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

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The discussion centers on the concept of how the real numbers (ℝ) are said to "fill" the rational numbers (ℚ) in terms of absolute value. It clarifies that this notion relates to the density of ℚ in ℝ, meaning that between any two real numbers, there exists a rational number. Additionally, it explains that every metric space, including the rationals with the absolute value, has a completion, which is the real numbers. This completion is what allows ℝ to encompass all the limits of sequences of rational numbers. Understanding this relationship is crucial for grasping the structure of real analysis.
evinda
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Hi! (Cool)

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