# I The Density Theorem ℚ and ℝ

#### kris kaczmarczyk

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

Science Advisor
Homework Helper
Summary: Real number are uncountable nonetheless they always have Rational neighbor
No rational number has a "neighbor".
No real number has a "neighbor.

• dextercioby

#### Mark44

Mentor
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.
kris kaczmarczyk said:
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."
kris kaczmarczyk said:
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.
kris kaczmarczyk said:
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 $q$:
$q=0.2345968$
$q=0.2345969$
$q=0.2345970$
$q=0.2345971$
$q=0.2345972$

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