help-27

[rough sparrow]

@arctic field

[arctic field]

how are you getting the 2/16

[rough sparrow]

B intercept of a’ I think

@arctic field

[arctic field]

so using this

[rough sparrow]

1/2 times 7/8?

@arctic field

[arctic field]

plug in the numbers you know

[rough sparrow]

1/8 = 5/8 + x

Or do we want b

[arctic field]

we already know P(B)

[arctic field]

look here

[rough sparrow]

Mhm

[arctic field]

we know P(B n A)

thats 1/2 * 1/8 = 1/16

[rough sparrow]

So -.5

[arctic field]

no no

probabilities cant be negative

[rough sparrow]

I see hang on

[arctic field]

P(B) = 1/8 because P(B') = 7/8

then P(B n A) = 1/16

[rough sparrow]

Yes

[arctic field]

you should be able to work out P(B n A')

[rough sparrow]

1/16

[arctic field]

yes

exactly

so if we go back to this

we now know the numerator

and the denominator we also know

[rough sparrow]

13/16

Got it

Let me screenshot those formulas, thanks bunches

[arctic field]

have you ever seen like

two way tables before?

[rough sparrow]

I don’t think so

[arctic field]

  |  A  |  A'  |
--+-----+------+-----
B |     |      |
--+-----+------+-----
B'|     |      |
--+-----+------+-----
  |     |      |

the ones that kinda look like this

and like

you put the probabilities of the intersections in the middle 4 cells

and then you sum down the rows/across the columns and stuff

[rough sparrow]

Oh that would be nice

And simple and organized

I do have another problem that I need some assistance on, hopefully it will be quicker due to knowing this now

[arctic field]

its like this

you sum down the columns to get the last row

same thing for the rows to get the last column

[rough sparrow]

That’s neat actually

Never seen that beofre

[arctic field]

hmm

you should be able to fill in the entire table

using just the information theyve given you

[rough sparrow]

Could we use this as an example?

[arctic field]

uh

i guess so?

[rough sparrow]

Or I guess

How do we get p(E intersect F)

[arctic field]

     |    E    |    E'    |
-----+---------+----------+-----
  F  |         |          |
-----+---------+----------+-----
  F' |         |          |
-----+---------+----------+-----
     |         |          |   1

so heres the empty table

     |    E    |    E'    |
-----+---------+----------+-------
  F  |         |          | 13/24
-----+---------+----------+-------
  F' |         |          |
-----+---------+----------+-------
     |  17/24  |          |   1

we know these two because theyve been given

and then realise that

[rough sparrow]

Yes

[arctic field]

     |    E    |    E'    |
-----+---------+----------+-------
  F  |         |          | 13/24
-----+---------+----------+-------
  F' |         |    0     |
-----+---------+----------+-------
     |  17/24  |          |   1

so youve got this as the table now

the rest is just like

fill it in

its easy

cuz the rows/columns need to sum properly

     |    E    |    E'    |
-----+---------+----------+-------
  F  |         |          | 13/24
-----+---------+----------+-------
  F' |         |    0     | 11/24
-----+---------+----------+-------
     |  17/24  |   7/24   |   1

like first you can get those two on the sides

[rough sparrow]

So E and F’ would be 11/24

[arctic field]

     |    E    |    E'    |
-----+---------+----------+-------
  F  |         |   7/24   | 13/24
-----+---------+----------+-------
  F' |  11/24  |    0     | 11/24
-----+---------+----------+-------
     |  17/24  |   7/24   |   1

yeah then you can get those 2

cuz of the 0

[rough sparrow]

E and F would be 6/24

[arctic field]

and the final one is just

yeah

     |    E    |    E'    |
-----+---------+----------+-------
  F  |   6/24  |   7/24   | 13/24
-----+---------+----------+-------
  F' |  11/24  |    0     | 11/24
-----+---------+----------+-------
     |  17/24  |   7/24   |   1

and you're done

[rough sparrow]

Ok that’s way easy

[arctic field]

so then you can just calculate everything you need

shove the numbers into conditional probability formula

[rough sparrow]

Hehe yep

I will be studying that table for my exam

That’s probably the best thing I’ve learned in awhile

[arctic field]

lol

i mean

if you can get good at it quickly enough that is

dont want to confuse yourself with a new method

[rough sparrow]

I got like a week or so lol

[arctic field]

oh yeah

i guess thats enough time

[rough sparrow]

Yeah

Thank you once again

[arctic field]

np

[rough sparrow]

Wait so

Pr[E|F] = 6/26 / 13/24

[arctic field]

ye

[rough sparrow]

I mistyped 24

Ok that’s it

Thanks again

[arctic field]

conditional probability is just

[rough sparrow]

Lol

[arctic field]

you look at a single row basically

or column

and you take the intersection divided by the thing at the end of the row/column

[rough sparrow]

That makes it easy

[arctic field]

yeah

its super straightforward

[devout snow]

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[devout snow]

@tiny hedge Has your question been resolved?

[stuck cipher]

yeah, you could definitely do that

and you should notice that they all "converge" onto one value

or I guess "approach" could also work

[tiny hedge]

awesome ty 🙂

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[past pelican]

god pls help me

[past pelican]

f(x)
  
565 left parenthesis 1.0250 right parenthesis Superscript x

i really dont know what to do

[devout snow]

@past pelican Has your question been resolved?

[devout snow]

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[sand coral]

how do i find the third derivative

[ivory tapir]

what is f' and f''

[winter patrol]

pretty much the same way you found the second derivative,

differentiate the derivative before it

[sand coral]

f'(x)=-10/(1+5x)^2

f''=-10/(5^2)

f'''=-10/0?

