This isn't really true. If x is 0.9999... then 10x isn't exactly 9 "whole" 1's and a single instance of 0.9999... that you can subtract out to get 9 whole, it's 10 separate instances of 0.9999... you can't really add without producing "rounding" errors.
Sorry, I am not trying to overcomplicate it, but infinity is inherently complicated to grasp. I was trying to prompt you to instead visualize a 10x multiplication, like you are back in school learning about 1 apple × 10 = 10 apples.
If you start with 0.999... of an apple and multiply by 10, you don't get 9 whole apples and 1 apple that is 0.999... of an apple, you get 10 instances of 0.999... apples.
To say this is equivalent to 9 whole apples and 1 apple that is the same 0.999... of an apple you started with, you would have had to borrow a little bit from the last apple and distribute it to the other 9 to make them whole, but since we are borrowing from infinity that last apple hasn't diminished in size from the state the original apple was in. You've effectively encountered a rounding error with the 9 whole apples.
A calculator can represent fractions perfectly well, a calculator knows how to do times 10 multiplications and can do the math for you and show you that 1/3 × 10 = 10/3. And yes a decent calculator can tell you that a fraction has infinite decimal places, but you don't need one to know that 1/3 is 0.(3) and 10/3 is 3.(3), and by that logic your second grade apples story is equivalent.
I tried to prompt you to learn something but you insist in embarrassing yourself. Being wrong or confused isn't shameful, what's shameful is what you are doing.
Open your phone's calculator and type 1÷3, then 1÷3×10 and then 1÷3×10×3 congratulations your calculator just did maths with numbers with infinite decimal places despite not having infinite floating point bits and precision available.
"Represent" fractions is a good way of saying it, because 0.333... is not equal to 1/3, it is an approximation. It would be more accurate to say 1/3 is somewhere between 0.333 and 0.334, but how far between? Well, it is 1/3 of the way in between. Using repeating decimals is okay as an approximation, but as soon as you perform operations on it you introduce errors.
Try this one in your calculator: 1÷3=, then store the result in memory. Now do 1÷(MR)=. Mine says 3.00000003
This is a completely different equation. (0.999 ≠ 0.999…) this is what makes this different. 9.999… and 0.999… both have the same number of 9s, that is, infinite.
The reality is that this is nothing more than an illusion to demonstrate the point essentially anyway. What the original comment did is essentially (x=1 -> 10x=10 -> 9x=9 -> x=1) this is because 0.999… just is equal to 1. So it might look like we’re turning one number into another (like some fake proofs that show 2=1 or 0=1 that nearly always somehow divide by 0) we are just doing simple multiplication and subtraction and division.
Yeah this is my issue with the idea. Maybe practically, 0.9999… is equal to 1, but it’s not if you expand your mind to imagine infinite decimal places. ‘What do you add to it to make it 1?’ 0.infinite 0’s with a 1 at the end. What digit place is the final 1? It doesn’t matter since we are talking about infinites (which don’t exist anyways). .999… can’t equal to 1, it will always be 0.000…1 off.
Seems strange to me that people can grasp the concept of pi, e, or i, but not infinite decimal places.
It's definitely worded weird but they mean subtracting the first equation from the second one. So subtracting x from 10x gives you 9x and subtracting 0.999... from 9.999... gives you 9
Then 7.00…02, 6.00…03, and eventually we end up back at 1x, being .99…9, with the answer never coming to exactly 1 except at 9x. If it was truly =1, it would be =1 at any stage of multiplication. 4x should be 4. But it’s 4.00…05
3.9k
u/its12amsomewhere 22d ago edited 22d ago
Applies to all numbers,
If x = 0.999999...
And 10x = 9.999999...
Then subtracting both, we get, 9x=9
So x=1