MHB Why do the reals fill the rationals with the absolute value?

[evinda]

Hi! (Cool)

Could you explain me why $\mathbb{R}$ fills $\mathbb{Q}$ as for the norm absolute value? (Worried)

 

[mathbalarka]

What do you mean by $\Bbb R$ "fills" $\Bbb Q$? Do you mean to say that $\Bbb Q$ is dense in $\Bbb R$? Your question is void if you can't define the nonmathematical term "fill".

 

[Fallen Angel]

Hi,

I think you are asking why the reals are the complection of the metric space consisting on the rationals with the absolute value.

In essence, one can prove that every metric space has a complete space containing it, and we just name this complection the 'real numbers'.

Hope this can help you.