help-17

[cerulean brook]

alright nice

[pallid zenith]

but you have to make sure the answer makes sense

like we did before

[cerulean brook]

this helped a lot

made me understand it a lot more

lol

[pallid zenith]

when we tried to make the height 4 inches, it ended up too wide

so you have to check

[pallid zenith]

yea, if you solve this, we get a map thats 4 by (something bigger than 6)

[cerulean brook]

6.6 is what x is

so what do we do once we get value of x

[pallid zenith]

this told us that the image should be 4 by (something bigger than 6)

and we looked at the problem, and it said the map is going onto a 4 by 6 card

so, it wont fit

then, you just solve for the other dimension instead

$\frac{9}{15} = \frac{x}{6}$

[twin meteor]

jan Niku

[cerulean brook]

3.6

[pallid zenith]

yea

so (something smaller than 4) by 6

that will fit on a 4 by 6 card

there is a way to know which one to pick before you solve it, but its harder than just doing both IMO

you should just guess, and check, and if its too big, do the other instead

[cerulean brook]

uhh

[cerulean brook]

9/15 is 0.6

[pallid zenith]

yes

[cerulean brook]

and when u plug in x in that you get 6.6

so we know what its wrong

[pallid zenith]

,w Solve[ 9 / 15 == x /6 ]

[twin meteor]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by janniku

[pallid zenith]

,calc 18/5

[twin meteor]

Result:

3.6

[cerulean brook]

this one is right

[pallid zenith]

yea

[cerulean brook]

x = 3.6

[pallid zenith]

yes

[cerulean brook]

so what do we do once we learn that it's correct

[pallid zenith]

you have your answer

[cerulean brook]

the answer key is something different

[pallid zenith]

what does the answer key say

[pallid zenith]

this is the answer we got

[pallid zenith]

3.6 inches

by 6 inches

[cerulean brook]

uhhh

wait where was the 6 inches

[pallid zenith]

on the bottom

[cerulean brook]

this right here?

[pallid zenith]

yea

[cerulean brook]

oh

[pallid zenith]

when we write $\frac{H}{W}$, we mean "H by W"

[twin meteor]

jan Niku

[cerulean brook]

alright so

[pallid zenith]

so x/6 means "x inches by 6 inches"

[cerulean brook]

when i pick either height or width i need to completely remove that number and replace it with a variable

ok i understand this

imma just do one more problem similar

give me a second

[pallid zenith]

okay, just let me know

[cerulean brook]

yeah i got it down

thank you

[pallid zenith]

[cerulean brook]

do you think you can also explain this

is it 1/6 because 5+1

[pallid zenith]

yea, although you don't have to do it this way

this way is reasonable because you don't know how long that segment '1' will be

so you relate it to the total, rather than to the other piece (the 5)

you can see they correspondingly put 48 in the bottom of the fraction on the other side of the equals

(because 5+1 segments is 48 inches long)

[cerulean brook]

why is it x/48 instead of 48/x

[pallid zenith]

on the right, we wrote 1/6

this means like, this is the ratio between one segment, and the total

on the right is the same thing

[cerulean brook]

i see

[pallid zenith]

x/48

[cerulean brook]

so we need to put the total on bottom cuz we did that for the other

[pallid zenith]

ratio between x inches (the length of one segment, we dont know it), and 48 (the total)

[pallid zenith]

yea, the two fractions are saying similar things

[cerulean brook]

i have a test on this unit tmrw thats why im just reviewing everything i was taught over this week lol

[pallid zenith]

one way is saying it in inches, the other way is saying it in terms of ratio

i mean, theyre both ratios

but you know what i mean maybe

[pallid zenith]

good luck

[cerulean brook]

thank you

btw

for num. 32

wouldnt it be 85/34 = 5/2?

because its asking for the ratio of bigger to smaller

[pallid zenith]

oh, yea

the normal way of saying ratios in words is almost always top first, then bottom

4 by 6

A to B

thats 4/6 and A/B

[cerulean brook]

okay nice

hm

given a formula like (x+5)^2 + (y+3)^2 = 10, what is my next step if i want to find the center/radius?

just wondering

[pallid zenith]

you should just google this, lmc

[cerulean brook]

nooo

😭

i was joking

this was the first message u sent in the server

was just seeing if u remembered

[pallid zenith]

oh, yea

its funny i googled it back then and didnt understand it

so idk why i said to google it

[cerulean brook]

tbh i think im mostly good for the test tmrw i will just ask a friend to tell me what to be careful about and what to expect

thats what i usually do

[pallid zenith]

well good luck

[cerulean brook]

i cant wait for this year to be over

i was much better with algebra

and thank u

[pallid zenith]

math never gets easier it just gets cooler

[cerulean brook]

english just gets added to it

[pallid zenith]

FWIW

[cerulean brook]

over time

[pallid zenith]

and insane symbols

[cerulean brook]

😂

yeah

i will just end school at like pre calc

maybe ill do more if i enjoy it who knows

anyways thx for helping imma close ticket

[pallid zenith]

you never know

math gets pretty cool

can recommend if you enjoy it

[pallid zenith]

no prob, see ya

[cerulean brook]

pretty sure this is the obvious answer but i only enjoy math when i understand haha

.close

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[vocal sleet]

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[silver berry]

I don't have the answers on these so I if someone can confirm if it's correct that would be realy helpful:
a:
true
b: true
c: also true

[silver berry]

I don't think all can be true

but i can't think of any counterexample

[dull bear]

Why do you think a is true?

