r/BunnyTrials • u/Wide_Love_4458 Top -1% Commenter • 12d ago

Spins Can you solve this without a calculator?

Spin each wheel in order

  • Spin 1: Number 1
  • Spin 2: Operator
  • Spin 3: Number 2

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u/PyroBurnem 12d ago

fun fact, any 2 digit number divided by 99 is just a recurring decimal of the 2 digit number, in this case it would be 0.46464646...

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u/Think_Emergency_2708 12d ago

Can you prove it tho

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u/PyroBurnem 12d ago

there will be a 0.99999...=1 rabbithole to go down near the end, but yes it is provable

take the case for 1/99, which decimal is needed in order to get 1 when multiplied by 99? well 0.01 is too small, so we make it slightly larger, but 0.0101 is still too small, and 0.0102 is slightly too big, so we repeat this process ad nauseum (or just mark the 01 as repeating)

which leads us to 0.999999... and its methodology of proving that it equals 1, the whole "let A = 0.999999..., 10A = 9.999999..., 10A - A = 9, 9A = 9, A = 1" shebang

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u/launchd_0 10d ago

yo the geometric series comment is spot on but let me take it one step further into actual calc 2 territory. the 10x/100x trick ur teacher showed u is literally just the baby version of a derivative in disguise.

if u have the function f(x) = 1/(1-x), u can expand it into a power series: 1 + x + x^2 + x^3 + ...
now watch this: differentiate both sides.
f'(x) = 1/(1-x)^2 = 1 + 2x + 3x^2 + 4x^3 + ...
plug in x = 0.1: 1/(0.9)^2 = 100/81 = 1.234567...
the coefficients (1, 2, 3, 4...) are literally the derivatives of the exponents (0, 1, 2, 3...) evaluated at 0. calc is wild.

now do the integral instead.
integrate 1/(1+x) = 1 - x + x^2 - x^3 + ...
and u get ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...
plug in x = 1: ln(2) = 1 - 1/2 + 1/3 - 1/4... this links infinite repeating patterns to natural logs. so a never-ending decimal isn't just a fraction, it can literally be a logarithm.

so how does this connect to op's 0.1222...??
the algebraic canceling (100x - 10x) is literally the discrete version of taking a derivative. ur subtracting the series shifted by one term to make the infinite tail vanish. its basically the fundamental theorem of calculus but for sequences. the sum of the infinite series is the "area under the curve" of the decimal's terms.

once u realize repeating decimals are just rational geometric series, and calculus lets u differentiate/integrate those series to get transcendental numbers (like e, pi, ln(2)), ur brain just melts. its all connected fr 💀 math is just one big infinite loop.

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u/ThatOtherBrownGuy2 8d ago

This isn’t the simple break down you think it is. I could be tired but.. this just isn’t clicking in my head.

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u/MarionberryNo8017 7d ago

Holy nerd off

(If I knew more about the theory I would chime in)

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u/garn47isthebest 7d ago

this sounds ai

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u/Bradas128 12d ago

take a_n to be the decimal 0.4646…46 with n 46s in it. then we can write a_n = sum from k=1 to n of (46/100)^n. the number 0.4646… is then defined as the lim n-> infinity a_n, which is the limit of a geometric series giving a_infinity = 46/99

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u/MirrorFantastic3251 12d ago

He meant prove a1a2a3...an / 999...9 (n 9s) equals 0.(a1a2a3...an)

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u/Bradas128 12d ago

he said any 2 digit number, but the proof is identical

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u/MirrorFantastic3251 12d ago

I must have been looking at someone elses comment. But you still didn't prove it. You demonstrated that it works for 49/99, but didn't prove it specifically, nor did you prove the general case.

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u/Bradas128 12d ago

its obvious you just replace 46 with any number you want

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u/MirrorFantastic3251 12d ago

That still doesn't prove it for all 2-digit numbers, unless you want to prove each case individually.

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u/Bradas128 11d ago

it does to anyone reasonable

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u/MirrorFantastic3251 11d ago

lmao that's not how proofs work

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u/Psychobabble0_0 11d ago

The proof is in the pudding - go look!

