help-0

[alpine sable]

wait so this one is

[ionic jewel]

is?

[alpine sable]

8a -5b -5c

[ionic jewel]

yes

[alpine sable]

right?

[ionic jewel]

yes

[alpine sable]

okay ty

so three terms

[ionic jewel]

yes

[alpine sable]

ty

[sacred otter]

Anyone can please help me to solve This? I can't seem to wrap my head around the first move, if I can get the first move, the rest will follow. But I'm stuck

[alpine sable]

what's the question?

[sacred otter]

Find the sides

How did I get this sticker?

Anyway, find the sides of the big triangle

[alpine sable]

i think this can be solved with trig

[sacred otter]

Yea, just what's the first step. Maybe I'm exhausted and can't see it

[alpine sable]

I will look at it, give me some time

[sacred otter]

👍

[alpine sable]

you can use sin

sin(20) = 425/hyp
sin(20) * hyp = 425
hyp = 425 / sin(20)

,w calc 425 / sin(20)

[ocean seal]

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[alpine sable]

425^2 + b^2 = 1424^2

180625 + b^2 = 2027776

b^2 = 1847151

b = 429.784829886

[sacred otter]

The second triangle isn't right angled. The angle is only for the small triangle

[alpine sable]

u can use these formulas even though if the triangles are arbitrary or not

a^2 + 675^2 = 429^2

[alpine sable]

oh ye i forgot

well

u know this this then

u can also get the hypotenuse of the small triangle with sah coh tao whatever its called

tan(20) = mod/hos

tan(20) = 425/hos
tan(20) * hos = 425
hos = 425 / tan(20)

,w 425 / tan(20)

[ocean seal]

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[alpine sable]

a^2 + 675^2 = 1167^2

a^2 + 455625 = 1361889
a^2 = 906264

,w sqrt(906264)

[ocean seal]

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[alpine sable]

DONE

[sacred otter]

k

Thx

[tight locust]

K

[alpine sable]

it doesn't really look correct that the side is 951 lol

[tight locust]

anyone else need help i'm slightly bored.

[compact coral]

can you do justifying steps

😭

[alpine sable]

but i think it's correct

[oak chasm]

@alpine sable That tangent ratio only works in right triangles.

The triangle you're using it in isn't a right triangle.

[alpine sable]

Yh I was wondering as well

[oak chasm]

@sacred otter The two angles in the bottom middle add up to a straight line, or 180°. So, mark the left one A and the right one 180° - A. Mark the far right angle as B. Mark the top angle next to 20° as 180° - 90° - A = 90° - A . Mark the segment between A and the right angle as x. Mark the right side of the large triangle as y.

Now, the sum of the angles is the right triangle is 180°, so 20° + (180° - A) + B = 180°, so B = 180° - 20° - (180° - A) = A - 20°.

Now, we can use law of sines:

sin(A)/675 = sin(90° - A)/x = cos(A)/x
tan(A) = 675/x

sin(A - 20°)/675 = sin(90° - A + 20°)/(x + 42.5) = cos(A - 20°)/(x + 42.5)
tan(A - 20°) = 675/(x + 42.5)
(tan(A) - tan(20°))/(1 + tan(A)tan(20°)) = 675/(x + 42.5)

Now, we know from earlier that tan(A) = 675/x, so we fill that in:

(675/x - tan(20°))/(1 + 675/x tan(20°)) = 675/(x + 42.5)

Now you have x and a bunch of constants.

That should get you started.

[tight locust]

sure let's go into #help-8

[oak chasm]

Then you can use the Pythagorean theorem once you have x.

That will get you the other two sides.

[alpine sable]

what did i do wrong @oak chasm

[oak chasm]

@alpine sable Your tangent = side ratio thing was wrong, since you have a non-right triangle you're doing it in.

[oak chasm]

20° isn't one of the angles of one of the right triangles.

[alpine sable]

you mean this calculation?

[oak chasm]

No, this one before that ^

[alpine sable]

sin(20) = 425/hyp
sin(20) * hyp = 425
hyp = 425 / sin(20)

this?

[alpine sable]

oh it isn't the angle? what is it then

[oak chasm]

Yes, the 20° isn't part of a right triangle.

The angle to the left of it is part of the small right triangle.

The angle to the left of it plus the 20° is part of the large right triangle.

But the 20° itself isn't part of a right triangle.

[alpine sable]

oh well forget it then

[analog nymph]

question resolved*

[alpine sable]

is this the same thing as

6/5x

because the numerical coefficient is 6/5 so couldn't the variable go anywhere?

[limpid spade]

NOOO

caps

[alpine sable]

:0

[limpid spade]

(6x)/5 is the same as (6/5)*x

6/(5x) would be (6/5)*1/x

[alpine sable]

ohh

okie

ty

but the numerical coeffiuevciitn for that question is 6x/5 right?

[limpid spade]

?

[alpine sable]

*coefficient

i mean 6/5

sorry

[limpid spade]

what's the question

[alpine sable]

this

[limpid spade]

:/

[alpine sable]

the numerical coefficient

is it 6/5

[limpid spade]

yes

[alpine sable]

ty

the answer key says these are equivilant

can someone pls explain how these are equvilant

[limpid spade]

they're just rearranged

[tight locust]

a-b+c = c-b+a

[jade birch]

commutative property

[tight locust]

this is called the commutative property of addition.

A+B = B+A

this is true in all fields

[alpine sable]

okie

but what about the variables

and the powers

what do you do with those

[glass lichen]

nothing...?

$a+b+c=c+b+a$

[ocean seal]

Mosh

[alpine sable]

like in here

what do you do with the x and ^2

[glass lichen]

they're part of the "a"

cause it's all one term...

[alpine sable]

okie ty

[agile crescent]

hey guys just wondering if someone could tell me if this is right?

[glass lichen]

yes

that's the reciprocal identity of tan

[agile crescent]

thanks

[keen crane]

lim of cos(1/x) as x->0, why is it undefined?

[tight locust]

ok

let u = 1/x

[tawny condor]

ABCD is a square.
FD = 2CF
BE = CF

a) Find DG/CF (I found that it is 6)
b) Show that AHFD can be blocked inside a circle (done)
c) Find angle DHF (I need help with this)

[tight locust]

then x = 1/u. so you are taking the lim as u goes to inf

[tawny condor]

Can somebody help me with this?