[silver berry]

I was just testing a bit, so like a serie is divergent if it goes to infinity or -infinity I tihnk

it can't be alternating like a sequence

oh

so it's false

then?

but how can a serie be alternating between 2 numbers

[dull bear]

I mean, the fact I'm asking you, it probably is

[dull bear]

This isn't true, but that's not the reason why it's false either (for example, while not all positive, the series of (-1)^n is divergent: in fact, any convergent series must have its constituent sequence with a limit of zero)

[dull bear]

Speaking of counterexamples, can you think of sequences of positive terms that are divergent? Do you know any examples?

[silver berry]

sequences?

for sequences with positive terms there are many

[dull bear]

whose series are divergent* sorry

[silver berry]

oh

well geoemtric series

with absolute value r bigger then 1

or the p series

with p smaller then 1

[dull bear]

Uh-huh

1 + 1 = 2

[silver berry]

I guess it was not the answer u were expectign right

[dull bear]

It was an answer I wanted!

[dull bear]

[why do you think I'm being silly here? What do you think I'm trying to hint at?]

[silver berry]

uhh if I add 2 p series which are divergent u say it's convergent?

[dull bear]

!nosols arrrghhh

[vocal sleet]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[dull bear]

Well, multiply, but it's been given away already

[silver berry]

wait I didn't saw it

he rmeoved it

[dull bear]

@karmic forge get back here and do your thing

[silver berry]

this is hard

[karmic forge]

Like I sayin if we take series 1/n

[silver berry]

that is harmonic series

ooohh wait

harmonic series

[karmic forge]

Ya the divergent

[silver berry]

and the n

a_n = n and harmonic series as b_n

oh wait nvm

that is still divergent

[silver berry]

what is the other esries?

1 is harmonic

[karmic forge]

Both 1/n

[silver berry]

uhmm

1 / n^2

ohwait

ye

that is true

[karmic forge]

Multiply 1/n^2

[silver berry]

ye

that is def divergent

p series

2

1*

[karmic forge]

https://tenor.com/view/high-five-bleh-snow-crazy-person-duo-gif-13251996

[silver berry]

and for b it's prob false too then?

[karmic forge]

Ya

[dull bear]

https://tenor.com/view/minsi-gominsi-minka-explosion-fire-gif-24302500

[silver berry]

if i take harmonic series

and 1/n^2

it's 1/n^3

so b is fasle

and c is uhm

well c is true right?

it can't be that all 3 are false

[karmic forge]

These questions have no formula I think you can just prove by examples

[dull bear]

Bad reasoning

[silver berry]

well I did thought about it for a bit

[karmic forge]

I see these kind of questions teacher just say let take this series

[silver berry]

if it's convergent

it has t obe a number

so 2 numbers multiplied

it has to be an umber

is that correct

[karmic forge]

C is true

[silver berry]

my reasoning too?

[karmic forge]

C is a theorem

Like don't remembering right now

[dull bear]

In fact, there's an expression for the product of series if I recall correctly, which I always forget (but shouldn't!)

[silver berry]

I guess it isn't that important then right

if u keep forgetting it

[karmic forge]

Ya right I am also talking about that

Don't worry I also forgot 😂

[dull bear]

Well more that I rarely deal with products of series and always just look for it when I need it, I forget everything, I'd forget my own name if it weren't on my profile

See beloved and totally reliable [sic] https://en.wikipedia.org/wiki/Cauchy_product

[silver berry]

how do u get so good in math btw

[dull bear]

[of course, as each series is of positive terms, they both converge absolutely]

[dull bear]

Experience mixed with laziness for me I guess

[silver berry]

I do know cauchy schwarz

but I feel like it's differnet

[silver berry]

what does that mean

[dull bear]

it's slightly different, see the actual page

[silver berry]

so what do I do to get as good

[dull bear]

That for me, partly I've been doing math for quite some time, and also in a way if there's a shortcut to remembering/dealing with things, I'd try that, there was a quote somewhere (getting more philosophical than anything) that maths is good because you don't really have to remember that much, cause a lot you can derive from other things

(mind you, some people would argue that I'm not good, and rightly so )

But I guess just doing lots of problems, and to some extent (gasp) enjoying it was lucky to have the chance to teach myself stuff before university and found that enjoyable

@karmic forge how do you get good at math, I have no idea what to suggest myself

[silver berry]

oh u were good before uni alr?

for me I feel like the things being taught are sometimes so hard to understand

[dull bear]

I wouldn't say "good" more... I guess, that I had a chance to go over stuff through the situation I was in? Like I was lucky I got to self teach myself stuff, and that I also got to cover some stuff before I got into my course(?)
Honestly I would say it's mostly been luck how I've gotten this far, I have no idea

[dull bear]

And they can be sometimes, with a lot of things it can take time I'm terrible with this kind of advice but for me I've managed to get by mostly by trying to understand "everything", like if there's proofs and stuff trying to cover each step as I can

[dull bear]

Honestly, in general, I'm bad with these types of questions like the original one, I hate having to think up counterexamples, but in some cases there are natural ones that come up, for example for me, one of the first things I thought when looking at your thread was to see what your reasoning for each of them, and with not that much time after I realised you could think about the harmonic series and that the product of their terms makes a convergent series (hence I could react!)

[dull bear]

Won't even lie, I hadn't gotten around to thinking about (b), and you found a counterexample for it quicker than I could go back to read what it was asking So well done there!

