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Is the set of rational numbers closed under the operation "take the reciprocal"?

Class 8Rational NumbersRational Numbers
  • AYes — every rational number has a reciprocal
  • BNo — 0 is a rational number but has no reciprocal
  • CNo — the reciprocal of a fraction is never rational
  • DYes — but only for the positive rationals

Step-by-Step Solution

Closure demands that the operation work for EVERY member of the set. Taking reciprocals works for every rational except 0, so the rationals are not closed under it (the NON-ZERO rationals are). Reciprocals of fractions are certainly rational — gives — so the third option is simply false.
Answer: B) No — 0 is a rational number but has no reciprocal

Key idea: Identities and inverses

**Additive identity**: 0. For any rational a: a + 0 = a.

**Multiplicative identity**: 1. For any rational a: a × 1 = a.

**Additive inverse** (negation): For a, the additive inverse is −a. Their sum is 0. So a + (−a) = 0.

**Multiplicative inverse** (reciprocal): For non-zero a = p/q, the multiplicative inverse is q/p. Their product is 1. **0 has no multiplicative inverse** (1/0 is undefined).

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