[tight locust]

and cos(inf) is not a well defined expression

the most you can say about it is that it is between -1 and 1

[tight locust]

alright this is very simple.
triangle BCF is similar to triangle GAB

[tawny condor]

yeah

[tight locust]

Do you have any exact measurements?

[tawny condor]

Nope

You can calculate angle AGB though

with inverse tan in triangle AGB or DGF

atan(1/3) which is 18.43494882

[keen crane]

i looked at the answer, and it said it DNE

wouldnt there still be an answer, because it still approaches a number, its just that there would be a hole there

[tight locust]

no it does not approach a number

cos(1/0) = cos(inf)

there is no such thing as cos(inf)

it literally does not converge to a value

,calc cos(1/(0.1))

[ocean seal]

Result:

-0.83907152907645

[tight locust]

,calc cos(1/(0.01))

[ocean seal]

Result:

0.86231887228768

[tight locust]

,calc cos(1/(0.001))

[ocean seal]

Result:

0.5623790762907

[tight locust]

,calc cos(1/(0.00001))

[ocean seal]

Result:

-0.99936080743769

[tawny condor]

can I get some help aswell

[tight locust]

what exactly are you having trouble with proving?

[tawny condor]

finding angle DHF

c) Find angle DHF (I need help with this)

[carmine notch]

Whats the difference between round brackets and square brackets when determining domain and range

[tawny condor]

square brackets is including the number, parenthesis are not including

so [0, 3) will be from 0 to 3, including 0 but not including 3 (there is a hole at 3)

[tawny condor]

bump

[carmine notch]

like a whole where the coordinates are located?

hole*

[tawny condor]

yes

not including that specific number

[carmine notch]

alright thanks

[tight locust]

@tawny condor you said that AHFD is cyclic, correct?

whew this was a tricky one but i found the answer

or what exactly did you mean by "blocked inside a circle?"

,w arctan(2/3)

[ocean seal]

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Requested by EndTimes#8053

[alpine sable]

what is cos 😭

[sand jetty]

uh

[alpine sable]

are those just variables put together?

[sand jetty]

thats a long one, sure you have to do this exercise and you didn't learn what cosine is?

[tawny condor]

I don't understand this

Oh

wait

I seeeeee

that's actually very simple

ty

[tight locust]

@tawny condor

[tawny condor]

yes I got it already

:D

[tight locust]

yeah just explaining how

it's tricky if you don't know the cyclic quadrilateral theorems

[tawny condor]

yeah dw I remember that rule

[alpine sable]

what is cosine im just learning polynomials and then this was my last question 😭

[tawny condor]

I know everything unfortunately

tysm

[alpine sable]

k imma just try smth and see how it goes lol

[tawny condor]

@alpine sable you aren't supposed to know that at all then

have u learned trigonometry?

[alpine sable]

yeah it was just a bunch of letters

that were variables put together

idk

it just says this

[tawny condor]

Oh

why would they put cosine there lmao

[alpine sable]

lmaooo

what even is cosine 😭😭

[tight locust]

the question is whether f(x) = cos(x) + x is a polynomial

which it is not

well it sort of is but not really. infinite polynomials are not polynomials

[alpine sable]

me = confusion

[tight locust]

don't worry about it. a polynomial is just something that looks like this:

[alpine sable]

hehe

[tight locust]

$\sum\limits_{k=0}^n a_kx^k = a_0x^0 + a_1x^1 + a_2x^2 + ... + a_nx^n$

[tawny condor]

why would u use sigma

[limpid spade]

pretty sure they didn't see sum

[alpine sable]

daveie

[ocean seal]

EndTimes

[limpid spade]

Dave is in 7th grade

[tight locust]

where the sequence $a_k$ are the coefficients of the powers of x

[ocean seal]

EndTimes

[limpid spade]

I mean strawberry

[tight locust]

sigma notation is very easy. i can teach you how to use it if you want

[limpid spade]

teach they integrals afterwards 😂

[tight locust]

well if you insist. reimann sums are not too difficult to explain

[limpid spade]

and u can end with topology

[tight locust]

klein bottle

WOAH

[limpid spade]

and if they want a little challenge

u can teach them Inter-Universal Teichmuller Theory or Prime Dimensional Matrices

[tawny condor]

wtf

[alpine sable]

can anyone confirm the last one? pls

[vague coral]

it's possible tho

this function is bijective

[alpine sable]

that's not injective, clearly there are several x's for the same y

oh wait

miss read n with 1 right away

[vague coral]

f(n) = n/2 if n is even
f(n) = -(n+1)/2 if n is odd

[alpine sable]

sorry

interesting

very nice

thanks!

[vague coral]

np

[ocean seal]

69420

[alpine sable]

wdym?

[ocean seal]

69420

[alpine sable]

it can

[vague coral]

what are u talking about

[alpine sable]

but ye xd

[glass lichen]

post the question

and dont ping helpers right away

Ok.. also it's very unclear what's in what

secsecx-tanx for example

[somber osprey]

Mathematics

[alpine sable]

@everyone

[somber osprey]

Lol

[knotty sierra]

how do we do b bit?
n/4
is used to find lower quartile
but the number are starting from 150
so i did 190-150
which is 40
40/4
=10
150+10=160
and 160 is included in 150<h<=160
but the answer key says thats wrong
please tell me where i made a error

[west tartan]

Help, I need to know what X is. And I need to understand how you got it because I need to transfer that into JS for a project I'm working on.

[thorn kindle]

3.67X + X = X(3.67+1) = 4.67*X

[west tartan]

ok hold on i messed something up here

[thorn kindle]

Oh really?

[west tartan]

8720=(X*0.0367)+X

the final number is the sum of the unknown number plus 3.67% of the unknown number

[thorn vortex]

$8720 = X(0.0367 +1) \implies 8720 = X(1.0367)$

[west tartan]

It just defines x as 1?

[thorn kindle]

You are getting your numbers mixed up.

Is it 0.037 or is it 3.67%?

Those values are different

[west tartan]

its 3.67%. i thought i could write 3.67% as 0.0367. is this incorrect?

[thorn vortex]

you wrote 0.037 instead

[thorn kindle]

Yes that is correct. But 3.67% is NOT 0.037

[ocean seal]

Pealover

[west tartan]

Am i stupid lol, 1 is 100% 0.1 is 10% 0.01 is 1% correct?

oh no

i am stupid ok, ok i see

hold on

[thorn vortex]

yes

[west tartan]

3.67%

im sorry

i misunderstood his question

8720=(X*0.0367)+X is the final question i need help with.

