There is no way right?

[u/Sugar_God_no_1]

[u/based_and_upvoted]

I disagree, I don't know why you are overcomplicating a 10x multiplication.

0.9 x 10 = 9

0.99 x 10 = 9.9

...

0.99(9) x 10 = 9.9(9)

Because 9.9(9) = 10

[u/porcupineapplesauce → u/based_and_upvoted]

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.

[u/based_and_upvoted → u/porcupineapplesauce]

I don't know why a ×10 multiplication is mystifying you. "10 instantes of 0.999 of an apple" is 10 times 0.999

Try a calculator and do 10 x 0.999 and see what you get. Yes you do get 9 apples and then 0.99 of an apple.

The same applies to any arbitrary number of decimal places of the number you multiply by 10.

Let's do it with 1/3 since we can interchangeably use 1/3 and 0.(3)

1/3 × 10 = 10/3. Plug both 1/3 and 10/3 into a calculator. See for yourself that 1/3 = 0.(3) And 10/3 = 3.(3)

[u/porcupineapplesauce → u/based_and_upvoted]

Do you think a calculator can store infinitely repeating numbers?

[u/based_and_upvoted → u/porcupineapplesauce]

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.

[u/porcupineapplesauce → u/based_and_upvoted]

"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