[silver berry]

ye so I guess if I just make more question it should be fine

but I don't mind making more questions it's just when I don't know them thats the furstrating part

[dull bear]

Sometimes you learn the most from those ones and as always there is this server and other places you can ask if you get stuck and can't figure it out!

[silver berry]

right, thanks for helping

[dull bear]

A pleasure (though I didn't do that much!) Have a good one

[vocal sleet]

@silver berry Has your question been resolved?

[vocal sleet]

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[lethal heart]

I don't understand what's there, I write after "Since"

[hybrid flicker]

Since g(a) = ...

[lethal heart]

yes

[hybrid flicker]

since ... then by the existence

[lethal heart]

yes

[hybrid flicker]

so what is the issue exactly?

[lethal heart]

the signs of inequalities

[hybrid flicker]

ok

well recall where lies y0

[lethal heart]

yo ∈ (a,b)

[hybrid flicker]

uh no

that's not true

[lethal heart]

ah

yo ∈ (f(a),f(b))

[hybrid flicker]

that is true

so what does it mean in terms of inequalities

[lethal heart]

Doesn't it depend on the function?

like the graph

[hybrid flicker]

ok just

suppose f(a) < f(b)

so (f(a),f(b)) is a valid interval

then what does "yo ∈ (f(a),f(b))" mean in terms of inequalities?

[lethal heart]

?

nothing

[hybrid flicker]

well how is it different from yo ∈ R then?

[lethal heart]

only that y in f(a) < y in f(b)

[hybrid flicker]

huh?

[lethal heart]

the value of y in f(a)???

[hybrid flicker]

what does "y in f(a)" even mean?

alright you're getting confused

[lethal heart]

the value of y in f(a)???

[hybrid flicker]

?

f(a) is a number

the value of a number in a number?

[lethal heart]

😦

[hybrid flicker]

Ok let's start from the beginning

what is (x,y) when x < y?

[lethal heart]

wait

y=f(x)

so in the point a

[hybrid flicker]

that's not it

[lethal heart]

y=f(a)

[hybrid flicker]

no let's start over

you know the notation [x,y]?

[lethal heart]

yes

[hybrid flicker]

what does it mean or represent?

[hybrid flicker]

?

what am I supposed to understand from this?

[lethal heart]

from point a to point b are the x's, from f(a) to f(b) are the y's

so [x,y] ->[(a,b),(f(a),f(b))]

like something

[hybrid flicker]

nonono

I'll use different notations then

A and B are two numbers such that A < B

[lethal heart]

yes

[hybrid flicker]

what represents the notation [A,B]?

[lethal heart]

a coordinate

[hybrid flicker]

no

[lethal heart]

ah ok

an interval

[hybrid flicker]

The answer is literally in the FIRST sentence here

yes an interval

more precisely, which numbers does it contain?

[lethal heart]

im confused

i dont know

a and b

[hybrid flicker]

no, that's not all numbers

What is an interval?

whip out your definition if you need it

[lethal heart]

the interval is a set of numbers

[hybrid flicker]

yes

what property does it have that makes it different from other sets of numbers?

[lethal heart]

which is unlimited?

[hybrid flicker]

what do you mean by that?

[lethal heart]

which has infinite numbers

[hybrid flicker]

N has infinite numbers, is it an interval?

[lethal heart]

yes

[hybrid flicker]

{0,1,2,...}

no it's not

[lethal heart]

(0,+infinity)

[hybrid flicker]

ok even your notion of intervals is flaky, so I'll start from the beginning

A set of numbers is an interval if it doesn't have holes

N is not an interval because there's a big hole between 0 and 1

[lethal heart]

yes

[hybrid flicker]

now

you should know that all intervals can be written as:

[a,b] OR [a,b) or (a,b] or (a,b) where a < b can either be real or +/- infinity

[lethal heart]

yes

[hybrid flicker]

So wait

can you describe all numbers in (0,+infinity)?

[lethal heart]

yes

[hybrid flicker]

what does a number x in (0,+infinity) verify?

x compared to 0 is...

[lethal heart]

?

what do you mean?

[hybrid flicker]

Is 0 bigger than x, smaller, bigger or equal, ...?

[lethal heart]

bigger

[hybrid flicker]

YES

[lethal heart]

x is bigger

[hybrid flicker]

so any number in (0, +infinity) is BIGGER than its left bound

[lethal heart]

yes

[hybrid flicker]

so maybe you can infer what a number x in (a,b) verifies?

[lethal heart]

which is greater than a and less than b

[hybrid flicker]

yes

so a < x < b right?

[lethal heart]

yes

[hybrid flicker]

Can I now ask you this question one more time?

[lethal heart]

f(a)<y<f(b)

[hybrid flicker]

Yes!!!!

[hybrid flicker]

so can you see where the inequalities come from?

[lethal heart]

no

because

wait

y0 is a number?

[hybrid flicker]

yes

[lethal heart]

but how can y0 be certainly greater than f(a)

[hybrid flicker]

because we picked a y0 in (f(a),f(b))

[lethal heart]

but like

[hybrid flicker]

so y0 IS by definition greater

[lethal heart]

if y0 is a number

in the red circle f(a)>y0 or not ???

[hybrid flicker]

but we didn't choose y0 that way

you're looking at y0 = f(x0) aren't you?

