The Real Number System Is Contained in the Imaginary Number System

Terminological note: In this essay, “imaginary number system” refers to the complete complex numbers a+bi, not the narrower set of pure imaginary numbers.

The imaginary number system is the origin of the number systems; real numbers are merely its derivation and projection. The complex number system z = a + bi is one of the most complete number systems in modern mathematics. In this space, real numbers do not exist independently — they are merely the degenerate case when the imaginary part b = 0, a projection of the complex world onto the real axis.

· Imaginary numbers give rise to real numbers: simply set the imaginary part to zero, and the imaginary number system naturally collapses, generating all real numbers.
· Real numbers cannot generate imaginary numbers: In the real number domain, no matter how you perform addition, subtraction, multiplication, division, root extraction, or even limit operations, you can never spontaneously produce the imaginary unit i = \sqrt{-1}.

This is an irreversible evolutionary relationship. Whoever contains the other, whoever generates the other — that is the matrix. From this perspective, real numbers are nothing but a special, observable cross‑section of the imaginary world.

Modern physics has already revealed:

· The deepest level of the world must be described using complex numbers;
· The vacuum is not empty; matter is an excited state of fields;
· Measured real values are merely projections of complex wavefunctions under specific operations.

Most mainstream physicists cautiously avoid ontological conclusions. But if we follow the reasoning based on the facts above, we are compelled to adopt a position: the essence of the world is an imaginary (complex) structure; real numbers are only appearances.

From a philosophical perspective

Laozi said, “Being arises from non‑being.” Zhuangzi spoke of “gathering emptiness as the Way.” Eastern philosophy has long held the intuition that “emptiness is fundamental, reality is derivative,” but previously this remained largely at the level of metaphor.

This essay attempts to map that intuition onto mathematical structures:

· The imaginary (complex) number system = non‑being, origin, the Way
· Real numbers = being, appearance, the myriad things

In my view, this is a clear anchoring of the highest Eastern wisdom in the foundations of modern mathematics — a mathematization of philosophy and a philosophization of mathematics.

Real numbers are not fundamental; imaginary numbers are not unreal.
The vacuum is not empty; it is the true source.

From the perspective of mathematical structure, real numbers are a subset of complex numbers. From the perspective of fundamental naming, the complex domain is the complete imaginary number system.

Therefore, the real number system is contained in the imaginary number system — this assertion: its structure has been revealed by those before us, but here I clarify its philosophical standing.