How real no. Is the combination of rational and irrational numbers?
Answers
Rational numbers are those which can be written as the quotient ab where a and b are integer with b≠0 (for b=1 you have the integer) and Irrational numbers can be defined as the complement of rational ones in the set of real numbers. There are no other real so the answer to your question is NOT
The answer (to the question in the title) is that yes, that covers all of them, by definition.
We take the real numbers R and the rational numbers Q. We note that Q is a proper subset of R, so we take all the real numbers that aren't rational and call them irrational numbers, i.e. we define I=R∖Q.
There are other splits possible. In particular, we can take all the real numbers that are roots of polynomials with integer coefficients, and call them the algebraic numbers (sometimes notated as A, although that also sometimes includes the complex algebraic numbers). So we can also consider the set of what's left, and that's what we call the transcendental numbers R∖A. Since any rational number pq is the solution to qx−p=0, we can soon confirm that Q⊂A, i.e. the rational numbers form a proper subset of the algebraic numbers, and conversely the transcendental numbers form a proper subset of the irrational numbers.