[lethal heart]

ah ok

i understand

wait

[hybrid flicker]

it's possible that f(a) > f(x) for some x

that's not what matters

we want to show that IF f(a) < y < f(b), THEN y = f(x) for some x

[hybrid flicker]

IF THEN is not IF AND ONLY IF

[lethal heart]

the y0 is in f(a) and f(b)

therefore it cannot be outside the range, as in the previous photo

i understand

thank you so much

.close

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[drowsy citrus]

Can someone help me check this answer?

-12582912 is the decimal number

In binary its 0110 0000 0000 0000 0000

2's complement is 1010 0000 0000 0000 0000 0000

So when I convert it to hexadecimal is it 00200000 or 00A00000?

[sly sierra]

if it's a negative number, the most significant bit should be 1, no?

[drowsy citrus]

Oh so would it be 00A00000 then?

I fixed it so then the MSB is 1

[sly sierra]

the MSB is in the first digit

so your 8 digit hex number can't start with a 0 if it's negative

[drowsy citrus]

oh you are right

Would FFA00000 be correct?

[sly sierra]

nope, first two digits are right, third is not

[drowsy citrus]

Would FF400000 be correct?

[sly sierra]

yes

that is correct

[drowsy citrus]

thank you so much I appreciate it!

.close

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[clever mason]

I need help understanding divisibles, multiples, and factors.

[clever mason]

Its not a difficult idea but i keep getting confused between them

My best understanding of definitions is this. A is divisible by B if B|A, making B the divisor of A, making A divisible by B. Somehow divisible and multiples are related but my best understanding of multiples is that A is a multiple of B if A = C*B, such that C is an element of the set of integers. I dont even fully understand off the top of my head factors but if i can remember its the opposite of divisible so if A is divisible by B then B is a factor of A

im losing my mind trying to differentiate these

[misty belfry]

"A is divisible by B" is the same as "A is a multiple of B"

[dark kiln]

yeah it's the same thing

[clever mason]

yup, im writing it out right now

its just weird cause cause you would think their different

one has div (ision) and the other multi (plication) which sounds dumb

[misty belfry]

yeah the dividing part refers to A and the multiplying part refers to B

[clever mason]

but my best understanding is that they are the same because, A divisible by B if B|A, A multiple of B if A=CB, therefore for them to be equal A/ B has to equal C right?

[misty belfry]

correct

[clever mason]

lets go

[misty belfry]

and if A is divisible by B then B is a factor of A

[clever mason]

i was just about to mention factor

thats a weird one

i actually thought it was the exact same as divisble until i really read into it

i guess i should ask, what does factor mean?

[misty belfry]

factor=divisor

[clever mason]

wow

that actually makes sense

so they're the exact same thing?

[misty belfry]

yes

[clever mason]

that makes sense, cause factors are mostly used in expressions like 2x-2 and to say youre factoring 2 you simply divide out the 2 from it 2(x-1)

2 being the divisor

that answers my questions. thank you!

.close

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[left elk]

I need help on this question..... I just need explanations to get a hang of it

[left elk]

Plus hows the tabular form going to be like

[frozen bobcat]

tabular form just means you list the elements

[left elk]

oooh Starting to get it..... so how do i go about a

@open sundial

[left elk]

@stone birch @high abyss

Any help????

[frozen bobcat]

what are some natural numbers that also solve x^2 -5x + 6 = 0?

[left elk]

sorry i dont get

[left elk]

sorry i replied late

[frozen bobcat]

your first set in (a) is
A = { x such that "x is a natural number" and x^2 - 5x + 6 = 0 }
so you just need to find those numbers and list them

[left elk]

will i solve the quadratic equation to find the numbers?

[left elk]

You there?

.close

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[vast shale]

hii

[vast shale]

i need help

[vast shale]

so

so

[formal pond]

Look like chemistry

[vocal sleet]

@vast shale Has your question been resolved?

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[buoyant edge]

can someone help me transfer this equation into y = mx + b

[buoyant edge]

3x + 2y = 459

i was doing a question but my final answer was negative when it shouldnt have been

[buoyant edge]

so i think the root of my problem was here at the start

[buoyant edge]

i did -2y = 3x - 459, then divided vertyhing by -2

so y = -1.5x + 229

not sure what i did wrong

[rugged lintel]

looks right to me except the division of 459 which can't be whole

[buoyant edge]

wdym

[rugged lintel]

it's odd

[buoyant edge]

oh yea i rounded it but its not a big deal, my issue is something

shouldnt be negative

im just giong for a rough answer

hold on let me show the rest of my process work maybe i did something wrong later on

[proper igloo]

Are you sure this is the correct equation then? Because written this way X will always be negative

[buoyant edge]

i wrote it myself from a word problem

il send it rn

at the very bottom

x + y = 181
3x + 2y = 459

[proper igloo]

So you would want to solve the system of equation here

[buoyant edge]

i did this after but my handwriting is pretty bad

[proper igloo]

Use these first one and plug it into the second one by solving for one variable

[buoyant edge]

i just subbed in and got -96 instead of 97

[buoyant edge]

i subbed y = 181 - x and this together

is that correct

[proper igloo]

You missed a negative sign somewhere

[buoyant edge]

yea not sure where

should i just do everything again or will i make same mistake

cuz idk if its a brain fart or if im actally doing something wrong

[buoyant edge]

i think

in the second line of subsituting

when i moved 229 to other side

i put it before the 181

[proper igloo]