[winter patrol]

no

[sand coral]

im confused

[winter patrol]

why is your f'' a constant

[sand coral]

because we substituted 5 for x

[winter patrol]

what's your f''(x)

(before subbing in 5), after subbing in 5, you'll have f''(5)

[sand coral]

i know its 100/(1+5x)^3 but that dosnt make sense

where did the 100 come from

shouldnt it have gone to 0

[winter patrol]

chain rule

no

why would it go to 0

show work

[winter patrol]

how did you "know" the second derivative,
where's the work for it,

(the third derivative is wrong btw)

[sand coral]

i thought i saw a pattern where youd just square the top and bottom

[winter patrol]

no

[sand coral]

then i used it for the third as well

becaseu i googled the asnwer

[winter patrol]

don't make assumptions like that

differentiate the first derivative to get the second derivative

[sand coral]

so i have to differentiate -10/(1+5x)^2

[winter patrol]

yes

\verb|i thought i saw a pattern where youd just square the top and bottom|
$$5 \times 5 = 25$$
it looks like multiplying by 5 means adding a 2 in front of a number
$$\therefore 5 \times 0 \wthonk 20$$

[woven radish]

ℝamonov

[sand coral]

how

where does the u come from

[winter patrol]

that's the chain rule

where u is the inner function

[sand coral]

as it relates to (1-5x)/(1+5x)

i dont see an inner function

[winter patrol]

well you wouldn't really need chain rule for that

[sand coral]

but to find the 2nd and 3rd derivative

[winter patrol]

well for the second derivative, like you said earlier,
you're trying to find the derivative of
-10/(1+5x)^2

[sand coral]

so i use the quotent rule

[winter patrol]

consider
$$y = -\frac{10}{(1+5x)^2}$$
let $u = 1 + 5x$
$$y = -\frac{10}{u^2}$$
and you can find $\dv{y}{x}$ using the rule above

[sand coral]

consider
$$y = -\frac{10}{(1+5x}^2$$
let $u = 1 + 5x$
$$y = -\frac{10}{u^2}$$
and you can find $\dv{y}{x}$

[woven radish]

space

ℝamonov

[winter patrol]

made a typo earlier

[sand coral]

np

find dy/dx using quotent rule

can you show me an example of how you would use the chain rule to find the derivative of -10/(1+5x)^2

im not getting it

i'll ask a tutor tmrw i guess thanks anyway

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[jaunty grove]

Hi I just like to know if I’m in the right track, and if I’m right what’s next to this step?

[devout snow]

Please don't occupy multiple help channels.

[jaunty grove]

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[reef cloud]

f(x) = 1_
x , translated 5 units to the left
and 4 units up

[reef cloud]

y - 4 = f (x + 5)

how would the answer be x if f(x) = 1/x?

[wooden wraith]

If f(x) is 1/x

Then what is f(x+5)?

[reef cloud]

wdym?

[wooden wraith]

You have

y - 4 = f(x + 5)

[reef cloud]

yea

[wooden wraith]

So if f(x) = 1/x, what is f(x + 5)?

[reef cloud]

1/x?

[wooden wraith]

That's just f(x)

A few examples:

f(x) = 1/x
f(a) = 1/a
f(2) = 1/2

What is f(x +5)?

[reef cloud]

erm

1/(x+5)?

[wooden wraith]

Yeah, exactly!

So

y - 4 = f(x + 5)

becomes what?

[reef cloud]

y-4=1/(x+5)

[wooden wraith]

👍

[reef cloud]

wait so in this one we are actually just solving for x?

of the f(x)

[wooden wraith]

Oh

I'd need to see the actual question

[reef cloud]

ooh no like that was the actual question

f(x) = 1/x , translated 5 units to the left
and 4 units up

or each transformation, identify the
values of h and k. Then, write the
equation of the transformed function
in the form y - k = f(x - h).

sry t his was

[wooden wraith]

Yes

Then that's it

[wooden wraith]

This answers that question

[reef cloud]

k so for this question

or each transformation, identify the
values of h and k. Then, write the
equation of the transformed function
in the form y - k = f(x - h).

the f(x-h) is just the x of the f(x)?

[wooden wraith]

I'm not sure what you mean by this

f(x-h) means you plug x-h into the function f

So if f(x) = 1/x, then f(x-h) = 1/(x-h)

[reef cloud]

ye was trying to say this

alr man thx for the help dude

[wooden wraith]

Sure thing

"Identify the values of h and k"

Don't forget to do that part lol

[devout snow]

@reef cloud Has your question been resolved?

[devout snow]

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[eager crystal]

How do you know which graphs are not polynomial?

Oh continuous and smooth

If we lack any of them

[restive river]

ye

[eager crystal]

It’s not a polynomial

I see thanks

[restive river]

so d is not polynomial already

[eager crystal]

Right

Also b

Not smooth

Its straight

[restive river]

yes

well

polynomials can be straight on every part

but if it is then its linear

[eager crystal]

Yeah but it’s like a v

[restive river]

not like straight on side

and pointing to another direction on another

its probably |mx-c| graph

[eager crystal]

Oh i see

|mx-c|=y is not polynomial?

[restive river]

no

[eager crystal]

Oh poly means more than 2 terms

[restive river]

no

[eager crystal]

Like 2x+5

No?

[restive river]

polynomial function is any function that can be written in the form $f(x)=C+\sum_{i=1}^{n} c_i * x^i$

[woven radish]

maths>>>physics

[restive river]

for $n \in \mathbb{N}$

[woven radish]

maths>>>physics

[eager crystal]

Lol i don’t know all of those

[restive river]

you havent seen $\sum$?

[woven radish]

maths>>>physics

[eager crystal]

Oh that means sum?