[thorn kindle]

0.0367X + X = X(0.0367+1) = 1.0367*X

[west tartan]

Wait what did you do here? 0.0367X + X = X(0.0367+1)......how did you getX(0.0367+1)

[thorn vortex]

factorization

[thorn kindle]

This is called the "distributive property"

[thorn vortex]

$ab + ac = a(b + c)$

[ocean seal]

Pealover

[west tartan]

Where did the 1 come from

[thorn kindle]

X/X=1

[thorn vortex]

because $X/X = 1$

[ocean seal]

Pealover

[west tartan]

so we are left with 8720 = 1.0367*X

[thorn vortex]

yes

[west tartan]

so divide both sides by 1.0367?

[thorn vortex]

exactly

[west tartan]

OH MY GOD

i knew it had to deal with 100%+ but i lost my train of thought and couldnt get it back thank you so much

[thorn vortex]

np

[remote lodge]

haven't done this in a while, so without giving me the answer how do i do this?

[jagged imp]

try write sqrt(32) in terms of sqrt(2) so you can combine the two terms. if you need guidance on how to do that say it

if its the rationality part you're struggling with then say that

[remote lodge]

I just forgot how to simplify

[thorn vortex]

try to split the number into product of prime numbers

@remote lodge

[remote lodge]

Yeah that's what i was thinking

[proud estuary]

How would I work this out, not the answer, just how do I work it? One bag of fertilizer will cover 1,225 ft2 of lawn. A front lawn is approximately 115 ft by 100 ft, and the back yard is approximately 130 ft by 100 ft. How many bags of fertilizer will be needed to fertilize both the front and back yards

[surreal meadow]

you are given a few key pieces of information:
how much area a bag can fill
the side lengths of one of the areas
the side lengths of the other area

[proud estuary]

yes that I understand but what do I do with it, I have not had to do this yet?

do I add? divide what?

[surreal meadow]

think about it for a second

you know how much each bag can fill

so the first thing we want to find

is how much area we are looking to fill up

how would you go about that?

[proud estuary]

I don't know this is why I am asking

not sure how to find it

[surreal meadow]

How many bags of fertilizer will be needed to fertilize both the front and back yards

reading that^

what are we tring to fertilize?

[proud estuary]

the front and back yard

[surreal meadow]

right, so it follows that we want to find out the areas of the front and back yards

[proud estuary]

so how do I find the areas?

[surreal meadow]

read the question and find the information given

A front lawn is approximately 115 ft by 100 ft, and the back yard is approximately 130 ft by 100 ft.

it's safe to assume these are rectangular areas

[proud estuary]

not sure where your going

[surreal meadow]

do you know how to find the area of a rectangle?

[proud estuary]

no

[surreal meadow]

i'm assuming you're in algebra 1?

[proud estuary]

yes but Ineed to undertand can you tell me the steps I need to take?

[surreal meadow]

yes

the question asked us "How many bags of fertilizer to fertilize both the front and back yards"

so our solving process is as follows:

1. find out the total area we want to fertilize

1a. this is composed of the front and back yards, so we must find the area of both of those

1b. sum up those areas

2. create an equation that tells us how much area X bags will give us

3. solve that equation when the area we want to fill is equal to that of the front and back yards together

so let's look at point 1.

the front and back yards are rectangular, so to find their areas we need to do length * width

A front lawn is approximately 115 ft by 100 ft
the length is 115ft, and the width is 100ft. @proud estuary what is the area?

its not 11 everywhere

[proud estuary]

11,500

[surreal meadow]

these comments are a little disruptive

😐

[surreal meadow]

yes, and get used to adding units at the end

in this case: 11500 ft^2

[thorn vortex]

yes

[surreal meadow]

alright now @proud estuary do you think you can find the area of the back yard?

[proud estuary]

13,000

[surreal meadow]

this is disruptive

can you not @dense scaffold

[surreal meadow]

yes

[surreal meadow]

@proud estuary what would be the total area then? (look at what i'm replying to)

[proud estuary]

24,500

[surreal meadow]

great

[surreal meadow]

this is the tricky part

how much area can we fertilize with 1 bag?

[proud estuary]

1225

[surreal meadow]

exactly, so how much would 2 bags be able to fertilize?

[proud estuary]

2450

[surreal meadow]

yup, so i think you understand how that works

so we can say that X bags will fill up an area of 1225 * x ft^2

so we create a function A(x) = 1225x

[gray isle]

<@&268886789983436800>

[surreal meadow]

which tells us the area fertilized by x bags

[surreal meadow]

the final step is finding the number of bags from that function

[proud estuary]

@surreal meadow thank you so much I got it, it is 20 bags

[surreal meadow]

awesome 🙂

[proud estuary]

Got time for one more?

[surreal meadow]

yes i do

[proud estuary]

A commercial airline plane with 250 passengers on a 3250 km flight from Seattle to Detroit averaged 650 kilometers per hour and used fuel at a rate of 5000 liters per hour? What is the fuel consumption in gallons per passenger (gal/passenger)

[surreal meadow]

thank you @ namington

[surreal meadow]

ok, after you've read the question, can you tell me how much fuel the plane uses per hour?

[proud estuary]

5000 liters

[surreal meadow]

great

and how long is the flight?

or the distance that is

[proud estuary]

3250km

[surreal meadow]

ok so you understood the info given. let's move forward

we know that the plane travels at 650 km/h

and the flight is 3250km long

so how many hours will it take for the plane to get to its destination?

[proud estuary]

5hr?

[surreal meadow]

yup

so how much fuel will the plane use up?

[remote lodge]

[surreal meadow]

(given that it uses 5000 liters per hour)

[proud estuary]

25000?

[surreal meadow]

exactly

so now we want to find the final ratio, gallons/passangers

can you write a ratio of liters/passanger from what we've solved so far?

[proud estuary]

this part I do not understand

[surreal meadow]

alright, so the question is asking us for the fuel consumption per passenger

and we know the following: 25000 liters of fuel were used up to transfer 250 passengers

so the ratio of fuel (in liters) to passengers is 25000:250 or $\frac{25000}{250}$

[ocean seal]

maximo

[surreal meadow]

does that make sense/look familiar?

[proud estuary]

no sorry it does not

[surreal meadow]

let's think of it the other way around then

250 passengers traveled with 25000 liters of fuel

so 250 passengers used up 25000 fuel, how much fuel did each passenger use up?

