Question

Is 2.2360679774998... a rational, irrational, natural, whole, integer or real number?

Answer 1

It's real.
It's NOT integer NOR whole (same as integer), NOR natural (positive integer).

Explanation:

It has decimal fractions, so it cannot be integer nor whole, nor natural.

Whether it is rational or irrational depends on what the ellipses at the end stand for.

Rational numbers in decimal notation are either finite or infinite periodical, while irrational numbers are infinite non-periodical.
It's not clear from the condition of this problem whether a fraction is periodical or not. My guess is that, since the period is not clearly marked (usually with parenthesis or a horizontal bar above it), it's more like non-periodical fraction, which makes an irrational number.

You can find it useful to study these issues at Unizor items "Algebra - Numbers".