Einstein's riddle - can you solve it?

[Atlblkz06]

The easy way:
x = .999…
10x = 9.999…
10x - .999… = 9
9x = 9
x = 1
.999… = 1

Other way:

Lets say y is the .99999 number of finite number of 9's (n).
For any y, there is a y1 such that y<y1<1.
We know that y limits to 1 as n goes to infinity, and therefore so does y1 and they're both 1.

If you pick a number thats closer to 1 than my .999999 (n times), I'll simply pick a number closer to 1. We'll do this infinitely until we both end up with 1.

The kicker is that you're doing this infinitely and thats something thats hard to imagine.

[Magnus213]

The easy way:
x = .999…
10x = 9.999…
10x - .999… = 9
9x = 9
x = 1
.999… = 1

Other way:

Lets say y is the .99999 number of finite number of 9's (n).
For any y, there is a y1 such that y<y1<1.
We know that y limits to 1 as n goes to infinity, and therefore so does y1 and they're both 1.

If you pick a number thats closer to 1 than my .999999 (n times), I'll simply pick a number closer to 1. We'll do this infinitely until we both end up with 1.

The kicker is that you're doing this infinitely and thats something thats hard to imagine.

I still don't like this. y and y1 will continually subdivide and approach 1 and all that, but "and they're both 1" and "we both end up with 1" don't prove that they become one. The number will just get infinitely close to 1 and infinitely longer on paper, but never actually get there.

I feel like something fishy is going on with the numerical proof as well. Let's assume that n = 6, so x = 0.999999. Pretty close to 1, clearly not infinitely close, but let's go with it.

x = 0.999999
10x = 9.99999 (notice that you lose a decimal place value)
10x - 0.999999 = 9.99999 - 0.999999 = 8.999991
9x = 9(0.999999) = 8.999991 =/= 9
x =/= 1

I don't see how you can fundamentally say that two distinctly different numbers have the same value.

[Atlblkz06]

I still don't like this. y and y1 will continually subdivide and approach 1 and all that, but "and they're both 1" and "we both end up with 1" don't prove that they become one. The number will just get infinitely close to 1 and infinitely longer on paper, but never actually get there.

I feel like something fishy is going on with the numerical proof as well. Let's assume that n = 6, so x = 0.999999. Pretty close to 1, clearly not infinitely close, but let's go with it.

x = 0.999999
10x = 9.99999 (notice that you lose a decimal place value)
10x - 0.999999 = 9.99999 - 0.999999 = 8.999991
9x = 9(0.999999) = 8.999991 =/= 9
x =/= 1

I don't see how you can fundamentally say that two distinctly different numbers have the same value.

For part A, think in terms of limits, not any finite point in time. You cannot "freeze" the process at any point and check to see if it works.

Part B, same thing. the 9s never stop repeating (hence the 3 dots)
10x-x = 9x, not 8.999x

Good points though - but let me assure you that this stuff is true. I did NOT just create this proof, its a "well known fact". Juts check online or ask any math professor.

[hondachik]

lmao @ IA turned into a math website. Srsly. Not many people are good at math =)