Rational and Irrational Numbers

We are all familiar with rational and irrational numbers. For example 5/4=1.25 is rational,
the square root of 2=1.41421356237309504880168872420... is irrational, 11/4=2.75 is rational, PI=3.1415926535... is irrational.

So here is my question for which I have not worked on a proof one way or another.

Are there ever two irrational numbers where one comes right after the other or is there always at least one rational number in between?

Similarly, are there ever two rational numbers right together or is there one or more irrational numbers in between?

Basically. If we could draw an actual number line with all the numbers, what would it look like?
-----------------------------------------------------------------
The answer looks like NO. Thanks with a little help from my friends...

Is it always possible to find a rational number between any two irrational numbers.
Is it always possible to find an irrational number between anytwo irrational numbers.


The desired rational can be constructed by "truncating" the decimal expansion of the larger irrational at a certain point and then considering the average of the two rational numbers.

or for the other,

if the average of the two irrational numbers is rational, an irrational number satisfying the requirement is the average of one of the irrationals and the rational average of the two irrationals. The desired irrational between rational numbers can be found by adding a sufficiently small irrational number to the smaller rational number.