You can now sort single numbers into rational and irrational. What happens when you add or multiply them? Some answers are always the same, and some depend on the numbers.
Start with what is certain. Rational numbers are closed under the four operations: add, subtract, multiply or divide (not by zero) two fractions and you get another fraction. For example, $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$ and $\frac{2}{3} \cdot \frac{9}{4} = \frac{3}{2}$.
Por qué funciona
| Combination | Result | Example |
|---|---|---|
| rational $\pm$ rational | always rational | $\frac{1}{2} - 3 = -\frac{5}{2}$ |
| rational $\pm$ irrational | always irrational | $4 + \sqrt{3}$ |
| nonzero rational $\times$ irrational | always irrational | $5\pi$ |
| irrational $\pm$ irrational | it depends | $\sqrt{2} + \sqrt{2} = 2\sqrt{2}$, but $\sqrt{2} + (-\sqrt{2}) = 0$ |
| irrational $\times$ irrational | it depends | $\sqrt{2}\cdot\sqrt{3} = \sqrt{6}$, but $\sqrt{2}\cdot\sqrt{8} = 4$ |
Worked examples
Ejemplo resuelto: Rational plus irrational
- $\frac{3}{4}$ is rational. $7$ is not a perfect square, so $\sqrt{7}$ is irrational.
- Rational + irrational is always irrational.
Respuesta: Irrational.
Ejemplo resuelto: Two irrationals whose product is rational
- Each factor is irrational, so no rule decides it. Compute.
- $\sqrt{3}\cdot\sqrt{12} = \sqrt{36}$.
- $\sqrt{36} = 6$, a rational number.
Respuesta: Rational.
Ejemplo resuelto: Irrational parts that cancel
- Group like parts: $(5 + 1) + (\sqrt{2} - \sqrt{2})$.
- $\sqrt{2} - \sqrt{2} = 0$, so the sum is $6$.
Respuesta: Rational, even though both numbers being added are irrational.
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