Which one of the following is/are correct?(A) Every irrational number…

The correct answer is Option (3) → (A), (C) and (D) only

The real numbers are made up of two groups: the rational numbers, which can be written as \(\frac{p}{q}\) with integers \(p, q\) and \(q\ne0\), and the irrational numbers, which cannot (such as \(\sqrt{2}\) and \(\pi\)).

(A) Every irrational number is a real number. Correct. The irrational numbers are one of the two groups that make up the real numbers, so each of them is real.

(B) Every real number is an irrational number. Incorrect. Real numbers such as 2 or \(\frac{1}{2}\) are rational, not irrational; only some real numbers are irrational.

(C) The sum or difference of a rational number and an irrational number is an irrational number. Correct. If \(r+x\) were a rational number \(s\), then \(x=s-r\) would be rational too, which is impossible for an irrational \(x\). For example, \(2+\sqrt{3}\) is irrational.

(D) The product or quotient of a non-zero rational number with an irrational number is an irrational number. Correct. If \(r\times x\) were a rational number \(s\), then \(x=\frac{s}{r}\) would be rational. The rational number must be non-zero, because \(0\times\sqrt{2}=0\) is rational.