[proud estuary]

this is likely wrong but 1 gallon?

[surreal meadow]

let me ask a different question, if 2 people ate 10 apples, how many apples did each person eat? (assuming the apples are distributed equally among the people)

[proud estuary]

5

[surreal meadow]

right

so now instead i'll ask

2 people used up 10 liters of fuel

how much fuel did each person use?

[proud estuary]

5

[surreal meadow]

perfect

now let's use bigger numbers

25000 liters of fuel were used for 250 passengers

how many liters were used for each passenger?

[proud estuary]

100

[surreal meadow]

yeah

[remote lodge]

[surreal meadow]

so the fuel consumption in liters per passenger is 100 liters/1 passenger

did that make sense?

[proud estuary]

no

[surreal meadow]

what about it didn't make sense?

[proud estuary]

nvm it did I read it wrong

so the answer is 100

[surreal meadow]

not quite

[proud estuary]

ohhh

[surreal meadow]

we now have a fuel consumption in liters/passenger

but we want gallons/passenger

do you happen to know the conversion rate of liters->gallons?

[proud estuary]

no

[surreal meadow]

yeah same here, but i'm assuming the question provided it

[surreal meadow]

^ that right there is a good starting point

we know that 1 gallon = 3.78541 liters

so how many gallons are in 100 liters?

[proud estuary]

26.4172

[surreal meadow]

looks about right

[proud estuary]

so the answer is 26.4

[surreal meadow]

26.4 gallons/passenger

[proud estuary]

@surreal meadow I wanted to say thank you, your a life saver, I would not have figured these out without your help, you are very appreciated.

[surreal meadow]

here to help

[remote lodge]

[alpine sable]

how can I solve this?

plz

[placid zinc]

Might be easier to figure out what volume the tetrahedron doesn't take up

Realize there's 4 of the exact same "missing volume"

[alpine sable]

yeah but how do I do that?

[placid zinc]

Can you find the length of any of the sides of the tetrahedron?

[alpine sable]

1st Diagonals of the square. 2nd sides of the square. 3rd Angle between diagnal and the sides are 45 degrees (maybe use sin or cos here...) Then you can calculate the volume of the one of the "missing volumes"

pythag>

[surreal meadow]

A tetrahedron fits perfectly inside a unit cube

[placid zinc]

Kek

[surreal meadow]

do you know what each of those words mean?

[placid zinc]

I guess the fit isn't really perfect

[surreal meadow]

oh lol ig not

A tetrahedron fits perfectly inside a unit cube
@alpine sable

[alpine sable]

hmm

i still don't get it

is it 1/3?

[surreal meadow]

a unit cube has side lengths of 1

[alpine sable]

a unit cube is 1x1x1 right?

[surreal meadow]

if you see "unit (shape)" chances are it's defining feature is of length 1

[alpine sable]

thx

I solved it

it's 1/3

[fresh bolt]

Find the inverse s of -1959 modulo 979 such that 0 ≤ s < 979. You must show all the detailed steps.

[alpine sable]

it's correct

[zealous fractal]

how to prove this cos^4 x + cos^2 x sin^2 x = cos^2 x

[tidal sparrow]

Take the common factor out

Left with answer😂

[zealous fractal]

oh jesus my brainnn

[rough zinc]

How would I write 4x^(10/3) with a root and whole exponent? I'd assume it to be curt(4x^10) but I'm really not sure--sorry.
I have a really icky foundation in math because I moved from Sweden to the United States when I was young. My american schools have had a lot more rigorous math programs than my Swedish school did.

[tough hatch]

10/3 = (a/3) + (b/3)

where a is the greatest integer less than 10

[old sable]

Curt(4x^10) is correct I believe

[gray isle]

assuming curt is supposed to mean cube root, it is not

[tough hatch]

where n and a are integers

[rough zinc]

I don't quite understand what the bot is doing but yes, something like this

[tough hatch]

edited for brevity

but you are familiar enough with the rules of exponents, yes?

@rough zinc

[rough zinc]

I mean

I obviously have some glaring holes with exponents

so I guess the answer would be not entirely

[tough hatch]

x^(a+b) = (x^a)(x^b)

10/3 is the sum of _ and _

both are something over 3

one of them is the greatest integer less than 10

the other is made obvious once you know the first addend

[rough zinc]

am i missing something or is that just nine

[tough hatch]

yes that's correct

10/3 = 9/3 + (?)/3

[rough zinc]

oh so 30

[tough hatch]

eh?

[rough zinc]

fuck me what am i doing

10/3=3+1/3

[tough hatch]

yes

you got it

you see what to do now?

[rough zinc]

but how does that help?

[tough hatch]

x^(a+b) = (x^a) (x^b)

[tough hatch]

probably didn't pay attention to this earlier

[rough zinc]

I checked with a dummy number and it is this

or?

[tough hatch]

do you not understand what this is saying?

[rough zinc]

I do but i don't see how it helps?

[tough hatch]

4x is your x

a is 3

and so on

a+b is the sum you mentioned earlier.

[rough zinc]

4x^3+cbrt(4x)?

[tough hatch]

no

why +?

and

[rough zinc]

oh sorry that's not meant to be there

but that doesn't make sense factor it in and it's 9 not 10

[rough zinc]

ok yeah right

[rough zinc]

here there's a three and a three but how does that make a 10 and a three?

[tough hatch]

because cubrt(4x) = (4x)^(1/3)

[rough zinc]

that's a three and a three

no 10

[old sable]

Because 10/3 is equal to 3 and 1/3

[rough zinc]

Oh sorry

i'm an idiot thank you so much for being so patient

[rough zinc]

also sick bot

[tough hatch]

yeah, you forgot already about how you got 3 (from 9/3)

[rough zinc]

Yies

[tight locust]

wow!

great job!

[untold warren]

Got a quick q

trying to isolate for n on the n^3 summation formula so i can create a program, but just wondering how I could isolate for n to see if I can solve n for some summation of n^3

sorta like this

the line above is where i try to see if I can find a way to isolate for n to solve that n = 2 in that case

[dire nacelle]

Please help

[gray isle]

where are you stuck

[fringe robin]

int x^n is x^n+1/n+1 and for cot^2 x is 1+cosec^2 x and int for cosec^2 x is -cot x

[hollow cypress]

what's the Magma for the empty set?