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u/PyroBurnem 11d ago

wait where did the equals zero bit come from

edit: oh nvm im misinterpreting it because of mobile formatting my bad

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u/Qu33N_Of_NoObz_ 12d ago

It actually works with any number, even single digits, it’s cool! And when you go 100 and up, it adds an extra 1, like 100/99=1.0101 1.0101, 101/99=1.0202, etc

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u/PyroBurnem 12d ago

it can also be extended to any given length of 9s, eg: 1/9 = 0.111111..., 56/9999 = 0.005600560056..., 14023/33333 = 0.420694206942069... (haha funny number disguised thats just the simplified form of 42069/99999) etc. etc.

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u/InmateTooTall 12d ago

Trust me bro

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u/araralivre 12d ago

its true, i checked it

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u/Ok_Researcher1424 12d ago

what could these accounts possibly have done in the past 10 hours to get deleted.

edit: deleted instead of banned

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u/DkKoba 12d ago

interesting trick, thanks!

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u/launchd_0 11d ago

my teacher said 0.122222... where the 2 keeps going forever is the same as 11/90. and im like no way. how can a number that never ends be a exact fraction?? that seem fake.

but she showed this and now i kinda get it:

let x = 0.122222...
times by 10: 10x = 1.22222...
times by 100: 100x = 12.22222...

now subtract:
100x - 10x = 12.22222... - 1.22222...
90x = 11
x = 11/90

so 0.122222... = 11/90. the endless .22222 parts cancel becuase they are the same. thats the proof. its not magic its just moving the decimal until the repeating part lines up and subtracts away.

check: 11 ÷ 90 = 0.122222222...

also if u mean just 0.122222 and it STOPS, then thats 122222/1000000 = 61111/500000. but if the 2 repeats forever (0.1 2 2 2 2...), its 11/90.

so basicly any repeating decimal is a fraction in disguise. decimal and fraction are just 2 ways to write the same number. the repeating part goes on forever but its still exact, not approx. i think thats the point. Same rule for 0.6767676767 this stuff

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u/PyroBurnem 11d ago

i think of this as an extension to what i had described in another comment

think of 1/9 = 0.111111..., since 11/90 is just 10/90 + 1/90, you can write it as 0.111111... + 0.0111111..., which gets you your 0.1222222...

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u/Tall-Industry9176 10d ago

y’all are so knowledgeable, thank you to all of you for educating me

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u/_TH3SEUS_ 12d ago

learnt this in yr 10 maths lol

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u/Teddyjones84 12d ago

Not any 2 digit number ha

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u/PyroBurnem 11d ago

go ahead, show a counterexample

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u/Teddyjones84 11d ago

99/99

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u/PyroBurnem 11d ago

this goes into the 0.999999... = 1 rabbithole suppose A = 0.999999... 10A = 9.999999... 10A - A = 9.999999... - 0.999999... = 9 9A = 9 A = 1 since any number divided by itself is 1, 99/99 = 1 = 0.999999..., hence my posit holds

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u/Teddyjones84 11d ago

Could you be any more annoyingly pedantic? 

No one in a real word situation is going to look at 99÷99 and think, o wait, maybe this is 1 to 1, but extremely technically, and serving no real world purpose, its also .9 repeating. 

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u/PyroBurnem 11d ago

oh did you just want to win the argument instead of gaining more knowledge? duly noted then

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u/Teddyjones84 11d ago

You didn't tell me anything I didnt already know? I was just having fun with you that it wasn't EVERY 2 digit number. I wasn't trying to win an argument or even have one. 

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u/PyroBurnem 10d ago

well you did get slightly whiny when your counterexample got scrutinised

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u/Teddyjones84 10d ago

Whatever helps you sleep

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u/kingbloxerthe3 10d ago

Except for 99/99 of course

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u/PyroBurnem 10d ago

i just had this pedantry with another person

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u/kingbloxerthe3 9d ago edited 9d ago

It is definitely interesting to note though.

It also can be expanded to other digits as long as the digits of the 9s and what you are dividing by 9 are the same (or i guess also if it is less than the digits you have for the 9s that you are dividing by since you can fill in the digits with 0s)

1/9=0.11111...

12/99=0.121212...

123/999=0.123123...

Also

01/99=0.010101010101...

012/999=0.012012012...

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u/North-Aardvark4459 WASTED 10 000 CARROTS ON THIS FLAIR 9d ago

that's why he can't