[restive river]

yes

add up

[eager crystal]

Ohh

[restive river]

sum from i=a to b of f(i)=f(a)+f(a+1)+f(a+2)+...+f(b)

[restive river]

and this means n is a natural number

[restive river]

many students who just learn functions make a mistake they think all functions with more than 2 terms are polynomials

but thats very wrong

[eager crystal]

What are some examples

[restive river]

you know log?

[eager crystal]

Of functions that are not polynomial when its more than two terms

Yeah

[restive river]

ok log(x)+log(sqrt(x))+2^x

3 terms

but is not polynomial

another is 1/x+1/x^2+x

and another last one is sqrt(x)+1/x+2^x

[restive river]

do these examples clear it?

[eager crystal]

Did i write it correctly?

[restive river]

yes

base doesnt matter

as long as its positive

[restive river]

see if u can write it in this form

[eager crystal]

One sec

[eager crystal]

Hold on im confused by this

Confused on pamdas

[restive river]

ok i will latex it

[eager crystal]

This?

[restive river]

$f(x)=\frac{1}{x}+\frac{1}{x^2}+x$

[woven radish]

maths>>>physics

[eager crystal]

How come does that tell you it’s not a polynomial?

Oh one variable?

[restive river]

no

[restive river]

it cant be written in this form

thats what makes it not a polynomial

[eager crystal]

So with that sum symbol you add up the top and bottom numbers ?

[restive river]

wait i will write it without that

$f(x)=C+c_1x+c_2x^2+....+c_n*x^n$

[woven radish]

maths>>>physics

[restive river]

so can u now write the function i gave in this form?

[eager crystal]

Yeah

[restive river]

how?

the powers of x are positive integers

keep in mind

[eager crystal]

Wait no

Cuz the x is on the denominator

[restive river]

hence this is not polynomial

[restive river]

yes

[eager crystal]

The variable needs to be kinda by itself

I mean

At least on the numerator

[restive river]

there are some general types of functions u will encounter in school polynomials functions with 1/x trig functions logarithms exponentials hyperbolic functions

[eager crystal]

And whole number exponent

[restive river]

yes

u might sometimes manage to simplify it

[eager crystal]

Yeah such important basics

Alright back to the original question

[restive river]

so now u have got the result i think

[eager crystal]

|mx+b| = y graph is not polynomial because… of the absolute value

Because it makes it one

You can’t just take things out of the absolute value

It’s just one

Like how x is one

[eager crystal]

Yeah you gotta +_ the other side of the equation and get rid of the absolute value first

Yup

Division is not allowed but multiplication is allowed

For example you can’t have 3/x but you can have 2x because 2x =x +x

You can just take x at that point

And put it in the function you gave me @restive river

Yup i think i understand now

Thank you for your help 🙂

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[eager crystal]

Proving is one of the most fun part of math

[eager crystal]

Oh sorry

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[topaz portal]

i need to simplify sin⁶x+3cos²xsin²x+cos⁶x, i keep finding 1 - 4sin²xcos²x but is that correct?

[odd elbow]

,w simply (sinx)^6+3(cosx)^2(sinx)^2+(cosx)^6

[restive river]

no

[sonic smelt]

I'd try factoring sin^6 + cos^6

[restive river]

yes

(sin^2)^3+(cos^2)^3

then sum of cubes formula

[odd elbow]

Ye open them up in the a^3+b^3 formula

[topaz portal]

a+b (a²-ab+b²)?

im confused whats the answer

[devout snow]

@topaz portal Has your question been resolved?

[sonic smelt]

Yes use that, except a+b should be in ( )

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[lunar condor]

Given that the Euro (€) closed yesterday with the following results; 1.00€ = $1.4642 CAD, how many
$CAD will it take to purchase 750€? I've been stuck on this question on a test review

[plush remnant]

1 euro = 1.4642 cad which would mean

1/1.4642 euro = 1 cad

[lunar condor]

yes exactly

[odd elbow]

Now multiply both side by 750

[lunar condor]

1098.15

That the answer?

[novel cedar]

If you mean 1098.15 CAD then yes

[lunar condor]

perfect thank you I appreciate the help! 🙂

[novel cedar]

no problem

[lunar condor]

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[rapid socket]

(P(X),U)

[devout snow]

@rapid socket Has your question been resolved?

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@rapid socket Has your question been resolved?

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@rapid socket Has your question been resolved?

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[restive river]

Oranges sell for 12 oranges for a dollar. How much will 27 oranges cost?

[restive river]

its ratios

[amber valley]

Each orange costs 1/12 of a dollar, so 27 oranges cost 27*1/12=2.25 dollars

[restive river]

thank you!

[devout snow]

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[tranquil lintel]

i need help with this

and step by step explaination if you can

its easier for me to understand

[cinder mural]

you know all the angles of a triangle equal 180 degrees right?

[hazy sleet]

Which means we know that 3a+c=180

[cinder mural]

do you have any more information than that?

[tranquil lintel]

yeah

[tranquil lintel]

no i just need help with it

i got C = 180 - 3a but the system says its wrong

[cinder mural]

hmmmm, the problem doesn't give any information about the triangle like if its a right triangle or something?

[tranquil lintel]

I have to go now but i will figure it out later

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[deep quartz]

I have developed an obsession with irrational numbers, such that I have begun working to devise an alternative system of mathematics in which all numbers may be expressed rationally. That has led me here.

I know this formatting is very dense, but please bear with me. I hope that someone will check my work for mistakes. I feel that I have come across an amazing conclusion. Most of the work is just me narrowing down √2, until I cannot narrow it down any further. I believe I have done that. Skip to the lower text barriers for conclusions without reading the works which resulted to them.