[placid zinc]

I like to call it George

[hollow cypress]

okay, aside from the startle, since that's my name, i'm out of the loop...
woosh it goes

[sly mantle]

an empty function

[hollow cypress]

i guess this exists ⊥ → ⊥ → ⊥ but it's kind of weird. are there any uses for it?

[sly mantle]

no

[placid zinc]

Oh wow I swear I didn't know your name haha

[copper heath]

Can someone help me? There's this question that was asked after the lecture where i can't rlly wrap my head around so insights are welcome

You are given an infinite countable set X and you define X^X is this still countable?

What do they mean by X^X how do you take the power of a set?

My feeling is that it's uncountable take X = N then you have N^N that seems uncountable but it's not rigorous in any way

[vale wigeon]

$X^Y$ is the set of all functions from $Y$ to $X$

[ocean seal]

Ann

[vale wigeon]

and $X^X$, accordingly, is the set of all functions from $X$ to itself

[ocean seal]

Ann

[copper heath]

aah u need to read it like that okay that makes sense

so it's basically a set made up of bijections

aldus countable since they're all bijections?

[vale wigeon]

no

not bijections

i said all functions

not all bijections

some functions from X to X are not bijections

[alpine sable]

I'm 13

Going into yth

7th

This makes no sense to me

[rigid smelt]

What doesn't make sense? Do you feel like going to 7th grade is too fast for you?

[jagged imp]

make a wish kid has the right skills for 7th grade at least

[covert agate]

hi

what’s the point of modular inverse if you can solve congruences using extended euclidean algorithm

[placid zinc]

What does it mean to "solve a congruence"?

Are you talking solving something like:
5x = 2 (mod 23)?

[covert agate]

yea

[placid zinc]

Bad example there's an obvious solution to that one haha

[covert agate]

just make it into a diophatine equation and solve for x

[placid zinc]

You make it sound easy lol

[covert agate]

no inverses needed

[covert agate]

eh at least there’s a way

for ax ≡ b (mod n)

i see modular inverse is usually introduced after linear congruences in number theory courses

whats the use of it if you cant solve eqns with ir

it

[placid zinc]

You can solve equations with it kek

I'm wondering what method you're currently thinking of

5x = 2 (mod 167)

I suppose you could just do euclid algo to get:
5m + 167n = 2

But then what if you get asked
5x = 27 (mod 167)?

If you did the exact same amount of work to get the modular inverse of 5, you could do both questions with no extra effort

But if you get 5m + 167n = 2 directly, you now have to repeat the process for 5m + 167n = 27

I guess that's pretty fringe haha, but in general the modular inverse is more info than other alternatives

[mellow shard]

hey i need some help with my maths project anyone ready for it?

[alpine sable]

i was going through linear algebra gone wrong:

does anyone know how to do example 1.5? there are no solutions for it 😦

[sly mantle]

find a 2 by 3 matrix $A$ so that $A+M=M+A=M$ for all $M\in M_{2\times 3}$

[ocean seal]

Dreadful Encore of Twisted Karma

[hybrid patio]

Hello, in a lesson I was learning beginner trig and in this part of the equation i put 5sin30 and got -4... and my teacher said it was 2.5 not sure how?

oh wait nvm

was using google's calc and it messed it up

i used my normal calc and it works

[placid zinc]

@alpine sable
You can likely guess it. What matrix in M2×3 would best represent the idea of 0?

[alpine sable]

a matrix with all 0s?

[placid zinc]

Exactly haha

[alpine sable]

... what a useless question 😠

[placid zinc]

Not at all!

Note what Dreadful has said above, we call this matrix a 0 because it doesn't change any other matricies with addition

This 0 has to exist, or else this isn't a vector space

[alpine sable]

I guess it is just trivial imo

also what does \mathbb F here mean?

field? what does that mean?

[sly mantle]

F is an arbitrary field

[alpine sable]

a vector space of bases?

[placid zinc]

What form 0 takes is usually trivial, yes.
0's implications are not trivial at all

[vale wigeon]

"a vector space of bases" what

[alpine sable]

what is a field?

[placid zinc]

You might have missed fields in your book, take a quick sec to look over it

[vale wigeon]

loosely speaking, a field is a set upon which you can do arithmetic

[alpine sable]

i know of galois fields with mod arithmetic

[vale wigeon]

i.e. add, subtract, multiply and divide (except by zero)

[alpine sable]

but what does that have to do with vectors

[vale wigeon]

...so you know the axioms that define a field, yes?

[alpine sable]

you can divide in fields?

[vale wigeon]

of course you can divide in fields

[alpine sable]

😮

[vale wigeon]

you can take reciprocals and you can multiply

R is a field lol

as is C

as is Q

[alpine sable]

no... i learned in a weird order 😅

[placid zinc]

A vector space is two structures squashed together.
The vectors form a group.
The scalars form a field.

[vale wigeon]

the axioms themselves are bureaucratic

kinda

[alpine sable]

but ik them now

i just searched it up

ty!

[vale wigeon]

mostly to ensure arithmetic works as you would expect it to

[placid zinc]

"squashed" means the vectors and scalars interact in specific ways

[alpine sable]

i think i got it? is this right:
a single vector is in a field^n as it is a collection of scalars
but a collection of vectors is a group/vector space?

[vale wigeon]

a single vector is in a field as it is a collection of scalars
no

a vector shouldnt be thought of only as a list of scalars

[alpine sable]

do fields have an inherent ordering then?

[placid zinc]

It ends up being the case that all finite dimensional vector spaces can be represented this way haha

But no, that's not why something is a vector

[covert agate]

so it provides solutions for all ax ≡ b (mod n) provided a, n unchanged?

[placid zinc]

@covert agate
Yaya. It solves for all b at once

[patent swan]

0hai =]

How often do division-derived branches of the Fibonacci sequence outgrow the main sequence?

for example:

1 1 2 3 5 8 13 21 34
21 / 3 = 7 ; 7 * 5 = 35 ; 35 > 34

1 1 2 3 5 8 13 21 34 55 89 144 233
144 / 8 = 18 ; 18 * 13 = 234; 234 > *233

P.S.: I wish Markdown had colors for highlighting math sequence members for sequence property math... IRC had %C commands...

[covert agate]

what’s the point of that

[placid zinc]

Kek I guess there isn't much point of that

[covert agate]

if it’s a single problem

[placid zinc]

More info with the same amount of work

[covert agate]

proving things?

