1 = 0.999...

[Agent Smith]

So we have the problem \(\displaystyle 2 \div 2\) and the answer is \(\displaystyle 1\) (A), but the answer is also \(\displaystyle 0.999... = 0.\overline 9\) (B).
Comments ... please

 

[Dr.Peterson]

View attachment 38409

So we have the problem \(\displaystyle 2 \div 2\) and the answer is \(\displaystyle 1\) (A), but the answer is also \(\displaystyle 0.999... = 0.\overline 9\) (B).
Comments ... please

You've just demonstrated that 0.999... = 1. There's nothing wrong with that. It's a well-known fact.

 

[Agent Smith]

You've just demonstrated that 0.999... = 1. There's nothing wrong with that. It's a well-known fact.

Si, I was tinkering around with the Hindu long division algorithm after having visited a page on \(\displaystyle 1 = 0.\overline 9\).

Confiteor, I don't think I understand the Hindu long division algorithm.

Second, in the case of B, the division is nonterminating as the remainder \(\displaystyle 2\) recurs. It's like \(\displaystyle \frac{1}{3} = 0.\overline 3\).


The above alternative seems similar to what's in the OP. Just as \(\displaystyle 0.8 + 0.2 = 1\), \(\displaystyle 0.9 + 0.09 + 0.009 + \dots = 0.\overline 9 = 1\)