Yea when you moved around the 181 and the 229

you forgot a sign

[buoyant edge]

sorry for wasting ur time

ty

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[mystic pumice]

how to solve this

[pallid forge]

you would first need to find where it isn't continuous

for example, anything under a square root would have to be greater than or equal to 0

[tawny nacelle]

^

also, the denominator cannot equal 0

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[inland flower]

help pls idk what to do other than it saying sub x=2

[tawny nacelle]

if a function is continuous at a point, it's left and right limits as x approaches that point must be equal to each other

since you have a piecewise function made of polynomials, it is continuous everywhere except maybe at the "break" point, which is x = 2 in this case

you wish to find a number c such that the piecewise is continuous at x = 2 as well

so you want to set lim x->2- f(x) = lim x->2- f(x)

from there, you can sub in the respective polynomials for f(x) on both sides of the equation, find the limits by plugging in x = 2, and then solve for c

@inland flower

[inland flower]

oh ok lemme give this a try

actually im confused

[inland flower]

how is this written out

do i take xc^2+3x and equal it out to something

[tawny nacelle]

lim x->2- f(x) will use the definition of f(x) when x < 2, while lim x->2+ f(x) uses the definition of f(x) when x >= 2

thats because taking the limit is asking what the function approaches when x approaches a certain value

in the case where x -> 2-, we're asking "what happens when f(x) approaches 2 from the left hand side"

the behaviour of f(x) from the left hand side of x = 2 is just the behaviour of the piecewise section of f(x) when x < 2

so when i said to set lim x->2- f(x) = lim x->2- f(x), i'm saying "set the limit of f(x) as x approaches 2 from the left equal to the limit of f(x) as x approaches 2 from the right"

on the left, you have f(x) = cx^2 + 3x and on the right, you have f(x) = x^2 - cx

i think i'll leave the rest to you

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[boreal bane]

This is the solution.

I need help understanding the solution

How did they get p1p2 > n^1/2 * n^1/2

Wouldn't we assume they're equal?

[surreal basin]

did you read the second line

"since they cannot be equal to one another, at least one must be greater than √n."

so that then means in p1 p2 that one of the factors is > √n instead of ≥

[boreal bane]

yeah but either way they both equal N

[surreal basin]

having two prime factors p1 and p2 means two primes that are different numbers, not the same

[boreal bane]

from what i understand thats true for p1 and p2 seperately

how did we assume that p1p2 together is larger than root n squared

n = p1p2

[surreal basin]

your real problem here is how the second line is true

please rephrase the question

[boreal bane]

i get that one factor will be larger than root n at least

but how did we get from that to p1p2 > root n * root n

[surreal basin]

so if one factor is larger than root n

and you multiply both sides by the other factor

how is going to be that p1 p2 = √n √n

if p1 > √n

if you have 4 > 3

and you multiply both sides by the same positive number

youre not going to get an = sign

you will remain an > sign

thats still true if you choose to multiply the bigger side by a slightly bigger number than you do the other side

[boreal bane]

p1p2 = n though

and we know that

[surreal basin]

yes theyre doing a proof by contradiction

do you know what a contradiction is

[boreal bane]

prove the negate to be false

[surreal basin]

no

do you know what a contradiction is

[boreal bane]

erm thats legit the definition in the book lmao

[surreal basin]

no

I want the actual word

[boreal bane]

a statement which opposes another statement

[surreal basin]

thats just "opposite"

what do contradictions do

not enough

the statements oppose another statement and both are true

thats a contradiction

if one of them was false then it wouldnt be much of a contradiction

[surreal basin]

contradictions are a very specific kind of wrong

a proof by contradiction makes you assume something to be true, then finding a contradiction that would happen if you did

in this case you have two facts that need to be true

• p1 p2 = √n √n because it said so
• one of p1 or p2 needs to be greater than √n

the fact that theyre true but go against each other means a contradiction

that means one of them is wrong

p1 p2 = √n √n cant be wrong

one of p1 or p2 needs to be greater than √n should be

but that came from the original assumption that p1 and p2 are ≥ √n

therefore one of p1 or p2 is < √n

thats the proof theyre doing

[boreal bane]

okay that makes much more sense

[surreal basin]

you oughta google the big words youre seeing next time

[boreal bane]

idk im just going by the book

[surreal basin]

and does the book tell you what a contradiction is? not really

[boreal bane]

i just started proofs today

fair enough

[surreal basin]

a few times theyll use regular english words inside of the math jargon

this is one of those times

so if a word seems too big, chances are its not invented, its borrowed from somewhere else

that gives you a chance to make sure the book isnt BSing you on this

[surreal basin]

as for what they said, this is in a way what ends up happening

[boreal bane]

appreciate it

[surreal basin]

you proof that the negative is false but they leave out the "because two opposite statements have to be true"

which is the "contradiction" part of it

negative being false ==> positive being true isnt really doing anything with contradictions
(I think I mightve forgotten the specific words for it, I think its called a "contrapositive"(?))

