Question

Which of the following statements are incorrect?

• A. The sum, product, division of two rational numbers is always a rational number
• B. The sum, difference, product, division of two irrational number is an irrational number
• C. Irrational numbers can be represented on the number line
• D. The sum or difference of a rational number and an irrational number is irrational

Answer 1

Answer: (A), (C), (D)

The correct options are
A

The sum, product, division of two rational numbers is always a rational number

C

Irrational numbers can be represented on the number line

D

The sum or difference of a rational number and an irrational number is irrational

Multiplication of two rational numbers is always a rational number.
This is because every rational number can be expressed as a $\frac{p}{q}$ form. Therefore, multiplying two rational numbers would result in multiplying two numbers in their $\frac{p}{q}$ forms, with the denominators not equal to 0, in each case. This would result in another number which is also in its $\frac{p}{q}$ form, with both the numerator and denominator being integers.

The sum of two irrational numbers needs not always be irrational.
For example, consider $2+\sqrt{3}$ and $2-\sqrt{3}$, both of which are irrational numbers. Adding these gives us 4, which is a rational number.

Irrational numbers are real numbers too. Since we know that the number line constitutes all the real numbers, irrational numbers form part of it too.

The sum or difference of a rational and an irrational number is always irrational.