Also read ‘Cardinality of the Reals’

Do Negative Rationals Change the Cardinality?

Short answer: No. Adding negative rationals keeps the set countably infinite, the same cardinality as the integers \mathbb{Z}.

1) Positive Rationals vs. Integers

The positive rationals \mathbb{Q}^+ can be listed (e.g., via Cantor’s diagonal), giving a bijection with \mathbb{N}:

    \[ |\mathbb{Q}^+| = |\mathbb{N}|. \]

2) Adding Negative Rationals

The nonzero rationals split as \mathbb{Q}\setminus\{0\}=\mathbb{Q}^+\cup\mathbb{Q}^-. The map q\mapsto -q is a bijection \mathbb{Q}^+\to\mathbb{Q}^-, so

    \[ |\mathbb{Q}^-| = |\mathbb{Q}^+| = |\mathbb{N}|. \]

The union of two countable sets is countable, hence

    \[ |\mathbb{Q}^+ \cup \mathbb{Q}^-| = |\mathbb{N}|. \]

Adding the single element 0 does not change cardinality.

3) Conclusion

    \[ |\mathbb{Q}| = |\mathbb{Q}^+ \cup \mathbb{Q}^- \cup \{0\}| = |\mathbb{N}|. \]

✅ Including negative rationals does not increase the cardinality; the rationals remain countably infinite, like the integers.