๐ง Disproving Irrational Sum โ Generalized Polarity Trick
A common misconception is that the sum of two irrational numbers is always irrational. This module formalizes a systematic counterexample techniqueโthe Generalized Polarity Trickโwhich constructs irrational pairs whose sum is rational via algebraic cancellation.
๐ Core Insight
If two irrational numbers contain equal and opposite irrational components, their sum cancels the irrationality and yields a rational result.
This works regardless of sign, magnitude, or algebraic complexityโso long as the irrational parts cancel and the remaining components are rational.
๐งฉ Formal Construction
Let:
- $x = r_1 + j$
- $y = r_2 - j$
Where:
- $r_1, r_2 \in \mathbb{Q}$ (rational)
- $j \in \mathbb{R} \setminus \mathbb{Q}$ (irrational)
Then:
- $x + y = (r_1 + r_2) + (j - j) = r_1 + r_2$ โ rational
โ Rational Closure Justification
- Rational numbers are closed under addition: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \in \mathbb{Q}$
- Integer operations are closed under addition and multiplication
- Therefore, $r_1 + r_2 \in \mathbb{Q}$
๐งช Audit Examples
| $x$ | $y$ | Sum | Rational? | Notes |
|---|---|---|---|---|
| $\sqrt{2}$ | $2 - \sqrt{2}$ | $2$ | โ | Positive irrationals |
| $-1 + \pi$ | $3 - \pi$ | $2$ | โ | Mixed signs |
| $5 + \sqrt{7}$ | $-2 - \sqrt{7}$ | $3$ | โ | Negative irrational cancel |
| $\sqrt{2}$ | $\pi$ | $\sqrt{2} + \pi$ | โ | No cancellation |
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