusty python]

(unary is actually a used term for that)

[sly cape]

what is dusgusting?

[twilit jetty]

read this again

this looks like 2 * 2 * 4 + 4

[sly cape]

it doesnt

the brackets show the results of multiplication

[twilit jetty]

if ou had to explain it, it isnt clear

[sly cape]

what even is unary expression

[twilit jetty]

in the middle of trying to be clear, youve used brackets, which are themselves a part of math

maybe you can say "which is"

2 * 2 (which is 4) + (which is also 4)

[sly cape]

well but it was an example to show you

[lusty python]

Let me explain this one to him

[twilit jetty]

if you just write 2 (4), thats short for 2 * 4

[lusty python]

I can try this one

[sly cape]

you misunderstood here.

this time

[twilit jetty]

oval, yes I did, and its still your fault because that was an ambiguous statement

(2)(4) means 2 * 4, its another way to write down multiplication

[lusty python]

Hey, I can try explaining this but I don't have much time

[sly cape]

im aware of that , i just wanted to show how it was done..

[twilit jetty]

make the most of it

[lusty python]

May I have your permission?

[lusty python]

Alr thanks

So let's define a unary expression as this

It has an operator of + or -

There's no expression to the left of it

And it's right expression is exponents or parentheses

[sly cape]

example?

[lusty python]

-7

[sly cape]

okay thats negative 7

[lusty python]

Or +1

[twilit jetty]

yep, not "minus 7" since theres no 0 to the left of it

when you use - in this way, its called a negation

[lusty python]

-x^2 is also an example of a unary expression

[sly cape]

so would - 7 be different?

[lusty python]

We treat unary expression as a number

[twilit jetty]

well we dont see a 0 to the left of it, so - 7 and -7 are the same

[lusty python]

It makes no difference

[twilit jetty]

youd need something on the left to see the difference: 0 - 7

0 - 7, 0 -7, and 0- 7 are also the same

[lusty python]

So for example: -2 * -2

Since we treat -2 as a number, then we perform multiplication on both sides

[sly cape]

okay

its 4

[twilit jetty]

yep, its 4

[lusty python]

Yep

Now try -2^2

[sly cape]

which equals to -2 and -2

[lusty python]

Look at this definition

[sly cape]

duplication of a number

[hybrid rain]

Can someone help me with cypher ?

[twilit jetty]

which order are we using, oval's version or the conventional version?

[lusty python]

Look at this definition and think again

[lusty python]

!occupied

[midnight plank]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[sly cape]

but if we did +5^2 then its +5+5

[twilit jetty]

thats not how you read the + symbol here

^2 does not mean duplication in that sense

for example, if I did (1 - 1)^2, which would be 0,

would you really be doing 1 - 11 - 1?

thats -11

[lusty python]

Try looking at this definition and calculate -2^2

[sly cape]

but -2 is alone in there

[twilit jetty]

try answering this question before they leave

[lusty python]

I think I can give enough context for you to help

[sly cape]

so its -2 * -2

[lusty python]

-2^2 is a unary because to the right of it, there's an exponent

And we treat unary as a number

So to calculate a unary

[sly cape]

idk what is unary

[last slate]

erm acktually

-2^2 = -(2 * 2)
if you wanted -2 * -2
it wld be (-2)^2 :3

[lusty python]

You have to calculate the right one first

[lusty python]

@sly cape @twilit jetty

[lusty python]

Then perform what the sign says

[sly cape]

yeah so unary = - 7 and ary = - 7

[lusty python]

So to calculate -2^2

You have to calculate 2^2 first

Then change the sign of it

[sly cape]

but -2 is a NEGATIVE number!!!