√2 = ~1.414
1.41 * 1.41 = 1.9881
1.411 * 1.411 = 1.990921
1.412 * 1.412 = 1.993744
1.413 * 1.413 = 1.996569
1.414 * 1.414 = 1.999396
1.4141 * 1.4141 = 1.99967881

1 41/100 * 1 41/100 = 1 .41

1.415 * 1.415 = 2.002225

√2 is between
1.41421.4142=1.99996164
and
1.41431.4143=2.00024449

1.4132² = 1.99996164
1.41325² = 1.9972755625
1.413275² = 1.99734622563
1.41330² = 1.99741689

1.4134² = 2.00024449

1.41330² = 1.99741689
1.41335² = 1.9975582225
1.41338² = 1.9976430244
1.41339² = 1.9976712921
1.4133999² = 1.99769927732
1.41339999999999999999999² = 1.99769956
1.41340² = 2.00024449

1.4133999² = 1.99769927732
1.41339999999999999999999² = 1.99769956
1.41340² = 2.00024449

1.41338² = 1.99764867792 (???)

%%%%%%%%%%%%%%%%%%%%%%%

1.413399999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999² = 1.99769956

[deep quartz]

1.413399999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999² = 1.99769956

1.41339999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999² = 1.99769956

Therefore, 1.41339[...]² = 1.99769956

1.41339² = 1.9976712921
1.41340² = 2.00024449

Therefore, √2 must be between:
1.41339[...] & 1.41340

So this raises the conundrum that what number could possible exist between between 0.9 repeating and 1? Why, it's 0.01, of course.

Not to even mention those who assert that .9[...] = 1.

%%%%%%%%%%%%%%%%%%%%%%%

1.41340 - 1.41339 = 0.00001
1.41340 - 1.413399 = 0.000001
1.41340 - 1.4133999 = 1e-7 = 0.0000001
1.41340 - 1.41339[...] = 1e-7 = 0.0[...]01

I am not sure what further conclusion(s) to draw at this point.

If I have made mistakes, do not ridicule me, but correct me instead. But please prove the validity of your corrections with the submission of said corrections. Thank you.

[midnight dirge]

WOW

another one

[deep quartz]

Does that really surprise you at this point?

heh

[sand dove]

Do you have a particular question then ?

[deep quartz]

Yes. Have I made any mistakes?

[sand dove]

A rigorous proof that 0.99999999... = 1 ?

[deep quartz]

I know that it will be tedious to check.

[torn vessel]

Interesting. So .1 is between .99 and 1?

[deep quartz]

Technically we're referring to 1.4133999_

[deep quartz]

I meant .01

[sand dove]

1.41340² = 1.99769956 and not 2.00024449

[deep quartz]

Thank you for the correct. I'll update my notes.

[deep quartz]

That can't be. I doubt checked that several times...

I must have typed something wrong each time.

[sand dove]

You'll realize that 1.41340² is the same value (up to a microscopic error) than 1.413399999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999²

[deep quartz]

Come again?

[sand dove]

The reason for this is that people who say 0.99999999... = 1 are correct, to a "limit" point of view

[deep quartz]

I don't quite understand what you meant by this.

[hollow pollen]

difference small

[sand dove]

1.41340^2 is almost the same as 1.413399999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999², with this exact number of nines

[deep quartz]

Ah. Understood.

[sand dove]

I think a proof of 0.999999... = 1 would enlighten you on why this is true

so

notice that :

0.9 = 1 - 0.1 = 1 - 1/10

0.99 = 1 - 0.01 = 1 - 1/100

0.9999999999999 = 1 - 0.0000000000001 = 1 - 1/1000000000000

and the difference between 0.9999999 and 1 gets increasingly smaller as you add more nines

so if you add an infinite amount of nines, you get :
0.999999... = 1 - 1/infinity = 1

[deep quartz]

1.41421^2 = 1.9999899241
√2^2 = 2
1.41422^2 = 2.0000182084

[deep quartz]

I am sorry. I just saw this. I am very interested. Let me read it.

@sand dove So wait. I mostly followed, but...

0.9[...] = 1 - 0.0[...]01 = 1 - 1/10[...]

Is this correct so far? @sand dove

[sand dove]

yes

But 10[...] going on forever is just infinity

It's like multiplying by 10 forever

the same as if you keep adding 1 + 1 + 1 + ... up until infinity, you indeed get infinity

[deep quartz]

Wouldn't all of them going on forever be infinity? And where does 1 - 1/∞ = 1 come from? I am very interested in that line in particular.

[midnight dirge]

well

[deep quartz]

Does every additional plus one limit that infinity in such a way that it cannot truly be infinite?

[sand dove]

So, we define 1/∞ = the limit as n goes to infinity of 1/n

[midnight dirge]

$\lim_{n \to \infty} \frac1n = 0$

[midnight dirge]

its how limits work

[sand dove]

basically, to understand what 0.999999... is all about, you need to understand limits

[deep quartz]

N + 1 =! N + 1 + 1 =! N + 1 + 1 + 1 + ...

[midnight dirge]

whats =!

[deep quartz]

I do not know how to read this. Will you tell me?

[deep quartz]

"does not equal"

[deep quartz]

Which I do not, presently. Haha.

[midnight dirge]

ive never seen it used that way but ok

[torn vessel]

That’s true for finite N

[midnight dirge]

as n approaches infinity, 1/n goes to 0

[deep quartz]

Yes. It is near-equivalent to =/=

[sand dove]

alright, here's an informal introduction to limits.
consider a sequence of numbers that we denote by this notation : Un references the nth number in the sequence

for example :
Un = 1/n

so the first term in the sequence is 1

then the second term is 1/2

and so forth...

[deep quartz]

But then infinite N would be made finite by N = ∞ + 1

i think?

[deep quartz]

Thank you.

