4 ms·

Another secondary school 'proof' x = 0.9999..... 10x = 9.9999..... (10x -x) = 9x = (9.9999.... - 0.9999....) = 9 x = 9/9 = 1

notRobot 7y ago
or, x = 0.9999... 10x = 9.999... 10x = 9 + 0.999... 10x = 9 + x 9x = 9 x = 1 Presented slightly more clearly https://en.wikipedia.org/wiki/0.999...#Algebraic_arguments https://en.wikipedia.org/wiki/0.999...#Algebraic_arguments
- hinkley 7y ago
  Yeah that’s way more complicated than it needs to be and I’m tempted to replace that whole section with: x = 0.9999... 2x = 1.9999... 2x - x = 1 x = 1
  - moonchild 7y ago
    It's much more intuitive that 100.9999...=9.9999... than that 20.9999...=1.9999...
    - ColinWright 7y ago
      You've met the formatting limitations. You mean: "It's much more intuitive that 10 * 0.9999... = 9.9999... than that 2 * 0.9999...= 1.9999..."
  - champagneben 7y ago
    How do you jump from x = 0.999... to 2x = 1.999..
    - saagarjha 7y ago
      It’s less intuitive, but still relatively straightforward from performing the “long” multiplication. You do need to convince people that the “8 at the end“ is irrelevant, however.
      - hinkley 7y ago
        You need to convince people there is no 8 at the end because there is no end. If you see an 8 it’s because you didn’t carry the 1. Keep going. Another way to think about it is that you have n digits, and I’m using the n and n + 1 digit at the same time. But since n goes to infinity, +1 hardly matters.
    - hinkley 7y ago
      Picture an infinite ticker tape in your mind. Write 0.9 + 0.9 —————- 1.8 Now keep extending the 9’s. You have to carry a one, so fix the carry and then add more nines. Where people keep getting tripped up is thinking you can stop when you get tired, or die, or when the universe ends in heat death. You don’t get to stop. You never get to stop. It’s nines all the way down.
pgtan 7y ago
> 10x -x subtracting infinities is dangerous, you can achieve any result from it https://www.youtube.com/watch?v=-EtHF5ND3_s https://www.youtube.com/watch?v=-EtHF5ND3_s
- OskarS 7y ago
  You're not subtracting anything infinite. Whatever you think of 0.99999... (and the correct thing to think is that it equals 1), I hope we can both agree that it is at least finite! I mean, we can agree that it's less than 2 and more than 0, right? That subtraction is just as valid as saying 0.333... + 0.333... = 0.666..., or that 1/3 + 1/3 = 2/3.
himinlomax 7y ago
I like this one better: 0.9 = 1 - 0.1 0.99 = 1 - 0.01 0.999 = 1 - 0.001 0.9999 = 1 - 0.0001 0.99999... = 1 - 0.00000... with a 1 at the end of the infinite series of 0
- jayd16 7y ago
  >[...] at the end of the infinite series of [...] ...uhh
  - himinlomax 7y ago
    "Eternity feels a bit too long ... especially near the end."
syockit 7y ago
The difficult part is to convince that (9.9999.... - 0.9999....) = 9. Someone else mentioned about Zeno's paradox involving Achilles and the tortoise. Since you multiplied x by 10 to get 9.9999...., you know that it's always going to be 1 decimal place slower in pace to reach convergence compared to 0.9999.... What you might get instead is (9.9999.... - 0.9999....) = 8.9999....

notRobot 7y ago

or, x = 0.9999... 10x = 9.999... 10x = 9 + 0.999... 10x = 9 + x 9x = 9 x = 1 Presented slightly more clearly https://en.wikipedia.org/wiki/0.999...#Algebraic_arguments https://en.wikipedia.org/wiki/0.999...#Algebraic_arguments

hinkley 7y ago
Yeah that’s way more complicated than it needs to be and I’m tempted to replace that whole section with: x = 0.9999... 2x = 1.9999... 2x - x = 1 x = 1
- moonchild 7y ago
  It's much more intuitive that 100.9999...=9.9999... than that 20.9999...=1.9999...
  - ColinWright 7y ago
    You've met the formatting limitations. You mean: "It's much more intuitive that 10 * 0.9999... = 9.9999... than that 2 * 0.9999...= 1.9999..."
- champagneben 7y ago
  How do you jump from x = 0.999... to 2x = 1.999..
  - saagarjha 7y ago
    It’s less intuitive, but still relatively straightforward from performing the “long” multiplication. You do need to convince people that the “8 at the end“ is irrelevant, however.
    - hinkley 7y ago
      You need to convince people there is no 8 at the end because there is no end. If you see an 8 it’s because you didn’t carry the 1. Keep going. Another way to think about it is that you have n digits, and I’m using the n and n + 1 digit at the same time. But since n goes to infinity, +1 hardly matters.
  - hinkley 7y ago
    Picture an infinite ticker tape in your mind. Write 0.9 + 0.9 —————- 1.8 Now keep extending the 9’s. You have to carry a one, so fix the carry and then add more nines. Where people keep getting tripped up is thinking you can stop when you get tired, or die, or when the universe ends in heat death. You don’t get to stop. You never get to stop. It’s nines all the way down.

pgtan 7y ago

> 10x -x subtracting infinities is dangerous, you can achieve any result from it https://www.youtube.com/watch?v=-EtHF5ND3_s https://www.youtube.com/watch?v=-EtHF5ND3_s

OskarS 7y ago
You're not subtracting anything infinite. Whatever you think of 0.99999... (and the correct thing to think is that it equals 1), I hope we can both agree that it is at least finite! I mean, we can agree that it's less than 2 and more than 0, right? That subtraction is just as valid as saying 0.333... + 0.333... = 0.666..., or that 1/3 + 1/3 = 2/3.

himinlomax 7y ago

I like this one better: 0.9 = 1 - 0.1 0.99 = 1 - 0.01 0.999 = 1 - 0.001 0.9999 = 1 - 0.0001 0.99999... = 1 - 0.00000... with a 1 at the end of the infinite series of 0

jayd16 7y ago
>[...] at the end of the infinite series of [...] ...uhh
- himinlomax 7y ago
  "Eternity feels a bit too long ... especially near the end."

syockit 7y ago

The difficult part is to convince that (9.9999.... - 0.9999....) = 9. Someone else mentioned about Zeno's paradox involving Achilles and the tortoise. Since you multiplied x by 10 to get 9.9999...., you know that it's always going to be 1 decimal place slower in pace to reach convergence compared to 0.9999.... What you might get instead is (9.9999.... - 0.9999....) = 8.9999....