Lorena Beard

2022-07-13

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

Expert

This is clearer to see when using fraction notation:
$0.\stackrel{˙}{3}=\frac{1}{3},\phantom{\rule{thinmathspace}{0ex}}0.\stackrel{˙}{7}=\frac{7}{9},\phantom{\rule{thinmathspace}{0ex}}0.3=\frac{3}{10},0.7=\frac{7}{10}$
So:
$0.\stackrel{˙}{3}+0.\stackrel{˙}{7}=\frac{1}{3}+\frac{7}{9}=\frac{10}{9}=1.\stackrel{˙}{1}>1\phantom{\rule{0ex}{0ex}}0.3+0.7=\frac{3}{10}+\frac{7}{10}=\frac{10}{10}=1$
Not only is it greater, it doesn't actually equal $1.000\dots$, but rather $1.111\dots$
Note that I've used dots above the number to repeated decimal expansion, for example: $0.\stackrel{˙}{1}=0.111\dots$

Michelle Mendoza

Expert

Yes, $0.33\dots +0.77\dots >0.3+0.7$
No, $1.000\dots \ngtr 1.0$

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