[placid zinc]

It does provide a bit of insight. That is, every number coprime to n has an "opposite". This helps wrap our heads around what's really going on, algebraically

This modular inverse pairs numbers together nicely

[patent swan]

so, does anyone know how often division derived branches of Fibonacci sequence outgrow the sequence itself? (see examples above, where I showed the first two such derivation points)

the first derivation point, 8, produces a very weak branch (8/2=4 ; 4*3=12 ; 12 < 13), 21 is the first derivation point where the branch outgrows the sequence

maybe this is the wrong place to ask this question...

[placid zinc]

Are you choosing your starting numbers at random?@patent swan

Like in that second example, 144 and 8

[patent swan]

no, I choose all Fibonacci numbers divisible by previous fibonacci numbers, 8 is the only way to derive a stronger branch out of 144

[placid zinc]

Ah I see. Then you multiply by the number after 8, and compare that to the number after 144

[patent swan]

divide 144 by 8, multiply result by 13 (next after 8), and compare to 233 (next number)

[placid zinc]

That's exactly the same as comparing
144/8 to 233/13

Except you only do this comparison when the first is an integer

[patent swan]

f_t+1-((f_t/f_d)*f_d+1)

dang, why doesn't Discord do super/sub?

[placid zinc]

Actually, that's exactly the same as comparing
144/233 to 8/13

[patent swan]

#imagineaplace, dang it hang it!

with non-integers like that, I'd think the outward branches would be more frequent than with an integer-only search

[placid zinc]

I mean you can choose to only include the search if 144/8 is an integer

But we can make the search more modular with this

[patent swan]

the first two integer outward branches (from 21 and 144) had their initial outgrowing by only 1, and were exactly 4 spaces apart within the sequence (members number 8 and 12)

[placid zinc]

I bet, actually, that this fraction is directly computable

Well, I guess it all is, haha

[patent swan]

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765
bold confirmed, underscore questioned, crossout means weak branch

10824 < 10946

this means 4 isn't quite accurate even for integers

this is important for the statistical mean risk of a double lash within the perimeter (not the twister core itself) of the cyclone, and other statistic mean intensity increases for things that exhibit the golden spiral pattern

it's also important for computing likelihood of double fronts of cyclones and anticyclones giving that extra strength to thunderstorms

living in a place with both sea and mountains nearby, the weather can be quite turbulent at times, and the forecast thing is a mess...

given the initial 1 1 gives a half circle and multiple spirals can emerge from a single completely circular core, I need to know the angular mean for outward branches of fibonacci spirals

what'd be the shortest formula for finding outward branches of the Fibonacci sequence/spiral?

[clear vessel]

Hey, if y = ax^k how do I write In y as subject?

[placid zinc]

Take the ln of both sides:
ln(y) = ln(ax^k)

Note the right can be expanded like so:
ln(y) = ln(a) + kln(x)

@clear vessel

[harsh steeple]

its known that AC = CD, how can i show that AC = CD = AD? it looks quite obvious

[median tendon]

well circles are symmetrical about the centre

[alpine sable]

Dude I'm horrendous at math I'm good at English

This stuff blows my mind

[alpine sable]

Like what does this even mean

[harsh steeple]

?

[alpine sable]

I'm going in 7the grade and this math stuff blows my mind I suck at math

[weary jolt]

is there any other info given?

[harsh steeple]

for everything i have studied it hugely depended on how i approached it, stuff that iwanted to learn came easy to me, stuff that i didnt, simply did not

[median tendon]

if u can somehow proove CQ=QD

[harsh steeple]

oh yeah sorry about that

it is

AP also equals to PC

[median tendon]

then u can proove using pythagoreas theorem

[harsh steeple]

oh

haha i feel so dumb now

thank you!

[median tendon]

np

[alpine sable]

Oh I understand

[weary jolt]

lol

[harsh steeple]

i was also like you, just approach it differently

enjoy math

[alpine sable]

Ill take a new approach

[harsh steeple]

im kinda stupid lmao, can also prove it by proving that ACQ is congruent to AQD

[ocean seal]

5'1 filipino subhuman incel

[median tendon]

what is your question

is it just solving the equation?

also both equations are satisfied, not just one of them

[alpine sable]

,tex He only tested $x = 6$ on
$$\sqrt{3x - 7} + \sqrt{2x - 1} = 0$$
and when it didn't work he said $x = 6$ satisfies $$\sqrt{3x - 7} - \sqrt{2x - 1} = 0$$ and didn't test $$-\sqrt{3x - 7} - \sqrt{2x - 1} = 0$$ or $$-\sqrt{3x - 7} + \sqrt{2x - 1} = 0$$

[ocean seal]

5'1 filipino subhuman incel

[median tendon]

maybe because when he did the first equation, he saw it was 11+11 and so putting a negative sign infront of one term would satisfy the problem?

[alpine sable]

Your third equation always works if the first works. And the fourth always works if the second works

[median tendon]

^ just multiply -1 by both sides to see that

[alpine sable]

Lmao I am dumb

I got it now, thanks

[median tendon]

np

[rigid smelt]

,rotate

[rigid smelt]

use logarithms

uhhh use logarithms? especially take advantage of the property that log_a(a^x)=x

[vivid comet]

i forgot what logarithms are or i just don't recognize the name in English. what do i do exactly?

[rigid smelt]

ok firstly tho, is that 1.02 or 7.02?

ok

so try to isolate (1.02)^n and then take the log base 1.02 on both sides

this is a giveaway at this point

[alpine sable]

First write your function as an equation: $k=2500 \cdot 1.02^n$

[ocean seal]

farbbohrer

[vivid comet]

then do i take 2500 over to the other side of the =?

[alpine sable]

yes

Do you multiply or add 2500 on the right side?

[vivid comet]

multiply so k/2500=1.02^n

[alpine sable]

yes

[vivid comet]

now what is this logarithm i have to do to get n

[alpine sable]

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:log-intro/v/logarithms

[covert agate]

it should be true for pairwise coprime n1 … nk

[covert agate]

but are there cases where its true without this

[vivid comet]

how do i enter this into my calculator? (ti 84)

i watch it and understand logs but idk how to enter it

[alpine sable]

You should have a log button on your calculator. Depending how your calculator works you either can do calculate the log with the correct base or you need to use a trick to calculate it with the correct base.

But you dont need to that for your current task.

You "just" have to rewrite it with logarithms.