[boreal bane]

yes

[surreal basin]

alr nice

[boreal bane]

contrapositive is the word

so essentially

in this example

we need to prove not 'one of p1 or p2 needs to be greater than √n'

or one of p1 or p2 needs to be less than root n

[surreal basin]

yep

[boreal bane]

and we can prove that if we can show that 'one of p1 or p2 needs to be greater than √n' isn't true

by reducing it to an absurd statement

like p1p2 > root n * root n when we know that both of them equal n

[surreal basin]

specifically a statement that goes against another true statement

[surreal basin]

oh other than big words they might also use latin

[boreal bane]

reductum absurdum

some shit like hat

[surreal basin]

that one yea

reductio ad absurdum

[boreal bane]

yeah that one

insane

proofs has so much more writing than even lettes or pronumerals

[surreal basin]

they break the steps down into smaller pieces so you end up having to make more of them

theres also that it has to go much slower than how you think

so itd be like thinking of a song then reconstructing it note-by-note

[boreal bane]

okay thank you very much tho

[surreal basin]

np

gl

[boreal bane]

i understand where i went wrong just flawed understanding

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[surreal basin]

then again the book made you misread an important word so not really

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[vast shale]

can someone help with these two

[waxen hawk]

For 1

You have to use the average velocity formula

[boreal bane]

dst

[waxen hawk]

Which is also known as the definition of average velocity

[vast shale]

so like distance=speed/time?

[boreal bane]

speed * tme

[vast shale]

oh

but how would i set it up..

[dusky willow]

45 minutes * 80 km/h = distance

[dusky willow]

you need to convert the km/h to km/minute

[boreal bane]

u have everything

[vast shale]

ohh

[boreal bane]

just extract information from the question + convert then sub in

[vast shale]

hmmmmm

i still dont get it..

[boreal bane]

did you read your question

[vast shale]

yessss

[boreal bane]

how long did the animal take

how fast was it traveling

[vast shale]

45 minutes to run and 80km per hour

[boreal bane]

yeah now convert 80kmph to kmpm

km per minute

[vast shale]

OHHH

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[solemn pike]

Need a lil help with qn12, unable to match answer

[solemn pike]

It's an ellipse

[sudden compass]

Do u have a diagram?

[sudden compass]

Also show work

[solemn pike]

Nvm it's done now

Just had a small calculation error

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[fiery oriole]

Could someone help me? I don't know the first steps to solving this problem

[fiery oriole]

basically what Im getting is that I need to make a recursive definition for this problem, but I dont konw how to get started

[vocal sleet]

@fiery oriole Has your question been resolved?

[surreal basin]

did you get what tn represents?

[fiery oriole]

the number of ways you can construct the a tower of n height using 1, 2, and 4 unit blocks

so far I have t1 = 1 t2 = 2 t3 = 3 ]

[surreal basin]

lets say you were trying to figure out t55

but you knew the values of t54, t52, and t50

[fiery oriole]

right

I guess I dont see the pattern

[surreal basin]

wait thats because I mistyped it

[fiery oriole]

should I keep going with manually getting values?

and try to find a pattern

[surreal basin]

no

hold on there

[fiery oriole]

okay

[surreal basin]

I just said I mistyped it

that doesnt mean you immediately give up

capiche?

[fiery oriole]

okay

[surreal basin]

hope that was cringe, thats gotta leave a mark

say you knew the values of t50, t52, and t53
and you needed to find the value of t54

you know that a 54cm tower can only be made out of 4cm, 2cm, and 1cm blocks
which means on top of the tower can only be a 4cm, 2cm, or a 1cm block

• if it was a 1cm block, you could remove it, and youd be left with a 53cm tower, and you know there's t53 possible 53cm towers
• if it was a 2cm block, removing it leaves you with a 52cm tower, and you know there's t52 possible 52cm towers
• if it was a 4cm block, removing it leaves you with a 50cm tower, and you know there's t50 possible 50cm towers
  adding the possibilities together,
  there are t53 + t52 + t50 possible t54 towers
  and so t54 = t53 + t52 + t50

this can be repeated for any height in cm ≥ 4,
so tn = t(n-1) + t(n-2) + t(n-4) is your recurrence relation

now you can give up but only after understanding the explanation

[fiery oriole]

ooh

it makes sense when you think of it backwards like that

[surreal basin]

ngl I wouldnt have thought of this, I heard of it from someone else

[fiery oriole]

Im still taking it in but it makes a lot more sense now

thank you

[surreal basin]

np

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[warped osprey]

hello how is this mean to be factorised?

[lapis marten]

Try expanding the first 2 squares

some things will cancel and then you can factor again

[warped osprey]

ok thanks

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[vast crow]

$\int{1}^{e} f(x)dx \geqslant \int{1}^{e} g(x)dx$
does that mean
$\frac{\mathrm{d} }{\mathrm{d} x}\int{1}^{e} f(x)dx \geqslant \frac{\mathrm{d} }{\mathrm{d} x}\int{1}^{e}g(x)dx$

[twin meteor]

Talot

[vast crow]

note: 1 and e are born of the integral

[tough crow]

Can anyone help me with d please it's urgent, gotta convert to exponential form

[full bobcat]

1^e is 1

[vast crow]

I just messed up writing the latex

[vast crow]

^

I need to know if two functions defined by same integral limits and have an inequality does taking the derivative reserve the inequality ?

[vocal sleet]

@vast crow Has your question been resolved?

[full bobcat]

ohh you meant from 1 to e?

like the integral from 1 to e?

well the derivative of a constant is 0

[vast crow]

yes

[arctic mantle]

so int of what from 1 to e

you can't have an integral of nothing

[full bobcat]

f(x)

[full bobcat]

at the end you are differentiating the definite integral

which is taking the derivative of a constant

so thats just 0

[arctic mantle]

so if int of f(x) in certain bounds is smaller than int of g(x) in said bounds, is the derivative of g(x) larger than the derivative of f(x)?

