Euler's Formula: The Imaginary Thing That Became My Most Real Tool
I was unusually good at math as a child.
Not in a dramatic prodigy way — just the quiet kind. Formulas made sense to me as wholes. Even when I forgot the details, the structure stayed. I could lose the steps and still remember the shape.
Euler’s formula was one of those.
I forgot it many times. But somehow, I never really lost it.
When Euler Felt Like a Party Trick
For a long time, Euler’s formula felt… fictional.
Beautiful, yes. Elegant, undeniably. But also suspiciously abstract — like something mathematicians invented to impress each other at conferences.
An equation involving an imaginary number, connecting exponentials to sine and cosine?
\[e^{i\theta} = \cos\theta + i\sin\theta\]At school, this lived in the same mental category as mathematical magic tricks: impressive, self-contained, and safely detached from anything I’d ever actually do.
I understood it. I could use it on exams.
But I didn’t need it yet.
The Suspicious Naming of Imaginary Numbers
The irony, of course, is in the name.
“Imaginary” suggests something optional. Decorative. A mathematical garnish you tolerate in coursework and abandon afterward.
Whoever named these clearly never worked with waves.
Because here’s the unsettling truth I learned later:
In wave physics, imaginary numbers aren’t imaginary at all. They’re bookkeeping devices for things that oscillate.
And oscillation, unfortunately, is extremely real.
The Moment Euler Became Useful
In optics — especially Fourier optics — you rarely work with fields that politely sit still.
Electric fields oscillate. Rapidly. Invisibly. Constantly. It’s their whole personality.
A real-valued field might be written as:
\[E(t) = E_0 \cos(\omega t + \phi)\]Perfectly physical. Perfectly real. Perfectly annoying to do calculus on.
The moment you want to differentiate it, integrate it, shift it, superpose it, or propagate it… things get ugly fast. Trig identities multiply. Signs flip. You start making mistakes.
Euler’s formula steps in quietly and says:
“Let me hold this for you.”
So we rewrite the same thing as:
\[E(t) = \text{Re}\left[ \tilde{E} e^{i\omega t} \right], \quad \tilde{E} = E_0 e^{i\phi}\]And suddenly, calculus becomes algebra.
No new physics appeared. No approximation was made.
We just chose a representation that respects how waves actually behave — instead of fighting them with trigonometry.
What the Complex Number Is Actually Doing
At some point, something clicks.
The complex number isn’t “fake.” It’s a container.
\[\tilde{E} = E_0 e^{i\phi}\]This single object carries:
- magnitude → how strong the field is
- phase → where it is in its oscillation
Nothing mystical. Nothing extra. Just two real quantities stored efficiently, like a tuple that graduated from math school.
When multiple waves interfere, you add these complex numbers and only take the real part at the end — like closing parentheses after a long calculation.
Euler’s formula isn’t introducing imagination.
It’s postponing reality until it’s convenient.
When Euler Became Infrastructure
The moment I started working seriously with Fourier optics, something changed.
Euler’s formula stopped being a trick. It became the language.
- Propagation? Phase accumulation.
- Interference? Complex addition.
- Fourier transform? Exponentials everywhere.
At that point, Euler was no longer optional.
Just like Fourier transform, Euler’s formula isn’t something you “apply.” It’s something you assume — the way you assume linearity or continuity.
It sits underneath everything, quietly enabling the math to match the physics.
You don’t think about it. You just use it. Like electricity, or caffeine.
In OCT, It’s Unavoidable
In coherent imaging and OCT, this becomes inescapable.
You’re not dealing with intensity alone. You’re dealing with complex fields.
Interference, phase stability, dispersion, reconstruction — all of it lives naturally in the complex domain. Trying to avoid complex numbers here is like trying to avoid water while swimming.
Intensity itself is defined as:
\[I = |\tilde{E}|^2\]Which is just another way of saying:
“The imaginary thing determines the real measurement.”
At that point, calling it imaginary feels almost rude.
The Funny Part
What still amuses me is this:
Euler’s formula once felt like pure abstraction — something elegant but detached from the real world. A mathematical flex with no practical purpose.
Now it’s one of the most practical expressions I use.
I don’t think about it. I don’t admire it. I just rely on it — the way you rely on a screwdriver without contemplating the philosophy of torque.
Like Fourier transform, it quietly moved from “theory I learned” to “infrastructure I assume.”
The Pattern I Keep Seeing
There’s a pattern here, and it’s starting to feel like a conspiracy:
- Things that feel abstract early on
- Become unavoidable later
- Once you start working with real systems
Euler’s formula didn’t become important because it’s beautiful.
It became important because waves exist, and waves don’t care whether you find complex numbers intuitive.
Acceptance (Again)
I no longer think of Euler’s formula as imaginary.
I think of it as honest.
It admits that oscillations are inconvenient to track directly — and gives us a way to handle them without lying to ourselves.
I forgot Euler’s formula many times.
But the idea stayed.
And eventually, it became one of the most real tools I have.
Summary
This isn’t a lesson on complex numbers. There are countless better explanations elsewhere.
This is a reflection.
Euler’s formula feels unreal when you first learn it. It feels inevitable once you start working with waves.
And like Fourier transform, it has a strange habit of waiting patiently — until you need it.
Then it shows up, acts like it was there all along, and makes you wonder why you ever thought it was optional.