[vivid comet]

now what?

[alpine sable]

Question: What is $log_1.02(1.02^n)$ ?

[ocean seal]

farbbohrer

[alpine sable]

Sorry let me reformat that

Question: What is $log_{1.02}(1.02^n)$ ?

[ocean seal]

farbbohrer

[alpine sable]

Forget your calculator

It can only calculator the logarithm for the base of 10. But our base is 1.02

[vivid comet]

then what do i use?

[alpine sable]

Your brain

[vivid comet]

breh

rn i got 2=1.02^n

[carmine lion]

lmfao

use ur log rules

u don't need a calc for this

https://cdn.discordapp.com/attachments/490557019623915520/878978266801799238/524261734396067840.png

let me ask a similar related question

Question: What is $log_2(2)$ ?

[ocean seal]

abe

[vivid comet]

what?

[alpine sable]

$log_2(16)$ is the question as to what power should I raise 2 to get 16.

[ocean seal]

farbbohrer

[carmine lion]

this channel is occupied

[alpine sable]

So what would $log_{1.02}(1.02^n)$ mean

[ocean seal]

farbbohrer

[tidal solar]

which channel can i get my answer from?

[vivid comet]

to what power do i have to raise 1.02 to get 5000/2500 or 2

[carmine lion]

any channel that isn't occupied

[carmine lion]

not quite

do you see that $log_2(2^2)=2log_2(2)$

[ocean seal]

abe

[alpine sable]

Maybe you should read or watch a bit more about logarithm

[vivid comet]

i just want a way to enter log1.02(2) into my calc cause i think i got that far at least.

[alpine sable]

I will you give a hint how to calculate the logarithm with your calculator (if there are no variables in it).

$log_{1.02}(2)=log(2)/log(1.02)$

[ocean seal]

farbbohrer

[vivid comet]

so they're the same?

[alpine sable]

right side is what you should put in in your calculator

[vivid comet]

got it but i don't quite understand the inputs yet.

so i have my answer but i don't get how i got there

in the vid u send he entered his formula differently.

[alpine sable]

$5000=2500\cdot 1.02^n$

[ocean seal]

farbbohrer

[alpine sable]

$5000/2500=1.02^n$

[ocean seal]

farbbohrer

[alpine sable]

then we use logarithm

$log_{1.02} (2) = log_{1.02} (1.02^n)$

[ocean seal]

farbbohrer

[alpine sable]

right side equals n

so we get $log_{1.02}(2)=n$

[ocean seal]

farbbohrer

[alpine sable]

Which you have to calculate with your calculator using the trick i posted

$\frac{log(2)}{log(1.02)}=n$

[ocean seal]

farbbohrer

[alpine sable]

So we get $n\approx 35$

[ocean seal]

farbbohrer

[scenic pulsar]

Hey there just wondering why we are using sin for i-direction and os for j-direction, can't seem to wrap my head around this

[alpine sable]

What is the definition of sin and cos?

[scenic pulsar]

wdym?

Like what they mean?

[alpine sable]

Yes. You use them in triangles and there they have a definition.

[scenic pulsar]

I mean sin=opposite/hypotenuse

cos=adjacent/hypotenuse

[ocean seal]

irf

[alpine sable]

at this point when i need to find the derivative, should i open the parentheses?

[median tendon]

Don't need to, use chain rule

[alpine sable]

whats that

[median tendon]

You dont know chain rule, quotient rule and product rule?

[alpine sable]

i probably do but i dont know how its called in english

ill google it rq

oh yeah i know what it is

ok ill use it thanks

[median tendon]

np

[alpine sable]

i know i dont need to open the parentheses but will it make it easier to solve?

[median tendon]

For that, might be like 10 sec faster. But for more complex equations, chain rule saves alot of time

[alpine sable]

oh ok

[rough tusk]

help

[vivid comet]

sry to just keep on asking but how did this happen?

[alpine sable]

@vivid comet 5000/2500=2. And we use the logarithm on both sides of the equation

[vivid comet]

thanks a lot. I think i got it for rhe most part.

[mystic sinew]

why can't you come up with a combination of the real and imaginary bits of a complex number such that it's equal to another complex number?

[vale wigeon]

what do you mean?

[mystic sinew]

like why can't we figure out some x +yi = zi
(x != 0)

I understand this is probably wrong? but I wanna know why it's wrong

[vale wigeon]

so you are essentially asking why two complex numbers that have matching real and imaginary parts are in fact equal?

[mystic sinew]

no

that's not what I am asking at all

[mystic sinew]

I am asking why we can't find some combination of x and y in this that works?

[median tendon]

Can this combination involve multiplication? Or just addition and subtraction?

[vale wigeon]

complex numbers are but points in the complex plane

you are asking whether or not it is possible to have (x,y) describe the same point as (0,z) without x being 0

assuming you intended for x, y and z to be real, of course

[mystic sinew]

I did

[median tendon]

If that helps visualise it for u

[mystic sinew]

that sounds like it has more to do with the construction of the human definition of complex numbers and not a "mathematical truth"

it's a correct answer though I guess

[alpine sable]

How does subtraction work?

[mystic sinew]

fair point mr far

[vale wigeon]

http://lmgtfy.com/?q=how+does+subtraction+work

[mystic sinew]

I think he's trying to make an argument

[alpine sable]

Things in math are equal if you subtract them from each other and the solution is 0

[mystic sinew]

gonna go for some meditation be back in 5

[alpine sable]

@mystic sinew In complex numbers it would be better to argue how can something be equal if there is a distance between those points. And we can easily show that the distance between such points is always greater than x.

[alpine sable]

What's the answer to this and where can I practice with these types of questions?

[remote heron]

3 to 4, i think

[ocean seal]

codeo

[calm solar]

There are 20 more men than women in a movie theater. If 10 more married couples had entered the room, the number of men would have been twice the number of women.

Express the two unknowns in the sentences given above with a system of equations.

Men = M
Women = F

M = F + 20
M + 10 = 2. (F + 10) =>>> Why is not 2.(M + 10) = F+10

[ocean seal]

codeo

[hearty cliff]

the first equation should be M = F+20

[remote heron]

https://www.onlinemathlearning.com/ratio-proportion-gre.html but if you look for quantitative reasoning ratio problems, these are very common and a lot of people struggle with these so theres a lot of practice out there

[hearty cliff]

the new number of men M+10 is twice the new number of women F+10

so M+10 = 2(F+10)

[alpine sable]

Thanks guys

[vale wigeon]

are we assuming all 10 entering couples are m/f?

