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Assertion (A): Every integer is a rational number. Reason (R): Rational numbers are closed under addition. Which is correct?

Class 8Rational NumbersRational Numbers
  • AA is false but R is true
  • BBoth A and R are true but R is not the correct explanation of A
  • CBoth A and R are true and R is the correct explanation of A
  • DA is true but R is false

Step-by-Step Solution

A is true: any integer can be written as , a ratio with non-zero denominator. R is also true — adding two rationals always gives a rational. But closure under addition says nothing about WHY an integer counts as rational, so R does not explain A. The trap is ticking "R explains A" reflexively whenever both statements happen to be true.
Answer: B) Both A and R are true but R is not the correct explanation of A

Key idea: Closure of rationals

Rationals are **closed under addition, subtraction, and multiplication** — the result is always rational.

But **division is closed only if you exclude division by 0**: for any rationals a and b ≠ 0, a ÷ b is rational.

This is more than integers (which weren't closed under division at all) and more than fractions (which sometimes had subtraction lead outside the fractions). Rationals form the smallest 'nice' number system that supports all four basic operations (with the 0-divisor exception).

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