A Song About Imaginary Numbers in Math

 

The mathematical concept of Imaginary Numbers baffles most people. How can there be Imaginary Numbers ,and why on earth do we need imaginary numbers? Here is a song titled "Are They Real?" composed with Suno. In the song, a student  is both baffled and curious about Imaginary Numbers and has many questions to ask his Professor.  

But first, here is an introduction to Imaginary Numbers

Imaginary numbers are a breakthrough mathematical concept designed to solve a puzzle that ordinary numbers cannot: finding the square root of a negative number. In everyday arithmetic, multiplying any number by itself always results in a positive value (for instance, \(2 \times 2 = 4\) and \(-2 \times -2 = 4\)). Because no traditional number can be multiplied by itself to equal a negative number, mathematicians defined a foundational unit called \[i\], which represents the square root of \[-1\]. An imaginary number is simply any regular number multiplied by this unit, such as \[3i\]. While ordinary numbers only allow us to measure distances along a one-dimensional left-to-right line, imaginary numbers introduce a vertical direction, creating a two-dimensional grid where complex real-world systems can be perfectly mapped. 

Real life uses of imaginary numbers

• Electrical Engineering: Alternating current (AC) electricity, which powers our homes, constantly changes direction. Engineers use imaginary numbers to calculate electrical impedance, combining resistance and timing shifts into a single formula.

• Signal Processing: The technology inside smartphones, Wi-Fi routers, and digital audio systems relies on splitting complex waves into simpler parts. Imaginary numbers make the mathematics behind filtering out background noise and transmitting wireless data manageable.

• Quantum Mechanics: The fundamental laws governing subatomic particles use equations that require imaginary numbers. Without them, physicists cannot accurately predict the behavior of electrons or light particles.

• Computer Graphics: Creating realistic 3D rotations, animations, and camera movements in modern video games uses extensions of imaginary numbers to turn objects smoothly in virtual space without distortion.

And here is a visualization of Imaginary Numbers.

Here are the lyrics of "Are They Real?"

[Intro]

Professor, wait—hold up, I’m lost again

You wrote i on the board like it’s my newest friend

 

[Verse 1]

You said a number needs a real straight line

I said, “Fine, then where does i reside?”

You said i squared equals negative one

Now my common sense is packing up to run

 

If real means a point I can measure and see

Then what kind of number is hiding from me?

Is it real, is it fake, is it somewhere between?

Or a useful invention dressed up in a dream?

 

[Pre-Chorus]

I’ve got a plus b i, letters in disguise

Two coordinates wearing one set of eyes

I’m asking for a reason, not a rule to repeat

Why build a second axis underneath my feet?

 

[Chorus]

Are they real? Tell me, are they real?

If they can move the world, then maybe that’s the deal

Why learn numbers that I cannot hold?

They steer the current, shape the signal, make the future unfold

Are they real? I’m still unsure

But the universe keeps using them to make its language work

From a square root’s impossible start

Complex numbers map the hidden parts

 

[Verse 2]

In the circuit, voltage turns and currents swing

Complex impedance tells the whole thing

A radio wave travels, carrying a call

Without those strange coordinates, the model hits a wall

 

In a rotating system, phase is the clue

A little multiplication tells what waves will do

Fourier takes a messy sound and splits it into tone

So music, images, data find a structure of their own

 

A plane takes a turn when you multiply by i

Quarter-circle motion—watch the arrow fly

Then i again sends it facing down below

Not imaginary motion—just a different way to show

 

[Pre-Chorus]

I’ve got a plus b i, not a magic trick

A point on a plane with a compact little script

The question keeps returning, stubborn and sincere:

If it works this well, why does it feel unclear?

 

[Chorus]

Are they real? Tell me, are they real?

If they can move the world, then maybe that’s the deal

Why learn numbers that I cannot hold?

They steer the current, shape the signal, make the future unfold

Are they real? I’m still unsure

But the universe keeps using them to make its language work

From a square root’s impossible start

Complex numbers map the hidden parts

 

[Bridge]

Maybe “real” is not the only test

Maybe useful truth can wear a different dress

A plus b i, a compass in the dark

A quiet extra dimension where the equations leave a mark

 

I don’t have to touch it to know it can be there

Like gravity’s pull or a field in the air

The name says imaginary, but the results arrive

In phones, machines, waves—things that keep us alive

 

[Verse 3]

So I’ll draw the real axis, then the imaginary side

Put the ordered pair together, let the two worlds coincide

Magnitude is the distance, angle tells the turn

Conjugate flips the sign—another lesson learned

 

I came in asking, “What is this supposed to mean?”

Now I see a broader map behind the numbers on the screen

Not fake, not fantasy, not a loophole in the art

Just a richer kind of arithmetic with room for every part

 

[Final Chorus]

Are they real? Now I hear the wheel

If they can model motion, then the question starts to heal

Why learn numbers I cannot hold?

Because they steer the current, shape the signal, make the future unfold

Are they real? More than I knew

A different kind of truth making old equations new

From a square root’s impossible start

Complex numbers map the hidden parts

 

[Outro]

Professor, write it slowly—I’m beginning to see

Real plus imaginary can still describe reality

The universe is strange, but the lesson’s finally clear:

Some numbers look far away, then bring the world near