Division of Two Complex Numbers

The Meaning of Division

Division of complex numbers is defined in a way that keeps the result a complex number. For any two complex numbers $z_1$ and $z_2$, with $z_2 \neq 0$, the quotient $\frac{z_1}{z_2}$ is defined as:

That is, dividing $z_1$ by $z_2$ means multiplying $z_1$ by the multiplicative inverse of $z_2$. This definition is natural — it mirrors the real-number idea that division is multiplication by the reciprocal.

The Method: Rationalising the Denominator

In practice, we never directly compute $\frac{1}{z_2}$ as a separate step. Instead, we use a technique that eliminates the imaginary part from the denominator — the same rationalisation trick you use for surds.

Given $z_1 = a + ib$ and $z_2 = c + id$ (with $c+id \neq 0$), we multiply numerator and denominator by the complex conjugate of the denominator:

The denominator becomes a real number: $(c+id)(c-id) = c^2 + d^2$. So:

Now expand the numerator, separate real and imaginary parts, and you have the quotient in standard form $x + iy$.

Worked Example from the Textbook

Let $z_1 = 6 + 3i$ and $z_2 = 2 - i$. Find $\frac{z_1}{z_2}$.

Step 1: Write the quotient as multiplication by the inverse:

Step 2: Find $\frac{1}{2-i}$ by rationalising:

Step 3: Multiply:

Step 4: Expand the product:

Since $i^2 = -1$:

Step 5: Divide by 5:

The General Formula for the Quotient

From the method above, we can write a direct formula. For $z_1 = a+ib$ and $z_2 = c+id$ ($c+id \neq 0$):

This formula is worth knowing, but in practice it's safer to remember the rationalisation procedure — it's less error-prone.

Key Properties of Division

The textbook does not list separate numbered properties for division alone — division inherits its properties from multiplication by the inverse. However, two results are central:

Property 1 (Uniqueness of the quotient): For given $z_1$ and $z_2 \neq 0$, the quotient $\frac{z_1}{z_2}$ is unique. This follows because the multiplicative inverse of $z_2$ is unique, and multiplication by a fixed complex number is a well-defined operation.

Property 2 (Division by a real number): If $z_2 = k$ is a real number ($k \neq 0$), then:

This is just the special case $d=0$ in the general formula — the denominator $c^2+d^2$ becomes $k^2$, and the numerator simplifies.

›Proof

Proof of Property 2:

Let $z_1 = a+ib$ and $z_2 = k$ (real, $k \neq 0$). Then:

$\frac{a+ib}{k} = (a+ib) \cdot \frac{1}{k} = \frac{a}{k} + i\frac{b}{k}$

This is immediate from the definition of division as multiplication by the reciprocal, and the fact that the reciprocal of a real number $k$ is $\frac{1}{k}$ (a real number).

Common Pitfall to Avoid

A frequent mistake is to treat division of complex numbers like division of real numbers — trying to "separate" the denominator term-by-term. For example:

$\frac{a+ib}{c+id} \neq \frac{a}{c} + i\frac{b}{d}$ …