[sand dove]

After that

we say that a sequence is "convergent" if it approaches a single finite value

for example :

[sand dove]

take this sequence Un

we know that for all n natural integer, Un >=0 (non negative)

yet, if I take any strictly positive real number, let's say a > 0

[torn vessel]

Nope. If I add sqrt(2) to the set of integers, that doesn’t suddenly mean there are a finite number integers.
There’s still infinite number of numbers in the new set.

[sand dove]

I can always find a rank N, such that 1/N < a, and all the ranks that come after that will have the same property

so, the difference between Un and 0 can be arbitrarily small, if we take n large enough

So Un HAS to approach 0

and thus the limit of the sequence Un (as n goes to infinity) is 0

[deep quartz]

Regarding this (I am still catching up lol): "So, we define 1/∞ = the limit as n goes to infinity of 1/n"

Could 1/∞ be phrased "how many times does infinity go into one?" as we do with other division? Infinity must go into 1 zero times unless it is bound/limited by 1. Right?

I really want to understand "1/∞ = the limit as n goes to infinity of 1/n"

[sand dove]

"1/∞" is not defined on it's own unless we talk about the limit of something that is defined

"1/∞" is used to represent the idea that we can "approach" infinity, thus talking about "limits". But we cannot reach it

[deep quartz]

This make sense. But then must infinity always be limited by the one which it encompasses? Isn't this Gödel's Incompleteness Theorem? Lol.

[sand dove]

YET, 0.999999... is defined as the limit of 0.9999 as the number of nines approaches infinity

so, in that sense, 0.99999... = 1

because we're not talking about reaching to infinity. We're talking about "what happens IF we could reach infinity"

So, IF we could put an infinite amount of nines in the number 0.9999..., then that number is strictly equal to 1

[deep quartz]

Musn't infinity be one (times [of] infinity)? Perhaps there is a limit on human comprehension.

[hollow pollen]

infinity is not a number

its an idea

you cant do math with infinity like that

[deep quartz]

So it cannot be expressed numerically?

[sand dove]

no

but it can be approached

which is the whole idea of limits

[deep quartz]

True infinity is boundlessness, is it not? It must be formless.

[midnight dirge]

...

[torn vessel]

Welp. I’m out.

[midnight dirge]

HAHAHHA

[hollow pollen]

you want to hear something cool

[deep quartz]

I have some comprehsion now. More than I did when we began.

[deep quartz]

Yes, please.

[hollow pollen]

some infinities are bigger than others

[sand dove]

yes

[deep quartz]

I have heard this also. But is not "big" quality of quantity?

[hollow pollen]

there are exactly as many rational numbers as there are integers

[deep quartz]

How do we define "big" innumerically?

[hollow pollen]

countable and uncountable

[deep quartz]

If an infinity can be bigger than another

[sand dove]

and more

[deep quartz]

then it can be numerically defined

[hollow pollen]

you cant define infinity numerically

[sand dove]

there are multiple ways to define the infinity you think of everyday, which is this infinity ∞

[hollow pollen]

well you could nobody can stop you but its not defined numerically

[midnight dirge]

well an example of what smt converges to as n approaches inf can be seen from

https://math.stackexchange.com/questions/195030/showing-a-sequence-is-convergent-using-the-epsilon-n-definition

epsilon N defn of conv

[deep quartz]

Then how can one be "bigger" than another? What constitutes big? If the members of its set, then numerical. If comparison to another infinity, then numerical.

Isn't it?

[midnight dirge]

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

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

[hollow pollen]

https://www.youtube.com/watch?v=OxGsU8oIWjY

here veritasium has a cool vid on it

[deep quartz]

I just saw that name for the first time earlier today reading about fractals. Neat synchronicity.

[sand dove]

This is the axiom defining the first infinity, or "countable infinity" :
"There exists a quantity "∞" that is strictly bigger than any natural integer".

That's it

Just, something bigger than any thing we can possibly imagine

[deep quartz]

How do we define big as a non-quantity?

Big implies more of something than another.

More times of something than another, even.

[sand dove]

yes, in that sense, ∞ is "more than any number"

[deep quartz]

And because "big" only exists by comparison, musn't it be subject to proportion and ratios?

[deep quartz]

More what than any number?

[sand dove]

ratios are only defined for numbers

infinity is a quantity, not a number we can manipulate to our ease

[deep quartz]

I chuckled at your edit. Lol.

[sand dove]

we can manipulate infinity in many ways, such as :
∞ + 1 = ∞
∞ + ∞ = ∞
∞ x ∞ = ∞

[arctic field]

oh god

[deep quartz]

But, what is ∞ more of than any number? I might even say "number below it".

[deep quartz]

I'll take that as a "hello, nice to see you."

[arctic field]

no no im talking about rafilou

[hollow pollen]

its not 10 more

its just more

its not any amount more

[midnight dirge]

[hollow pollen]

its just more

[deep quartz]

oh 🫢

[arctic field]

and possibly the other people here

[sand dove]

yeah, it's that time of the year again...

[deep quartz]

More what?

[midnight dirge]

WOW OK

[arctic field]

the thing is

[sand dove]

infinity theory in a nutshell xd

[arctic field]

its an unfortunate thing of mathematics

[midnight dirge]

i just talked about eps N

[arctic field]

to denote things that are similar by the same notation

and

unfortunately

there are many infinities which float around in mathematics

[sand dove]

yes

[deep quartz]

I wonder if all are equally valid.

[arctic field]

you can only work with them by how you define them

they arent all the same object

[sand dove]

I warned that "∞" here references ONLY the countable infinity

[arctic field]

the cardinality aleph null aka "countable infinity" is not really the same infinity when you say lim n->infinity

[sand dove]

ofc other infinities exist, which are even bigger

[arctic field]

nor is it really the same infinity when you talk about say 1/infinity = 0 in complex analysis

[deep quartz]

imo infinity can be defined as "incomprehensibly more or less than what I (can) comprehend".