^^

that what ur asking?

not necessarily true
take sin and cos

[vast crow]

this what I tried to ask

[arctic mantle]

not always true

[full bobcat]

both derivatives are 0

you are taking the derivative of a constant

[arctic mantle]

no...

[full bobcat]

yes

[arctic mantle]

he made a msitake

[arctic mantle]

he means this

[full bobcat]

derivative with respect to what then?

x?

[vast crow]

yes

[full bobcat]

in that case its still 0

the derivative of a constant is 0

[arctic mantle]

@vast crow imma ask again since those r different questions

if int from a to b of f(x) > int from a to b of g(x) is f'(a) > g'a) for all a

that's ur question right

[full bobcat]

well how can a function be bigger than another function?

what does that even mean

[arctic mantle]

x^2 and x^2 + 2

;-;

that better?

edited the message

[full bobcat]

well thats a nice example

not all functions are like that

you cant say one function is bigger than the other

it depends on the interval

[arctic mantle]

^^

[vast crow]

for all x between [ a , b]
is f'(x) > g'(x)

[arctic mantle]

ummm

no

[vast crow]

so fundamental theorem of calculus does not reserve the inequality ?

[arctic mantle]

what?

no-

[full bobcat]

we need a proof for this one

[arctic mantle]

take sine and cosine b/w 0 and pi

[full bobcat]

or a counter example

[arctic mantle]

the int of sine from 0 to pi is 2

[arctic mantle]

gotcha

[arctic mantle]

while the int of cosine from 0 to pi is 0

[vast crow]

makes sense

[arctic mantle]

blue is sine to clarify

[full bobcat]

cos(x) is bigger than -sin(x) from 0 to pi right

[arctic mantle]

no

[full bobcat]

oh yaa

[arctic mantle]

-1 and 0 at pi

slightly less before that

[full bobcat]

ya ez

[vast crow]

😄 thank you everyone

[arctic mantle]

!done

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[vast crow]

.close

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[vocal sleet]

Please don't occupy multiple help channels.

[boreal tundra]

can you get a better pic

@weak drum

[weak drum]

i cant but i can give the numbers

cd is 6

ad is 5

be is 1.5

@boreal tundra can u do it

[boreal tundra]

whats the question

[weak drum]

]find ed,

and ac

[boreal tundra]

ok gimme min

5-5/4 is what

[weak drum]

0/5?

[boreal tundra]

ok ed is 3.75 cm

[weak drum]

how do yk that

[boreal tundra]

so its similar triangle right

CD scaled to BE

is 4x smaller

[weak drum]

yea

[boreal tundra]

so same for that vertical line

[weak drum]

but the total is 5

[boreal tundra]

becomes 4x smaller to AE

which is now

5/4

[weak drum]

oh yeaaa

[boreal tundra]

because you made the base 4x smaller

then you subtract

total length from AE

[weak drum]

so x + 5/4 = 5

[boreal tundra]

yeah

so now AC?

[weak drum]

yea

[boreal tundra]

hm

[weak drum]

is it possible

[boreal tundra]

should be

idk how to get srry

[weak drum]

its fine ur first hald was rlly helpful

appreciate it

[boreal tundra]

gl 👍

[noble spindle]

This doesnt seem that hard

[weak drum]

can u do it

ac?

its grade 10 ib math

[noble spindle]

$\frac{AD}{AE} = \frac{6}{1.5} = 4$

[weak drum]

yea so 4

[twin meteor]

casework

[noble spindle]

$AD = AE + ED = 5$

[twin meteor]

casework

[noble spindle]

Calculate AE then ED

[weak drum]

how do i find ae tho

[noble spindle]

$\frac{5}{AE} = 4$

[twin meteor]

casework

[noble spindle]

Like we said

[weak drum]

oh 1.25

[noble spindle]

Yeah so ED = 5 - 1.25

Aka 3.75

[weak drum]

wb ac is it possible

[noble spindle]

You want AC?

[weak drum]

mhm

[noble spindle]

You have any more info?

[weak drum]

no

is it possible

[noble spindle]

Doesnt seem so

[boreal tundra]

not even an angle

[noble spindle]

As it depends on the angle

[weak drum]

how do u guys know sm

i js learned this and forgot how to do it

[noble spindle]

You can basically change the angle and AC will change

If i was good at desmos i would show you but idk how to do it

[weak drum]

bro ur good enough u did this question so easily and quick]t

[boreal tundra]

m a t h

[weak drum]

bro i thought i was good at math ill i joined this discord server

istg

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[night river]

Hi, I'm trying to find two linearly independent eigenvectors for unit eigenvalue of this matrix
$$
\mqty[
1&0&0&0&0\
0&1&0&0&0\
1/4&1/4&0&1/4&1/4\
0&1/2&1/2&0&0\
1/2&0&1/2&0&0\
]
$$
So I've got
$$
\mqty[
0&0&0&0&0\
0&0&0&0&0\
1/4&1/4&-1&1/4&1/4\
0&1/2&1/2&-1&0\
1/2&0&1/2&0&-1\
]\mqty[x_1\x_2\x_3\x_4\x_5] = \mqty[0\0\0\0\0]
$$
So
\begin{align*}
\frac{1}{4}x_1+\frac{1}{4}x_2-x_3+\frac{1}{4}x_4+\frac{1}{4}x_5&=0\
\frac{1}{2}x_2+\frac{1}{2}x_3-x_4&=0\
\frac{1}{2}x_1+\frac{1}{2}x_3-x_5&=0\
\end{align*}

[twin meteor]

Faq

[night river]

How can I proceed?

