Which one is correct
A. \(\sqrt{3}\) is irrational
B. Sum of two irrational numbers is always irrational
C. \(\sqrt{p}\), where \(p\) is prime, is irrational
D. Every composite number can be expressed as a product of primes
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A, C and D only
An irrational number is a number that cannot be written as \(\frac{p}{q}\) with integers \(p\) and \(q\ne0\). The square root of a whole number that is not a perfect square is irrational.
A. \(\sqrt{3}\) is irrational Correct. 3 is not a perfect square, so \(\sqrt{3}=1.732\ldots\) never ends or repeats and cannot be written as \(\frac{p}{q}\).
B. Sum of two irrational numbers is always irrational Incorrect. The sum can be rational: \(\sqrt{2}+(-\sqrt{2})=0\), and \((2+\sqrt{3})+(2-\sqrt{3})=4\). So it is not always irrational.
C. \(\sqrt{p}\), where \(p\) is prime, is irrational Correct. A prime has only the factors 1 and itself, so it is never a perfect square, and its square root is irrational (\(\sqrt{2}\), \(\sqrt{5}\), \(\sqrt{7}\), ...).
D. Every composite number can be expressed as a product of primes Correct. This is the Fundamental Theorem of Arithmetic: every composite number is a product of primes, in one way only apart from the order, e.g. \(84=2\times2\times3\times7\).