Question

If a set of six numbers that include both rational and irrational numbers is graphed on a number line, what is the fewest number of distinct points that need to be graphed?

Answer 1

If there are no duplications among the six numbers, then they sit at
six different points on the number line.

Irrational numbers are on the same number line as rational ones.
The only difference is that if somebody comes along, points at one of them,
and asks you to tell him its EXACT location on the line, you can answer him
with digits and a fraction bar if it's a rational one, but not if it's an irrational one.

For example:

Here are some rational numbers. You can describe any of these EXACTLY
with digits and/or a fraction bar:

--                    2
--                 1/2
--         (any whole number) divided by (any other whole number)
                             (this is the definition of a rational number)
--                 19
--         (any number you can write with digits) raised to
                                   (any positive whole-number power)
--               387
--                  4.0001
--       (zero or any integer) plus (zero or any repeating decimal)
--                13.14159 26535 89792
--        (any whole number) + (any decimal that ends, no matter how long it is)
             (this doesn't mean that a never-ending decimal isn't rational; it only
               means that a decimal that ends IS rational.
               Having an end is enough to guarantee that a decimal is rational,
               but it's not necessary in order for the decimal to be rational.
               There are a huge number of decimals that are rational but never end.
               Like the decimal forms of  1/3,  1/6,  1/7,  1/9,  1/11, etc.)
-->    the negative of anything on this list    

Here are some irrational numbers.  Using only digits, fraction bar, and
decimal point, you can describe any of these as close as anybody wants
to know it, but you can never write EXACTLY what it is:

--               pi
--             square root of √2
--             any multiple of √2 
--             any fraction of √2
--               e
--             almost any logarithm