[arctic field]

theyre all the same kind of thing

but theyre not really the same mathematical object

[midnight dirge]

[deep quartz]

but that is non-mathematical

[sand dove]

I'm trying not to scare him here XD

[deep quartz]

I would agree that 1/∞ = 0

[arctic field]

i feel like if we just talk about infinities here its too easy to construe two different "kinds" of infinity

not even in the sense of cardinalities

just

two different infinities used in different areas of maths

[deep quartz]

So let's unite them.

[arctic field]

real positive infinity is definitely not the same as complex infinity

[midnight dirge]

whats

cardinalities

[hollow pollen]

snow how far along are you in your education? you sound pretty advanced

[arctic field]

im actually over 3000 years old

[sand dove]

You never replied to my text in 868BC, where you been ?

[midnight dirge]

like this?

[deep quartz]

I actually believe you. I'm ~40,000 local years old. And just now learning about infinity lul.

[arctic field]

yea sorry been kinda busy

[hollow pollen]

okay snow

[arctic field]

D:

[hollow pollen]

is this solveable with markov chains?

[deep quartz]

But probably not really. But also maybe. lul.

[midnight dirge]

u sounds pretty old, u mean

[arctic field]

open your own channel

[hollow pollen]

okayy

[midnight dirge]

[deep quartz]

awwww, he/she/ze can hang out in here too ):

[arctic field]

actually yes but you might kinda neck

[hollow pollen]

i might die?

[deep quartz]

rip rip

[midnight dirge]

looks like a weird prob

[sand dove]

Ah, calculating the expected value of a random variable :)

[hollow pollen]

not just some variable

there is no border

[sand dove]

Yes, some variable

Infinite, for sure, but a variable

[hollow pollen]

i mean its a tough one

thats what i meant by not just some variable

[sand dove]

well, it is by definition some variable

xd

anywho

[midnight dirge]

what happened to

[midnight dirge]

this

the good stuff

[sand dove]

Guys

[deep quartz]

I wonder if we cannot comprehend infinity because to comprehend it is to contradict it. Then infinity would be that which must be true to its own nature.

Infinity must be truth by definition. Because if it is, then it is. And so truth is that which is.

[sand dove]

It's 3AM for me xd

[deep quartz]

Idek if that makes sense lol

[sand dove]

A mimir 💤

[deep quartz]

9:06 PM for me.

[sand dove]

see you guys

[deep quartz]

we'll get back to it xD

else I'll repost it. heh.

[deep quartz]

Sleep well, new friend. ¡Hasta luego!
then my brain said "pasta luigi" ...>___>

[midnight dirge]

LOL

[sand dove]

¡Gracias y hasta luego, señor !

[deep quartz]

Y tu, gran señor.

[deep quartz]

1.41421^2 = 1.9999899241
√2^2 = 2
1.41422^2 = 2.0000182084

We left off here.

[midnight dirge]

no

that was a joke

im not gettin involved

[deep quartz]

yeah well, it ain't now

1.41421^2 = 1.9999899241
√2^2 = 2
1.41422^2 = 2.0000182084

1.4142159^2 = 2.00000661181
1.4142151^2 = 2.00000434907

1.4142133^2 = 1.9999992579
1.4142140^2 = 2.0000012378

1.4142138^2 = 2.00000067211
1.4142135^2 = 1.99999982358

1.4142136^2 = 2.00000010642

1.4142135^2 = 1.99999982358
2^2 = √2
1.4142136^2 = 2.00000010642

1.41421355^2 = 1.999999965

so close I can taste it

1.414213559^2 = 1.99999999046

I somehow got 1.41421355921212[...]^2 = 1.99999999106

1.4142135591^2 = 1.99999999074

I'm not sure if it gets much closer than that to sqrt(2).

[torn vessel]

it's missing, roughly, an infinite number of digits, but sure it's close

[deep quartz]

So far I'm at 1.41421356238^2 = 2.00000000001

SO FUCKING CLOSE TO √2

Now let's start adding more zeros to that figure and see where we go lol

1.414213562361^2 = 1.99999999997
2^2 = √2
1.41421356237^2 = 2.000000000001

1.414213562371^2 = 1.99999999999

🥵🥵🥵

WAIT

1.999[...] = 2 RIGHT!!? @torn vessel @midnight dirge @sand dove @arctic field

then BAM sqrt(2) is rational

winrar winrar chimkin dinrar

I FUCKING DID IT

and it mysteriously already exists on Google

kinda pissed about how reality functions ngl

[arctic field]

,w 1.414213562371^2

[woven radish]

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[devout snow]

@deep quartz Has your question been resolved?

[deep quartz]

dammit

still closer though

,w 1.414213562372^2

[woven radish]

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[deep quartz]

,w 1.9^2

[woven radish]

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[deep quartz]

oh oops duh

,w 1.414213562373^2

[woven radish]

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[deep quartz]

,w 1.414213562375^2

[woven radish]

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[deep quartz]

,w 1.414213562374^2

[woven radish]

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[deep quartz]

,w 1.4142135623735^2

[woven radish]

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[deep quartz]

,w 1.4142135623731^2

[woven radish]

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[deep quartz]

,w 1.4142135623730^2

[woven radish]

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[arctic field]

unfortunately its not possible to have a terminating decimal representation of sqrt(2)

[deep quartz]

naysayer >:

There is something to be learned trying to discover one, though.

