What They Tell You About Complex Numbers
Open any mathematics textbook and you’ll find complex numbers introduced something like this: “A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit satisfying $i^2 = -1$.”
This definition is correct, but it misses the forest for the trees. It makes complex numbers sound like an abstract mathematical curiosity, something invented to solve the equation $x^2 + 1 = 0$. Students dutifully memorize the arithmetic rules, plot points on the complex plane, and wonder: When will I ever use this?
The Hidden Truth: Complex Numbers Represent Physical Reality
Here’s what they should tell you first: complex numbers are a natural way to represent any physical quantity that has both magnitude and direction.
Think about it. In the real world, many things aren’t just “how much” but also “which way”:
- A velocity isn’t just speed: it has direction
- A force isn’t just strength: it has direction
- An electric current isn’t just flow rate: it has phase
- A rotation isn’t just angle: it has orientation
The notation $a + bi$ isn’t some arbitrary algebraic trick. It’s a coordinate system. The real part $a$ and imaginary part $b$ are simply two perpendicular components, like $x$ and $y$ coordinates on a map.
Example 1: Projectile Motion - Throwing a Ball
Let’s start with something everyone understands: throwing a ball.
When you throw a ball, its velocity has two independent components:
- Horizontal velocity (how fast it moves forward)
- Vertical velocity (how fast it rises or falls)
We can represent this as a complex number:
$$v = v_x + iv_y$$For example, if you throw a ball with 6 m/s horizontally and 8 m/s upward:
$$v = 6 + 8i \text{ m/s}$$The magnitude (speed) is:
$$|v| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ m/s}$$The direction (angle above horizontal) is:
$$\theta = \arctan\left(\frac{8}{6}\right) \approx 53.1°$$As the ball flies through the air, gravity only affects the imaginary (vertical) component. At time $t$:
$$v(t) = 6 + (8 - 9.8t)i$$Notice how naturally this separates the two independent motions! The real part stays constant (no air resistance), while the imaginary part decreases linearly.
Let’s see a specific moment: At $t = 0.5$ seconds:
$$v(0.5) = 6 + (8 - 9.8 \times 0.5)i = 6 + 3.1i \text{ m/s}$$The speed at this moment is:
$$|v(0.5)| = \sqrt{6^2 + 3.1^2} = \sqrt{36 + 9.61} = \sqrt{45.61} \approx 6.75 \text{ m/s}$$The direction is:
$$\theta = \arctan\left(\frac{3.1}{6}\right) \approx 27.3°$$The ball has slowed down and is now traveling at a shallower angle.
Converting from Angle to Components
What if we’re told a ball is thrown at 10 m/s at a 60° angle above horizontal? We can use Euler’s formula to find the complex representation:
$v = |v|e^{i\theta} = |v|(\cos(\theta) + i\sin(\theta))$
$v = 10e^{i \cdot 60°} = 10(\cos(60°) + i\sin(60°))$
$v = 10\left(0.5 + i \cdot \frac{\sqrt{3}}{2}\right)$
$v = 5 + 5\sqrt{3}i \approx 5 + 8.66i \text{ m/s}$
This elegant notation shows that complex numbers in polar form ($re^{i\theta}$) naturally encode magnitude and direction! Now we have the horizontal component (5 m/s) and vertical component (8.66 m/s) that we can use to track the motion over time.
Example 2: Spinning a Wheel
Imagine a point on a spinning wheel. As it rotates, the point traces a circle. We can describe its position using a complex number:
$$z(t) = r \cdot e^{i\omega t} = r(\cos(\omega t) + i\sin(\omega t))$$where:
- $r$ is the distance from the center (magnitude)
- $\omega$ is the angular velocity (how fast it spins)
- $t$ is time
This single equation captures both:
- Where the point is (magnitude $r$)
- Which direction it’s pointing (angle $\omega t$)
Compare this to tracking $x$ and $y$ separately:
$$x(t) = r\cos(\omega t)$$$$y(t) = r\sin(\omega t)$$The complex form is more compact and reveals the rotational nature immediately. Multiplying by $e^{i\theta}$ literally means “rotate by angle $\theta$.”
Example 3: AC Electrical Circuits
In alternating current (AC) circuits, voltage and current oscillate. But they don’t always oscillate in sync—there’s often a phase difference.
An AC voltage might be:
$$V(t) = V_0 \cos(\omega t)$$While the current lags behind:
$$I(t) = I_0 \cos(\omega t - \phi)$$Engineers represent this using complex numbers:
$$\tilde{V} = V_0 e^{i\omega t}$$$$\tilde{I} = I_0 e^{i(\omega t - \phi)}$$The phase lag $\phi$ is now just a rotation in the complex plane. This makes calculations with capacitors, inductors, and resistors incredibly elegant—addition and multiplication of complex numbers automatically handle both magnitude and phase.
Example 4: Sound Waves and Interference
When two sound waves meet, they interfere. The resulting amplitude depends on both their individual amplitudes and their relative phase (timing).
Wave 1: $A_1 e^{i\omega t}$
Wave 2: $A_2 e^{i(\omega t + \phi)}$
The combined wave is simply:
$$A_{total} = A_1 e^{i\omega t} + A_2 e^{i(\omega t + \phi)}$$If they’re in phase ($\phi = 0$), they add constructively.
If they’re out of phase ($\phi = \pi$), they cancel.
Without complex numbers, you’d need trigonometric identities and careful tracking of phases. With complex numbers, it’s just addition.
Why Does This Work So Well?
Complex numbers work because perpendicular components don’t interfere with each other.
When you walk north, that doesn’t affect your east-west position. When a ball moves horizontally, gravity doesn’t change that horizontal velocity. When a wave oscillates, its horizontal and vertical components are independent.
The algebraic properties of $i$ (especially $i^2 = -1$) automatically encode the geometry of perpendicular directions and rotation. Multiplying by $i$ means “rotate 90 degrees.” Multiplying by $i^2 = -1$ means “rotate 180 degrees.”
The Big Picture
Complex numbers aren’t “imaginary” in the sense of being fake or abstract. They’re a powerful representational tool for describing:
- Two-dimensional quantities (anything with perpendicular components)
- Rotations and periodic motion (circles, oscillations, waves)
- Phase relationships (timing differences in oscillating systems)
Next time you see $a + bi$, don’t think “weird algebra trick.” Think “magnitude and direction” or “horizontal and vertical components.” The mathematics of complex numbers is simply the mathematics of geometry and rotation, dressed in algebraic clothing.
And that $i$? It’s not imaginary; it’s just perpendicular.
Want to explore more? Try representing your daily commute as a complex number (east + i·north), or think about how your phone’s accelerometer measures motion in perpendicular directions. Complex numbers are everywhere once you know how to see them.
