Wholly Real Number and Wholly Imaginary Number are Linearly Independent over the Rationals

Theorem

Let $z_1$ be a non-zero wholly real number.

Let $z_2$ be a non-zero wholly imaginary number.

Then, $z_1$ and $z_2$ are linearly independent over the rational numbers $\Q$, where the group is the complex numbers $\C$.

Proof

From Rational Numbers form Subfield of Complex Numbers, the unitary module $\struct {\C, +, \times}_\Q$ over $\Q$ satisfies the unitary module axioms:

• $\Q$-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:

$a z_1 + b z_2 = 0$

By the definition of wholly imaginary, there is a real number $c$ such that:

$c i = z_2$

where $i$ is the imaginary unit.

Therefore:

$a z_1 + b c i = 0$

Equating real parts and imaginary parts:

$a z_1 = 0$

$b c = 0$

Since $z_1$ and $c$ are both non-zero, $a$ and $b$ must be zero.

The result follows from the definition of linear independence.

$\blacksquare$