Family tree of real numbers

Start with the set of Natural numbers = N.Combine these with negative natural numbers and you get the set of Integers = Z.

Combine these with ratiOS of two integers, the second of which is positive, and you get the set of Rational numbers = Q.

Start afresh with numbers which are not rational, nor the roots of finite polynomial equations. This is the set of transcendental numbers.

Combine these with the non-rational roots of finite polynomial equations and you have the set of Irrational Numbers.

Combine the rational and irrational numbers and you have the set of Real numbers, R.