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

Question

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

Nicolas Calhoun · Accepted Answer

This is clearer to see when using fraction notation:  0.            3      ˙        =      1    3    ,    0.            7      ˙        =      7    9    ,    0.3  =      3    10    ,  0.7  =      7    10  So:  0.            3      ˙        +  0.            7      ˙        =      1    3    +      7    9    =      10    9    =  1.            1      ˙        &amp;gt;  1    0.3  +  0.7  =      3    10    +      7    10    =      10    10    =  1Not only is it greater, it doesn&#039;t actually equal   1.000  …, but rather   1.111  …Note that I&#039;ve used dots above the number to repeated decimal expansion, for example:   0.            1      ˙        =  0.111  …

Michelle Mendoza · Accepted Answer

Yes,   0.33  …  +  0.77  …  &amp;gt;  0.3  +  0.7No,   1.000  …  ≯  1.0