Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals
Let $z_1$ be a non-zero wholly real number.
Let $z_2$ be a non-zero wholly imaginary number.
- $\Q$-Action: $\C$ is closed under multiplication, so $\Q \times \C \subset \C$.
- Distributive: $\times$ distributes over $+$.
- Associativity: $\times$ is associative.
- Multiplicative Identity: $1$ is the multiplicative identity in $\C$.
Let $a, b \in \Q$ such that:
- $a z_1 + b z_2 = 0$
- $c i = z_2$
where $i$ is the imaginary unit.
- $a z_1 + b c i = 0$
- $a z_1 = 0$
- $b c = 0$
Since $z_1$ and $c$ are both non-zero, $a$ and $b$ must be zero.
The result follows from the definition of linear independence.