,w 1.4142135623715^2

[woven radish]

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[deep quartz]

,w 1.41421356237151^2

[woven radish]

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[deep quartz]

,w sqrt(1.99999999999999999999999999999999999999999999999999999999)

[woven radish]

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[deep quartz]

,w 1.415^2

[woven radish]

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[deep quartz]

,w 1.411^2

[woven radish]

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[deep quartz]

,w 1.412^2

[woven radish]

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[deep quartz]

,w 1.413^2

[woven radish]

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[deep quartz]

,w 1.414^2

[woven radish]

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[deep quartz]

How can the diagonal of a square possibly be irrational? lmao it makes no sense

,w (2√2)(2√2)

AHA

[woven radish]

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[deep quartz]

,w (1√2)(1√2)

[woven radish]

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[deep quartz]

oh duh

,w (3√2)(3√2)

[woven radish]

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[deep quartz]

,w (4√2)(4√2)

[woven radish]

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[deep quartz]

,w (5√2)(5√2)

[woven radish]

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[deep quartz]

,w (6√2)(6√2)

[woven radish]

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[deep quartz]

,w (0√2)(0√2)

[woven radish]

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[devout snow]

@deep quartz Has your question been resolved?

[torn vessel]

no.

[deep quartz]

Yeah, that seems about right.

[torn vessel]

yes.

[deep quartz]

And yet.

[torn vessel]

root(n) is only rational when n is a perfect square. 🤷 it is what it is.

[deep quartz]

If numbers can have a square root, can they have a triangle root too? How might we discover it?

[torn vessel]

tell me what a triangle root is first.

[arctic field]

square root is in the sense of squaring a number, i.e. a*a = a^2

[torn vessel]

the problem is 1.999.... is still a rational number, as is 2.
You're trying to write an irrational number as a rational number, which can't be done.
Your trick of write root(2) = 1.414... 99999..... can't work because if it did, then root(2) would be rational, but we know it's not.

[arctic field]

not square as in the geometrical object

"triangle root" has no meaning unless you define it

there is no "triangle" operation on numbers

[torn vessel]

well... it's the area of a square of side length a.
which is why a^3 is called a cubed, it's the volume of a cube of side length a

[arctic field]

thats a geometrical interpretation

[torn vessel]

root(a) is the side length of a square of area a

[arctic field]

because historically squaring was done by looking at areas of squares

but in modern algebra squaring simply means to multiply a number by itself

[torn vessel]

yes, which is why it's called square... the geometrical object.

[arctic field]

there is no sense to assign geometrical meaning to squaring in general

what meaning does i^2 = -1 have?

[torn vessel]

but, also, yes, triangle root is meaningless

[deep quartz]

Much, actually.

[arctic field]

what meaning does 3^2 when taking 3 as an element of Z/7Z have?

[deep quartz]

It says "I can conceive of this thing that I could not otherwise express."

[arctic field]

what meaning does (x^2 + x)^2 when taking x^2 + x as an element of R[x] have?

[deep quartz]

Such as a rational sqrt(2).

[torn vessel]

the area of a square of side length 3 (mod 7)
lol

[deep quartz]

Not a clue.

[arctic field]

thats completely nonsense

[violet wind]

jus count the unit squares and get rid of groups of 8

7*

[deep quartz]

What do you mean?

[arctic field]

i mean sure but thats not what we traditionally call area

[deep quartz]

Tradition is boring.

[violet wind]

modular math deals with remainders

[arctic field]

and area is part of measure theory

[violet wind]

if you add remainder 3 and remainder 4 mod 5, you get remainder 2

[arctic field]

can you assign a meaningful measure that says the square of side length 3 in Z/7Z is 2?

[deep quartz]

Wish this wasn't all gibberish to me. Sigh.

[violet wind]

Area was conceived of long before measure theory

[arctic field]

but measure theory is how modern mathematics formalises area

and abstract algebra is how modern mathematics formalises algebra

we arent living in the past anymore

[violet wind]

You can't have any shapes in Z/7Z

[arctic field]

exactly

so its quite meaningless

squaring is simply the operation of multiplying a number by itself

[violet wind]

ah yes measure theory is the only way to think about geometry

How could I forget

[arctic field]

well if you try to define area in other ways

it might not make sense

you can define things however youd like

but they might not be meaningful in any way

[violet wind]

you could define area for all polygons solely in terms of the area of triangles

[deep quartz]

So doesn't like that I asserted yesterday that a^2 = 4a

that's why y'all are arguing about this again

that I is a side, not a 1

[arctic field]

mmm sounds like measure theory but not as powerful

i mean

[violet wind]

sounds like moving goalposts

[arctic field]

the main problem why 3^2 = 2 doesnt make sense as area is because theres no order on Z/7Z

adding areas should ideally give you a bigger area

but whats bigger in Z/7Z?

it doesnt make sense

if we talk about complex numbers

its even worse

i^2 = -1

[violet wind]

I mean my example works as a way to represent it

[arctic field]

well thats not really a good definition of area then

[violet wind]

two things should ideally not multiply to zero if neither of them are zero

but that's not true of matrices

[arctic field]

well i mean

thats just because you're used to that from the real numbers

[violet wind]

same to you

you're just describing the way you're used to area working

[arctic field]

the concept of area is pretty well defined

the concept of multiplying is also pretty well defined

multiplying never stipulates that the product of two things shouldnt be 0

that is only in integral domains

[violet wind]

you're just appealing to what definitions mathematicians have generalized and what they haven't

[arctic field]

again you can define anything youd like

but it might not be meaningful

[violet wind]

but it is meaningful here

the meaning is obvious, not less valid then 1+1=0 mod 2

there's limited application for doing it but it's not just nonsensical

[arctic field]

limited application to not really being a proper area that can be worked with

[violet wind]

Proper is being used completely subjectively here

[arctic field]

it doesnt work so well in how youd want area to work

maybe you care about the remainder

in some sort of application

but then you wouldnt call it the area

you might call it the remainder

[violet wind]

still a visual representation of 3²=2 using actual squares

that's usable as a definition

[arctic field]

but still not something you'd ultimately call area

[violet wind]

No

[devout snow]

Channel closed

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[violet wind]

It's not something you want to call area

[arctic field]

well i mean sure you can call anything whatever you want

[violet wind]

I might as well stay negatives aren't something you'd ultimately call numbers

[arctic field]

but thats kind of not very cooperative

especially since people generally agree on certain things

[devout snow]

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[muted sierra]

these 2 are correct? right?

