Is $1.0000...$ ( $1$ with infinite zeros) greater than $1.0$?

Question

Given that $0.3333...$ is greater than $0.3$ and similarly $0.777...$ is greater than $0.7$, does it follow that the sum of $0.33...$ and $0.77...$ is greater than sum of $0.3$ and $0.7$?

Answer 1

The sum of $0.\overline{3}$ and $0.\overline{7}$ is equal to $1.\overline{1}$ and not $1.\overline{0}$. (where $\overline{d}$ means the digit $d$ repeated an infinite number of times).

Answer 2

Yes, $0.33\ldots+0.77\ldots>0.3+0.7$

No, $1.000\ldots\ngtr1.0$

Answer 3

This is clearer to see when using fraction notation:

$$0.\dot3 = \frac{1}{3},\,0.\dot7 = \frac{7}{9},\,0.3 = \frac{3}{10}, 0.7 = \frac{7}{10}$$

So:

$$0.\dot3 + 0.\dot7 = \frac{1}{3} + \frac{7}{9} = \frac{10}{9} = 1.\dot{1} > 1\\0.3+ 0.7= \frac{3}{10} + \frac{7}{10} = \frac{10}{10} = 1$$

Not only is it greater, it doesn't actually equal $1.000\dots$, but rather $1.111\dots$!

In general, if $a > m$ and $b > n$, then $a+b > m + n$

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Note that I've used dots above the number to repeated decimal expansion, for example: $0.\dot1 = 0.111\dots$

Answer 4

 0.3 = 0.300000...   
       0.333333... 

 0.3 < 0.33333...

 0.7 = 0.700000...   
       0.777777...

 0.7 < 0.77777...

   0.3   
 + 0.7
  ----
   1.0

   0.33333...
 + 0.77777...
   ------------
   1.11111....

 1.0 < 1.11111...