The Density Theorem ℚ and ℝ

[kris kaczmarczyk]

TL;DR Summary

Real number are uncountable nonetheless they always have Rational neighbor

count(ℝ) > count(ℚ) ; count(ℚ) == count(ℕ)

But still in-between any members of ℝ, we are quarantine to find element of ℚ

Can someone help me understand: were are these members of ℝ we cannot account for?

For reference: https://en.wikipedia.org/wiki/Rational_number

"The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions. "

 

[jbriggs444]

Summary: Real number are uncountable nonetheless they always have Rational neighbor

No rational number has a "neighbor".
No real number has a "neighbor.

 

[Mark44]

Summary: Real number are uncountable nonetheless they always have Rational neighbor

A better way to say this is that the rationals are dense in the real number line. That is, between any two real numbers, there is always some rational number.

count(ℝ) > count(ℚ) ; count(ℚ) == count(ℕ)

But still in-between any members of ℝ, we are quarantine to find element of ℚ

See above. Also, the word you want is "guaranteed," which is quite different from "quarantined."

Can someone help me understand: were are these members of ℝ we cannot account for?

What do you mean? No real numbers are missing. We can account for all of them.

For reference: https://en.wikipedia.org/wiki/Rational_number

"The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions. "

 

[SSequence]

If it helps, one (relatively) simple way to remember is it that given two real numbers $r_1,r_2$ (where $r_1<r_2$) we can always systematically extract a rational number $q$ so that $r_1<q<r_2$. I find this a helpful way to remember this result.

The precise description will probably get a bit complicated due to boundary cases, but here is the general idea via a specific example.
$r_1=0.23459678...$
$r_1=0.23459732...$
The dots indicate that either we don't know (or don't care) about the digits after that.

So to generate $q$ first we copy the initial part where the digits of $r_1$ and $r_2$ are equal. So the first five digits of $q$ (after the decimal point) will be the same as $r_1$ and $r_2$.

Now if you notice the next two digits in $r_1$, they are $6$ and $7$, while in $r_2$ they are $7$ and $3$.
So we could use any of the following as $q$:
$q=0.2345968$
$q=0.2345969$
$q=0.2345970$
$q=0.2345971$
$q=0.2345972$