By Any introduction to complex numbers and their operations follows a common pattern: the formulas are given without building any intuition. If you have ever wondered about why $j=\sqrt{-1}$, you can read about the origin of complex numbers. In this article, I will explain the intuitive reason behind why
- a product of two complex numbers multiplies their magnitudes and adds their angles, and
- a division of two complex numbers divides their magnitudes and subtracts their angles.
Let us start with the multiplication. The division scenario can be analogously derived with inverse operations.
Multiplication of Complex Numbers
The intuition behind multiplying two complex numbers lies in following the simple arithmetic of real number multiplications. Consider a number $z$ in a complex plane, shown as a blue dot in the figure below.
What happens when it is multiplied with another complex number $2+j3$? This operation can be written as
\[
(2+j3)z = 2z + j3z
\]
This is a multistep process.
- The first term $2z$ above is drawn in the figure by scaling $z$ by $2$.
- The second term involves the imaginary operator $j$. Since multiplication with $j$ rotates a complex number by $90^\circ$, the operation $jz$ is nothing but $z$ rotated by $90^\circ$.
- Next, this $jz$ is scaled by $3$ to produce $j3z$. This is equivalent to the line at the tip of $2z$.
- Finally, the two terms above are combined using addition rule (real + real and imaginary + imaginary) to generate the result $z(2+j3)$.
Now consider the magnitude and the angle of the output.
Magnitude
In the figure above, the shaded part is a right angled triangle. Therefore, using Pythagoras theorem, the magnitude of the result is seen as
\[
\text{Magnitude}^2 = |2z|^2 + |j3z|^2 = 4|z|^2 + 9|z|^2 = 13|z|^2 = |\text{Magnitude(2+j3)}|^2 \cdot |z|^2
\]
It is clear that the magnitude of the product of two complex numbers is simply the product of their individual magnitudes. This means the length of the first complex number is stretched or contracted based on the lengths of the second complex number.
Angle
In the figure above, the tangent of multiplier $2+j3$ is given by $3/2$. Observe the shaded triangle where the angle is still given by the ratio $3z/2z=3/2$. Since these are adjacent angles, the output is expressed as
\[
\text{Angle} = \text{Angle}(z)+\text{Angle}(2+j3)
\]
Therefore, the angle of the product of two complex numbers is simply the sum of their individual angles. In essence, we are combining their rotations on the complex plane. The angle is also known as the argument of the complex number.
Mathematical Perspective
This is how complex numbers are taught in every school and university. Consider two complex numbers $z_1 = r_1 e^{j\theta_1}$ and $z_2 = r_2 e^{j\theta_2}$. This is known as the polar form (see the intuition behind the polar form of complex numbers for details). Now we have
\[
z_1 z_2 = r_1 r_2 e^{j(\theta_1 + \theta_2)}
\]
This shows how the magnitudes multiply $(r_1 \times r_2)$ and the angles add $(\theta_1 + \theta_2)$ to form the result.
This elegant combination of scaling and rotation makes complex number multiplication particularly powerful in areas like digital signal processing, physics, quantum mechanics, computer graphics and many other fields.
Division of Complex Numbers
The intuition behind dividing two complex numbers is analogous to multiplications with inverse operations for their magnitudes and angles in the complex plane. Mathematically, using $z_1 = r_1 e^{j\theta_1}$ and $z_2 = r_2 e^{j\theta_2}$,
\[
\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{j(\theta_1 – \theta_2)}
\]
This implies the following for magnitude and argument of the result.
Magnitude
The magnitude of the result is the ratio of the magnitudes of the two complex numbers. As seen above, if $z_1$ has a magnitude $r_1$ and $z_2$ has a magnitude $r_2$, the resultant magnitude is $r_1 / r_2$. This effectively shrinks or expands the first complex number by the size of the second.
Angle
The angle of the result is the difference between the angles of the two complex numbers. If $z_1$ has an angle $\theta_1$ and $z_2$ has an angle $\theta_2$, the resultant angle is $\theta_1 – \theta_2$. This can be visualized as rotation of $z_1$ in a direction opposite to that of $z_2$.
Essentially, their division is about removing the effects of $z_2$ on $z_1$.