[twilit jetty]

that doesnt really answer why you couldnt just do the -2 first, just that youre supposed to "not do that"

[sly cape]

negative and positive number

[twilit jetty]

I think I know what to answer this with

youre aware we are using two different kinds of - here

one for denoting a negative number

and another for subtraction

[lusty python]

Use this definition to help him

[twilit jetty]

now you think this one should go first, before exponentiation

[sly cape]

okay

okay but this is a negative 2

[twilit jetty]

oval, listen to me

[sly cape]

it isnt two and then subtract

[twilit jetty]

yes or no, you can see the two different operations?

one that takes in one number and considers it negative

[sly cape]

yea

[twilit jetty]

and another that takes in two numbers and finds the difference

[sly cape]

but ultimately its the same purpose

[twilit jetty]

no it isnt

[sly cape]

it is

[twilit jetty]

and its entirely for a reason you gave:

now so far, you are saying that because the - sign is part of the number, it shouldnt be that an 'outer' operator can just take out part of it

[sly cape]

okay add 5 and -5 together

i can do 5 + -5 = 5 - 5

[lusty python]

True

[twilit jetty]

so for example, -2^2 should view the - first, so -2^2 is -2 * -2

[lusty python]

unary expression

[sly cape]

mhm thats my POV

[twilit jetty]

however, by doing this, you are treating the - signs differently

consider -5 - 5 for example

[sly cape]

BUT IN THIS EXAMPLE MINS IS WITH THE NUMBER NOT SEPERATED.

do you understand what i mean?

[lusty python]

consider -5 as a unary expression

[twilit jetty]

so tell me this

if I wanted to subtract -5 and -5 together, how would I write it

[sly cape]

-5 + -5

[twilit jetty]

if I wanted to multiply -5 and -5 together, how would I write it

[sly cape]

-5 * -5

[twilit jetty]

and if I wanted to add -5 and -5 together?

[twilit jetty]

and you just said this, as a reminder

[sly cape]

-5 - -5

[twilit jetty]

so adding no longer places a + sign between numbers now?

[sly cape]

wdym?

[twilit jetty]

wouldnt adding -5 and -5 be -5 + -5?

theyre both -5 and Im adding them

whats different here

[sly cape]

thats subtracting

because the 2nd subject is a minus

[lusty python]

That's subtracting -5 and 5

[lusty python]

||Use unary expressions (it's a real term btw)||

[sly cape]

this isn't right

[twilit jetty]

it isnt, yes

thats because the way youre doing things isnt very good

we have a better way around this

[sly cape]

adding -2 and -2 is
-2 - -2

[lusty python]

||This is the last resort||

[twilit jetty]

I think we've just stepped into a different kind of error that unary expressions arent going to solve

[lusty python]

Lol

[lusty python]

Yeah i see what's going on here

[lusty python]

To add -2 and -2

Avoid confusion

Add the parentheses first

Should be (-2) + (-2)

Good luck!

[sly cape]

that is subtraction

since the second 2 is a minus

[twilit jetty]

ok Im going to show you something that you need to know beforehand

what do you think the + operator does?

[sly cape]

fine

[sly cape]

add?

[twilit jetty]

you surely arent adding -2 and -2 here to get -4

youre turning the second number to always be positive then adding them

so if I were to start at -2, and try adding various numbers, Id get:

-2 + 5 = 3
-2 + 4 = 2
-2 + 3 = 1
-2 + 2 = 0
-2 + 1 = -1
-2 + 0 = -2
-2 + -1 = -3
-2 + -2 = -4
-2 + -3 = -5

any time I use the + operator, that is called addition

if the number we get is lower than before, it is still considered addition

in this case, we notice that adding a negative number makes you lower than before

does it simplify to subtraction? yes

but the initial step has used the + sign

[sly cape]

-2 + -1

and will this result get closer to negative or positive?

[twilit jetty]

not correct, oval

using the + sign is all you need to call it addition

signs do not care about the output of their result slanting a particular way

-2 + -1 is considered adding -2 and -1
-2 and -1 add to -3
-2 and 1 subtract to -3

-2 + -1 "negative 2 plus negative 1"
-2 - 1 "negative 2 minus 1"
-3 "negative 3"

[sly cape]

and whats the point?

[twilit jetty]

doesnt this go against what youre thinking

youre saying -2 plus -1 should be -2 plus 1 instead

even though thats not how the plus symbol is supposed to be interpreted as

[twilit jetty]

all of these are the same

[sly cape]

no?

[twilit jetty]

then why did you say this?

[sly cape]

its -1 not plus 1

[twilit jetty]

are you just saying -2 + -1 isnt considered addition because, when you simplify, theres a - sign?

[sly cape]

yes as it results to -3 which is <-2

[twilit jetty]

thats interesting, so whats the + sign doing there?