The Natural Numbers (a.k.a. Counting numbers) are 1, 2, 3, 4, ...
The Whole Numbers include 0, thus    0, 1, 2, 3, 4, ....
The Integers include the negatives, thus  ... -3, -2, -1, 0, 1, 2, 3, ...
The Rational Numbers are those obtained by dividing any integer by any other,
     e.g.,  5/1,  1/16, 3/7
   these may be integers, finite decimal fractions, repeating decimal fractions;
      all are Algebraic Numbers.
The IR-Rational Numbers are all numbers which are NOT Rational,
    e.g.  sqrt(2), e, pi,
   these will be NON-repeating decimal fractions;
      some are Algebraic Numbers, some are Transcendental Numbers.
The Algebraic Numbers are those which are the roots of a
  polynomial (of any degree) with rational coefficients,
     e.g.,  5/1,  1/16, 3/7, sqrt(2)
   these may be integers, finite decimal fractions,
       repeating decimal fractions, NON-repeating decimal fractions;
      some are Rational Numbers, some are IR-Rational Numbers.
The Transcendental Numbers are those which are NOT the roots of a polynomial,
     e.g.  e, pi,
   these will be NON-repeating decimal fractions;
      all are IR-Rational Numbers.

Between every two Rational, IR-Rational, Algebraic, or Transcendental numbers,
  there are an infinite quantity of all of the others.
All Integers are Rational, but not vice-versa.
All Rational Numbers are Algebraic, but not vice-versa.
All Transcendental Numbers are Irrational, but not vice-versa.
ALL OF THESE ARE REAL NUMBERS.
Real Numbers may be subdivided --
      -- into  the Algebraics and the Transcendentals.
          The Algebraics may be subdivided into
             the Rationals and SOME of the IR-Rationals.
 -- or into  the IR-Rationals and the Rationals.
          The IR-Rationals may be subdivided into
             the Transcendentals and SOME of the Algebraics.
        The Rationals may be subdivided into
            the Integers and Fractionals.
            The Integers may be subdivided into
                the Whole Numbers and the Negative Numbers.

Visually these relationships are ...
        ______Algebraic_______  __Transcendental__
       /                      \/                  \
      |  5/1        |         |                   |
      |  1/16       |         |   e, pi           |
      |  3/7        | Sqrt(2) |                   |
      |             |         |                   |
      |             |                             |
       \__Rational__/\____IR-Rational_____________/


Trigonometric functions may be in any of these areas
The Imaginary numbers are quite a story for another time, place, dimension.

Relatively UNrelated to the above is the discussion of ...
  Ordinal  Numbers are those used to specify order: 1-st, 2-nd, 3-rd...
  Cardinal Numbers are those used to specify quantity.

  Time  Flies Like an Arrow;
  Fruit Flies Like Bananas.