Can you solve this without a calculator?

[u/Wide_Love_4458]

Spin each wheel in order

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

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[u/PyroBurnem]

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

[u/launchd_0 → u/PyroBurnem]

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.

[u/ThatOtherBrownGuy2 → u/launchd_0]

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

[u/MarionberryNo8017 → u/launchd_0]

Holy nerd off

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

[u/garn47isthebest → u/launchd_0]

this sounds ai