Rational, algebraic, transcendental

Rationals. $\mathbb{Q}$: fractions $p/q$.

Algebraic. $\overline{\mathbb{Q}}$: roots of nonzero polynomials with rational coefficients. Examples: $\sqrt2$, $\sqrt[3]{2}$, $\sqrt2+\sqrt3$, $i$.

Transcendental. $\mathbb{C}\setminus\overline{\mathbb{Q}}$: not algebraic. Examples: $e$, $\pi$, $2^{\sqrt2}$.

Strict inclusion. $\mathbb{Q}\subsetneq\overline{\mathbb{Q}}\subsetneq\mathbb{C}$: $\sqrt2$ is algebraic but not rational; $\pi$ is transcendental.

Conclusion. The three levels form a strict hierarchy.