[versed bane]

oh wait yeah nvm i posted something and then deleted it but then it was actually right: if you want to solve (A-I) x = 0, then you are solving a linear system, so you do it like any other linear system

which is reducing the (augmented) matrix to RREF and solving from there

[night river]

Ah okay I'll try that. Thanks!

[versed bane]

once you hvae the entire space of guys, you won't have that much trouble finding two linearly independent guys

[vocal sleet]

@night river Has your question been resolved?

Channel closed

Closed by @night river

Use .reopen if this was a mistake.

[night river]

Yep, it worked. Thank you!

[vocal sleet]

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[magic wasp]

Ignoring the constraint, how do you go from Tn to T{n+1}?

[green crystal]

like for example going from T1 to T2?

T1 = 3, T2 = 8

[magic wasp]

Ignoring the constraint

[green crystal]

im not sure

[magic wasp]

How do you build a string of length n?

[green crystal]

Depends on the pattern

[magic wasp]

Do you not understand what ignoring the constraint means

[green crystal]

no, idk how to answer ur Q without including it

[magic wasp]

Can you list all the strings of length 3 over the alphabet {a,b,c}?

[green crystal]

thats a long list but its 27

[magic wasp]

Ok, now imagine, even if it's false, that all 27 of these satisfy the constraint (no substring aa)

How do you build all the strings of length 4 that also satisfy the constraint from there

I don't want the number, I just want the process

[green crystal]

i guess for each string, you would add a,b,c separatly

so aaa would be:
aaaa
aaab
aaac

[magic wasp]

Good start, now how to satisfy the constraint

[green crystal]

we cant have aa strings

[magic wasp]

Right, again, imagine all the 3-letter strings you have already satisfy the constraint

What do you need for the 4-letter ones

[green crystal]

i wanna say its something like n+1 but not sure

[magic wasp]

Not sure what you mean...

Take these 3-letter strings:
aba
bcc
cab
What 4-letter strings can you build from there?

[green crystal]

you can add a,b, or c to each string to make it 4

[magic wasp]

Yes, but...?

[green crystal]

are we taking aa free string into account here?

[magic wasp]

Now we are

[green crystal]

ahh ok

there cant be two a's beside each other when adding the fourth letter

such as for aba we can add b or c at the start or beginning of the string

[magic wasp]

start or beginning?

[green crystal]

(blankabcblank)

[magic wasp]

You mean start or end?

[green crystal]

its not letting me do it but yes

[magic wasp]

_ a b a _

Ok but if you consider both start and end you'll end up double counting

Adding c at the start of aba is the same as adding a at the end of cab

[green crystal]

right

so where will we add the letter?

[magic wasp]

Doesn't matter as long as you can build every string once and only once

[green crystal]

ok we'll just say then to add an additional letter i guess

as long as its aa free string

[magic wasp]

What's your native language?

[green crystal]

what?

[magic wasp]

to add an additional letter
That's a tautology, it doesn't say anything about where to add the letter

So it looks like you're not a native English speaker

[green crystal]

Thats technically math language but sure

[magic wasp]

That makes no sense

[green crystal]

anyways, can we continue?

[magic wasp]

To talk about maths you need a natural language, unless you want to do everything with symbols

I was just asking because it can help knowing what language the person you are speaking with is most comfortable with, that's all

[green crystal]

ok

[magic wasp]

Ok well if you take an n-letter string and append another letter to form a {n+1}-letter string, then you just need to throw away the cases where the n-letter string ends with 'a' and you append an 'a'

That's the recursion

That's all I can help you with

[green crystal]

how do i develop the expression where the string ends with a and we add a

[noble spindle]

Whats T_3?

[green crystal]

22

[noble spindle]

Seems like $T_n = 3T_{n - 1} - T_{n - 2} + 1$

Just a random guess

Try proving/disproving it

[twin meteor]

casework

[green crystal]

is T0 = 0?

if so T_2 = 7

[noble spindle]

For n > 2

Idk something like that

With this question the most important thing is to play with them

See what you get

[green crystal]

where does it state that?

[noble spindle]

I state that

Im guessing it

My guess may be wrong but its at least something

[green crystal]

T3 does give me 22

[noble spindle]

Try calculating T_4

And seeing if it still holds

[green crystal]

i get 59

[noble spindle]

T_4 is 59?

[green crystal]

yea

[noble spindle]

Well that is $3\times 22 - 8 + 1$

[twin meteor]

casework

[green crystal]

i calculated 59 through the formula

[noble spindle]

I mwan you got that by counting it or you got it using the thing i said?

Oh...

Well you should prob do it without it

Shouldnt be that hard

[green crystal]

is there a formula for that? its gonna take a long time to brute force it

[noble spindle]

Not yet

[green crystal]

its not hard just gonna take a very long time

unless im not thinking of a faster approach

[noble spindle]

Yea ik i mean this is the exploration route

If you have any other ideas you can take it

[green crystal]

do you have any idea on how to do it?

[noble spindle]

Well how many of aa free strings start with a

[green crystal]

without the constraint there are 3^4 possibilities

[noble spindle]

For n = 1 its 1 for n = 2 its 2 , for n = 3 its ???

[green crystal]

3

[noble spindle]

Gurantee you its not 3

[green crystal]

that start with a right

[noble spindle]

Yes

[green crystal]

9