The Last Post Wins!!!!!

[TwinStorm He/Him]

5 hours ago, OOKLA_the LIV said:

OH

so i get TWO NEW PARENTS YAY

 and a big sis i think

can I be your big brother?

3 hours ago, strmblsd said:

I  just started wheel of time I was reading and a character named wit was said and I legitimately gasped out loud in class then though oh my gosh I'm a idiot

I remembered having that reaction lol.

[Vyzkel He/Him]

25 minutes ago, Ookla the Dragonslayer said:

OKOKOKOKOKOKOKOKOKOKOKOKOKOKOKOKOKOK

It doesn't help that I cant type math characters.

ORIGINAL TEXT (for reference):

The limit as a approaches infinity of

((((1 - x^a)/(1 - x))^0.5)/(x^(a/2-1)

THAT EQUALS

x/(x-1)^2

AND

THE ONLY NUMBERS THAT GIVE RATIONAL ANSWERS ARE SQUARES PLUS ONE

BUT

17 is 16+1

AND

170 is 169+1

IT’S SO WONDERFUL 

 

SO!

The first time I noticed a pattern from this was...

11!

When you take the square root of any number with only ones as its digits, it results in one of two things:

• In the case that there are an even number of digits, it is equal to the same number of threes, with the decimal in the middle. Then it has a bunch of numbers after that, but those aren't important.
• In the other case, with odd digits, it just gives you a bunch of numbers, stating with a one and a zero. (It does this because when you add a digit, you are essentially [there is a bit more to it) multiplying by the square root of ten.)

AND SO!

I wanted to make and equation to model this, because why not!

I was kind of stuck on how to represent the number of digits as the input, but then... I remembered sigma notation

(don't kill me, that's what it's called)

To convert number of digits to the actual number, I used

   

__a__

\

  >            (10^n)

/

______

n = 0

 

I didnt have the symbol, so it's weird.

That essentially just equals 10^0 + 10^1 + 10^2 + 10^3 + ... + 10^a, which converts it.

Now, this is a geometric series, which means I can represent the sum without sigma notation pretty easily.

So that there is equal to

(1 - 10^a)/(1 - 10)

or (1 - 10^a)/(-9), which I'll defined as f(a)

Now that I had that, I wanted to pursue the threes thing.

So if I take the square root of f(a),

it should give me that pattern that I saw up there.

And it did, but I wasn't really satisfied yet, as I wanted the first 3 to always be in the ones place, as... i dunno. huh.

So i divide it by 10 to the a/2 - 1 power, giving the effect I wanted.

 

AND I REALIZED IT CAN BE ABRA KEDABRA-ed BY A LIMIT!

So if you put int in a limit as a approaches infinity, you get 10/3, or 3.33333333333333333333333 repeating forever.

But I wasn't satisfied. I thought, well, a is no longer changing, so why not change the base of the exponent (currently 10, ya know)

So I got the original equation, which was kinda complex, and it was:

The limit as a approaches infinity, of

((((1 - 10^a)/(1 - 10))^0.5)/(10^(a/2-1).

Substituting x in for the 10s,

 

The limit as a approaches infinity of

((((1 - x^a)/(1 - x))^0.5)/(x^(a/2-1)

Next, I notice that the value of this at 5 was 2.5 exactly, or 5/2

COOL!

But I WAS NOT SATISFIED

I wanted to see why specifically 5 and 10 gave coherent answers, while all the other ones gave... incoherent answers. I found that 20, 100, and others didn't work, but then I stumbled upon 17, which did, giving 17/4

So the rational values I get have been:

5/2, 10/3, and 17/4

The numerator increases by a shifting value each time (more on that)

but the denominator increased by a CONSTANT VALUE! Success!

Still, I wanted to find a pattern for the numerator, and soon found that the amount it increased by increased by 2 each time, indicating a quadratic relation!

But this didnt mean that ut was quadratic, only that the numbers that were rational happened to be a shift on perfect squares. This, in turn, indicated that there was a square root!

5 - 1 = 4!

square root to 2, the denominator of 5!

10 - 1 = 9!

square root to 3, the denominator of 10!

And this repeated itself...

I HAD FOUND MY PATTERN!

That weird limit equation was equal to x / (x-1)^0.5

SUCCESS!

And to top it all off, both 17 and 170 worked in the equation. So there.

YES!

I read, like, half of that. Alright Professor Wrath, that’s pretty cool.

[Through the Living Shadow he/him]

1 minute ago, Ookla the Invyzable said:

I read, like, half of that. Alright Professor Wrath, that’s pretty cool.

I feel another personality coming on…

also, I notice smt ELSE THAT IS ALSO AWESOME

if you convert the resulting number from base ten to the base of the original number, you get a cool number.

[Vyzkel He/Him]

Just now, Ookla the Dragonslayer said:

I feel another personality coming on…

also, I notice smt ELSE THAT IS ALSO AWESOME

if you convert the resulting number from base ten to the base of the original number, you get a cool number.