[hearty cliff]

yeah haha

[tranquil tulip]

how do u find the area of this?

[vale wigeon]

who pinged me

[median tendon]

Break it up unto smaller shapes

[tranquil tulip]

what shapes should i break it up to

[native temple]

Uhhh me saying let me google that for you is impossibly based….then deleted messages as this isn’t the place for discussion

[median tendon]

Into shapes that you know how to calculate the area of

[tranquil tulip]

ok thx

[median tendon]

I can see half a circle for example

[tranquil tulip]

We haven’t learned how to find the area of anything yet

i’m just gonna count

[median tendon]

You dont know how to calculate the area of a circle or triangle or square?

[tranquil tulip]

no i know how to do a square but i forgot circle and triangle

I did it like in 7th grade 3 years ago

[median tendon]

Circle area = $\pi r^2$

Triangle is just half of the area of a square

[ocean seal]

_Youssef_

[tranquil tulip]

do you know how to do 2/1 = -1/3x

[median tendon]

Ok first thing, what is 2/1?

[tranquil tulip]

2

[median tendon]

What do you need to multiply -1/3x to make it be just x?

[tranquil tulip]

ohhh

thanks i forgot algebra for a second

[copper wedge]

need help simplifying 32/27 / -1/6

the answer says its -64/9 but i dont know how to get there

[wary stream]

Do you know the rule when dividing with fractions?

[copper wedge]

its something like multiply the reciprocal right

[wary stream]

Yes

[copper wedge]

but i dont know how u would get a 9 on the bottom

nor would that equal -64 on the top

[sage bronze]

u have to simplify

[wary stream]

You have $$\frac{32}{27} * \frac{6}{-1}$$

[ocean seal]

dldh06

[alpine sable]

hi guys

[wary stream]

You can multiply across, like normal, then simplify or simplify parts of it now

[copper wedge]

technically yes and no

i have 32 Over 27 which is also OVER -1/6

[alpine sable]

how to manage our time?

[wary stream]

Channel busy

[alpine sable]

oh

[wary stream]

I did the multiply by the reciprocal rule

[sage bronze]

$$\frac {\frac {32}{27}}{\frac {-1}{6}} = \frac{32}{27} \times \frac{6}{-1}$$

[copper wedge]

so now my answer wouwld be 192 / - 27 / 6

[ocean seal]

HELLOBELLO

[wary stream]

I already told you, this channel is busy

[wary stream]

How?

[alpine sable]

oh wait

wrong channel

[copper wedge]

no nvm it would be 192/-27

[sage bronze]

yeah

[wary stream]

Now, simplify

[copper wedge]

omg

idk how i missed that

[wary stream]

$$\frac{32}{27} * \frac{6}{-1} \equiv \frac{32}{39} * \frac{23}{-1} \equiv \frac{32}{9} * \frac{2}{-1}$$

[ocean seal]

dldh06

[sage bronze]

which is -64/9

🙂

[wary stream]

Also, this is a math server. Don't know how this relates to math

[golden arch]

How do i check if this is a linear differential equation?

[uncut tapir]

Use a linearity test.
For linear functions
f(x + c) = f(x) + f(c)
f(ax) = af(x)

You want to rewrite the equation as
f(y'', y', y) = 0 and apply the tests

[golden arch]

Thanks

[torpid narwhal]

if i have a set like Q(sqrt(2)) where the elements are of the form a + b*sqrt(2)

(a, b are members of rationals)

if it were a field (which it is) how would i go about finding its multiplicate inverse?

i tried expanding 1 / a + b*sqrt(2)

by amplying with a - b*sqrt(2)

etc

but it's different from a solution i found online

[fervent gust]

what did you get

[torpid narwhal]

a - b*sqrt(2) / (a^2 - b^2)

am i bad at 6th grade algebra?

[fervent gust]

ye

[alpine sable]

if -4x + 3x = -x , what does -x equal to as a number

-1 ?

[fervent gust]

denominator should be a^2 - 2b^2 I think

[torpid narwhal]

omg yeah

thanks

[torpid narwhal]

i mean x is a variable

it can be anything

https://en.wikipedia.org/wiki/Free_variables_and_bound_variables

[alpine sable]

dont'know is there a general one

new here

xd!

[torpid narwhal]

you can evaluate that expression with anything

in the domain of the expression

@fervent gust now im being asked if "the set of all polynomials with integer coefficients forms a field"

but i've never thought of sets of polynomials before so im confused

[fervent gust]

err whats your problem?

you dont know what a set of polynomials is?

[untold marten]

hmmm helo

[torpid narwhal]

basically

[untold marten]

help

so i never take algebra before but i have an algebra test is coming, and its extremely imprtant to me

[fervent gust]

rip

[oak chasm]

@untold marten Algebra 1?

If so, you can learn it here: https://www.khanacademy.org/math/algebra. It has video lessons and practice problems.

If you get stuck on something, you can ask here.

[untold marten]

yay

thats wut im doing lol

[torpid narwhal]

@fervent gust so is it the set of elements of the form n, nx, nx^2, nx^3, nx^4, ..., (with n any integer)

[alpine sable]

khan academy also offers easy practice to build up confidence

that also helps me understand things i don't understand

[fervent gust]

more like $$a_0 + a_1 X + a_2 X^2 + a_3 X^3 ...$$ where $a_0, a_1, a_2...$ are integers

[torpid narwhal]

oh sure

good

[ocean seal]

uyitroa

[torpid narwhal]

so 1/x is not a polynomial?

[fervent gust]

nope

[torpid narwhal]

so x doesnt have an inverse

so not a field

cool

[fervent gust]

yea

[harsh steeple]

theres the trigonometric identity: cos a = sin (90 - a), can i also say that cos 2a = sin (90 - 2a) ?

im just confused

[night geyser]

Yes.

[harsh steeple]

alright, thank u

[night geyser]

You can replace a with whatever

As long as you do it on both sides, the identity holds

cos(goldfish) = sin(90 - goldfish)

[fervent gust]

damn now I can't unsee

[whole charm]

goldfish better be in degrees :p

[alpine sable]

Can you write it out as an equation?

[torpid narwhal]

how can i find the multiplicative inverse of a field's elements

that defines addition like (a, b) + (c, d) = (a + c, b + d)