[devout snow]

@muted sierra Has your question been resolved?

[devout snow]

@muted sierra Has your question been resolved?

[muted sierra]

<@&286206848099549185>

[devout snow]

@muted sierra Has your question been resolved?

[muted sierra]

.close

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[lost barn]

Hello i watched a video where a guy explained that 0.9999infinitly is simply 1 and i think its false and there is a mistake in his algebra that he wrote it up with i see where the problem could come from but i dont think it brakes any math rules here is his algebra

[supple knot]

it's correct. you're mistaken to think it's false

[lost barn]

it is?

[sand dove]

Hello, although this method "lacks" a little bit of rigor, it is true

[lost barn]

oh

[sand dove]

If you want to be completely rigorous you have to go through limits

[lost barn]

well im amazed

can you tell me how it works tho? like even if anything could fill that little gap and it would be one but still there is a gap between the 2 numbers

[sand dove]

Basically, the gap between the two numbers gets smaller and smaller as you add more nines. If you have an "infinite" number of nines, the difference between both numbers is 1/infinity, which is 0

[lost barn]

well i dont get it but thy very much

.close

[devout snow]

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[sand dove]

Funny enough, I had this conversation yesterday with somebody else

[lost barn]

oh lol

[devout snow]

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[buoyant ridge]

I am stuck on the following equation I keep looping between dividing because I want to detach the x from the ax

[supple knot]

what are the instructions

[buoyant ridge]

solve for x

[pseudo basin]

subtract x and add a to both sdies

that way you get ax - x = a + 1 => (a-1)x = a+1

[buoyant ridge]

How do you see that so quickly 0_o

.close

[devout snow]

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[buoyant ridge]

.reopen

[devout snow]

✅

[pseudo basin]

why close and then immediately reopen

what was that about

also do you want the self-aggrandizing answer or the hopefully-clarifying answer

[buoyant ridge]

Because someone was still typing so I thought maybe someone would still have some useful answer

[gleaming plover]

there I don't know how to format on discord so i just used mathpapa screenshot

it should explain pretty clearly

[pseudo basin]

steps 1-3 are exactly what i said, in fact.

[buoyant ridge]

Yess thank you! ^^

.close

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[proud prairie]

hi

[devout snow]

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[jolly lantern]

if they have the same remainder, does it mean that the remainder is just 0?

[jolly lantern]

so they're both the factors, i assume?

[novel compass]

use factor remainder theorem

it says quadratic therefore ax^2 + bx +c

use f(2) and f(-4) and equate both the equations i guess

[devout snow]

@jolly lantern Has your question been resolved?

[jolly lantern]

ah ok

.close

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[devout snow]

@lusty relic Has your question been resolved?

[devout snow]

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[dense cradle]

u1=1/81
u4=1/3
r=3

**Find the smallest value of n for which Sn >40 **

I need help solving this.

[prime egret]

What’s the formula for the sum of a geometric series

[dense cradle]

my bad

wait

sorry

[prime egret]

That’s the infinite sum

[dense cradle]

yes

u1(r^n -1)/r-1

[prime egret]

Yeah

[dense cradle]

yes

[prime egret]

So plug in the values

[dense cradle]

do i guess?

[prime egret]

You’d use logarithm

[dense cradle]

1/81(3^n -1) /2

[prime egret]

Yes that’s greater than 40

[dense cradle]

how

we dont even know what N is

[prime egret]

Then 1/81 (3^n -1) > 80

[dense cradle]

ohhh

right right

my bad

[prime egret]

3^n -1> 6480

[dense cradle]

you multiplied 80 by 81 right

[prime egret]

Yes

Then 3^n > 6481

Use log base 3

n> log_3 (6481)

Plug this in to your calculator

[dense cradle]

wait wait

how is it greater than 80

[dense cradle]

how did u know this was greater than 40

[prime egret]

That’s the question

Sn > 40

[dense cradle]

yes

ohh

oh

my bad

so sorry i read it wrong

okay

[prime egret]

Okay so what did you get?

[dense cradle]

okay let me solve

i took a moment to take in the info

7.9883

[prime egret]

N must be greater than 7.988…

N is an integer

So the smallest n could be is what?

[dense cradle]

ohhhhhhh

OH

OKAY

SO IT MUST BE 8

[prime egret]

Yes!

[dense cradle]

thank you so much man

[prime egret]

Np

[dense cradle]

there's another question

can u help me with it?

[prime egret]

Show me

[dense cradle]

consider x-3,x+1, 2x+8

One value for x to be geometric is 5

It's asking me *find the other value of x for which the sequence is geometric

[prime egret]

Okay so you know that there must be a common ratio between each consecutive terms

[dense cradle]

the common ratio is 3

[prime egret]

How do you know that?

[dense cradle]

if x=5

the ratio is 3

2,6,18

[prime egret]

Okay but it might not be the same

[dense cradle]

right

[dense cradle]

okay

[prime egret]

But how would find the common ratio

If your given 3 terms

Just in terms of x

[dense cradle]

x+1/x-3

[prime egret]

Yes

There’s another way

[dense cradle]

2x+8/x+1

[prime egret]

Yea and they must be equal

Because it’s the common ratio

So we can set them equal to each other and solve for x

x+1/x-3 = 2x+8/x+1

[dense cradle]

right

okay wait

im confused now

idk

i got