1. **Stating the problem:** We want to understand why the number $\pi$ is irrational. 2. **Definition:** A number is irrational if it cannot be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. 3. **Key fact:** $\pi$ is the ratio of a circle's circumference to its diameter. 4. **Historical proof:** Johann Lambert proved in 1768 that $\pi$ is irrational using continued fractions. 5. **Intuition:** If $\pi$ were rational, the decimal expansion would terminate or repeat, but $\pi$'s decimal expansion is infinite and non-repeating. 6. **Mathematical approach:** Lambert showed that the tangent function $\tan(x)$ is irrational for rational multiples of $\pi$, implying $\pi$ itself is irrational. 7. **Conclusion:** $\pi$ cannot be written as a simple fraction, so it is irrational. This is a fundamental result in mathematics with deep implications in geometry and analysis.