Which is…

[Through The Living Glass She/They]

8 minutes ago, Ookla the Irreplaceable said:

can I be your big brother?

. . . another family member?

5 minutes ago, Ookla the Dragonslayer said:

I feel another personality coming on…

 

*whispers*

They're multiplying!!

[TwinStorm He/Him]

1 minute ago, Through The Living Glass said:

. . . another family member?

*whispers*

They're multiplying!!

... it keeps getting bigger

[Existential]

this family tree thing has me lost

[Through The Living Glass She/They]

2 minutes ago, _Null_ said:

this family tree thing has me lost

hehehe

heyyyyyyyy againnnnnnnn

[Existential]

2 minutes ago, Through The Living Glass said:

hehehe

heyyyyyyyy againnnnnnnn

hiiiiiiiiiiiii

[Through The Living Glass She/They]

Just now, _Null_ said:

hiiiiiiiiiiiii

tiiiiiiirrrrrrrrrrrred

how's your fooooooooot

[Existential]

3 minutes ago, Through The Living Glass said:

tiiiiiiirrrrrrrrrrrred

how's your fooooooooot

resting

[Through The Living Glass She/They]

Just now, _Null_ said:

resting

That's gooooooood 

[Existential]

you should take a nap

 

if you're tired

[reisleK she/her]

Happy Koloss Head Munching Day people!

[Bird Furious she/her/un/important]

we should all take a collective nap

like the entire world

should decide

ok

from 1-3 Jamaican time we're all gonna take a nap

and the internet will just stop

and the fast food will just stop

and everything will just stop for two hours

and we can take a nap

sucks for the ppl who don't wanna sleep or for the ppl who are already asleep but ykw this ain't about them 

it's about EVERYONE ELSE

ehehehehehehee

[Through The Living Glass She/They]

6 minutes ago, _Null_ said:

you should take a nap

 

if you're tired

but I can't take napsssssss

Just now, Ookla the Hemalurgic Spike said:

Happy Koloss Head Munching Day people!

Happy KHMD

Just now, oOklA thE shInyYyy said:

we should all take a collective nap

like the entire world

should decide

ok

from 1-3 Jamaican time we're all gonna take a nap

and the internet will just stop

and the fast food will just stop

and everything will just stop for two hours

and we can take a nap

sucks for the ppl who don't wanna sleep or for the ppl who are already asleep but ykw this ain't about them 

it's about EVERYONE ELSE

ehehehehehehee

mm sucks for meeeeeeee

[Bird Furious she/her/un/important]

4 minutes ago, Through The Living Glass said:

but I can't take napsssssss

Happy KHMD

mm sucks for meeeeeeee

NO BUT YOU NEED A NAP THOUGH 

[Through The Living Glass She/They]

1 minute ago, oOklA thE shInyYyy said:

NO BUT YOU NEED A NAP THOUGH 

I DO

BUT I CAN'T

I'LL FEEL SICK

AND MORE TIRED

AND GROGGY AND HUNGRY AND LIKE I NEED A SHOWER

although I wouldn't mind a shower at all lol

it sounds really nice

[Bird Furious she/her/un/important]

2 minutes ago, Through The Living Glass said:

I DO

BUT I CAN'T

I'LL FEEL SICK

AND MORE TIRED

AND GROGGY AND HUNGRY AND LIKE I NEED A SHOWER

although I wouldn't mind a shower at all lol

it sounds really nice

THEN GET A FULL NIGHTS SLEEP INSTEAD I BELIEVE IN YOU PUEDES HACER MADRE

[Through The Living Glass She/They]

1 minute ago, oOklA thE shInyYyy said:

THEN GET A FULL NIGHTS SLEEP INSTEAD I BELIEVE IN YOU PUEDES HACER MADRE

*cough*

Mm. Only if you do, too

[Bird Furious she/her/un/important]

4 minutes ago, Through The Living Glass said:

*cough*

Mm. Only if you do, too

*sigh*

You're mean.

But I will try, just for you. 

[Through The Living Glass She/They]

3 minutes ago, oOklA thE shInyYyy said:

*sigh*

You're mean.

But I will try, just for you. 

aww

well

that's sweet

*hug*

thanks

[Existential]

19 minutes ago, Through The Living Glass said:

I DO

BUT I CAN'T

I'LL FEEL SICK

AND MORE TIRED

AND GROGGY AND HUNGRY AND LIKE I NEED A SHOWER

although I wouldn't mind a shower at all lol

it sounds really nice

BRUH YOU SHOULD"VE TOLD ME I COULD'VE HELPED 

[Through The Living Glass She/They]

8 minutes ago, _Null_ said:

BRUH YOU SHOULD"VE TOLD ME I COULD'VE HELPED 

With . . . the nap thing?

huh?

Edited December 19, 2024 by Through The Living Glass

[Existential]

1 minute ago, Through The Living Glass said:

With . . . the nap thing?

huh?

With all the above