❤️tayoro Art Heart Drawing 64 Where are imaginary numbers?

🌙 Introduction: Imaginary numbers are 'nowhere' yet they 'drive the world'

Yoro: Where do imaginary numbers exist? Is it true that they are orthogonal to the time axis? I've also heard they have a connection to parallel universes. And— 'Without imaginary numbers, modern technology would not exist.' What does this mean?

Once again, we will have A-ko, B-ko, Doku-ko, and quantum physicist Dr. Ryoko join us to explore the 'true nature of imaginary numbers'.

🎬 1. Is the imaginary axis orthogonal to the time axis?

A-ko 'Hey Doctor. Is the imaginary axis orthogonal to the time axis?'

Dr. Ryoko 'If we are talking only about mathematics, then "YES". On the complex plane, the real axis and the imaginary axis are perfectly orthogonal.'

B-ko 'So, does that mean the time axis equals the real axis?'

Doctor 'That is the interesting part. In mathematics, "real axis = time" is not always the case. However, in physics, if you introduce the concept of "imaginary time", the time axis rotates in the imaginary direction and becomes orthogonal to the spatial axis—that is how it can be treated.'

Doku-ko 'Time "rotating in the imaginary direction"...? It sounds a bit like chuunibyou, I like it.'

Doctor 'It is a technique called Wick rotation. Using this, the singularity at the beginning of the universe can disappear, or time can behave "like space". In other words, the imaginary axis can be treated as an axis representing "possibilities of other times".'

🌌 2. Do imaginary numbers suggest the existence of parallel universes?

A-ko 'So Doctor, are imaginary numbers proof of parallel universes?'

Doctor 'They are not proof, but they have become the key to treating parallel universes mathematically.'

B-ko 'What do you mean?'

Doctor 'The wave function of quantum mechanics is'

ψ=∣ψ∣eiθ

which is made of "complex numbers". This "iθ" part, in other words the phase of the imaginary number, represents "possibilities of the future that have not been observed yet".'

Doku-ko 'In other words, are imaginary numbers the direction of a "world that hasn't been determined yet"?'

Doctor 'Exactly. In the many-worlds interpretation, we consider that possibilities that were not observed actually exist as different worlds. What actually distinguishes those "different worlds" is the phase of the wave function containing the imaginary number.'

A-ko 'So are imaginary numbers like "shadows of parallel universes"?'

Dr. "That's a very good way to put it. An imaginary number is an 'axis of another possibility,' and you could say it is the shadow protagonist that allows us to handle parallel universes mathematically."

📱 3. Modern technology would not exist without imaginary numbers

B-ko "But Doctor, are imaginary numbers really that relevant to reality?"

Dr. "Actually, it's the opposite. Without imaginary numbers, your smartphone would just be a slab of plastic."

A-ko "Wait, really?"

Dr. "For example—"

🔧 Everyday technologies that wouldn't work without imaginary numbers

• Smartphone communication (4G/5G/Wi-Fi) Fourier transform = the world of complex numbers. All radio wave analysis uses imaginary numbers.

• Noise-canceling headphones Complex number angles (arguments) are essential for creating 'anti-phase'.

• GPS Distortion correction for radio waves is complex number wave analysis.

• Smartphone camera image processing 2D Fourier transform = decomposing and reconstructing images using complex numbers.

• 3D game/VR rotation (quaternions) Quaternions are an extension of complex numbers. They wouldn't exist without imaginary numbers.

• Electrical circuits (home appliances, power plants) AC calculations use complex numbers as the standard. Calculation is impossible without imaginary numbers.

• AI (Deep Learning) FFT (Fast Fourier Transform) is a complex number calculation.

Doku-ko "In other words, if there were no imaginary numbers, civilization would come to a halt."

Dr. "Exactly. Imaginary numbers are not 'numbers that don't exist in reality,' but rather another axis for driving reality."

🌱 4. Summary: Imaginary numbers are an 'axis of the future,' a 'shadow of parallel universes,' and the 'foundation of civilization'

To summarize our conversation so far, imaginary numbers are like a 'polyhedron' with the following aspects:

• In mathematics, another axis orthogonal to the real axis

• In physics, a tool for rotating time

• In quantum mechanics, a phase representing an uncertain future

• In cosmology, a coordinate for handling multiple universes

• In technology, the foundation of radio waves, images, AI, and communication

In other words, imaginary numbers are “an axis that is invisible, yet certainly moves the world”.

Yoro: So imaginary numbers really are “an axis that is invisible but certainly exists.”

🧪 From Dr. Ryoko, Quantum Physicist

Yoro-san, this theme really hits the essence of it.

Imaginary numbers are not “non-existent numbers,” but “the very structure behind the world”.

🌱 To our readers

Imaginary numbers are very similar to the movements of the human heart in that they are “things that are invisible but certainly at work.”

The possibilities of the future, emotions that have yet to be put into words, the shadows of choices not taken.

They may all exist in an “imaginary direction”.

The world of imaginary numbers is invisible to the eye, but it breathes quietly within our everyday lives.

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