Do imaginary numbers really exist? The history of imaginary numbers, which overturned the premise that undetectable things do not exist

"What cannot be observed, does not exist."

Living in the age of science, we are unknowingly bound by this premise. However, there was a time when humanity discovered that this premise was wrong.

That's whyImaginary number The invention of the number.

And then,Do imaginary numbers exist? This question is still being debated repeatedly on the internet. However, the true scope of this question is not limited to the discussion of imaginary numbers.

The birth of an "unacceptable amount"

16th Century Italy, a ghost appearing in the middle of calculations

The story dates back to 16th-century Italy.

Around the time the mathematician Cardano was perfecting the solution to cubic equations, strange quantities appeared in the intermediate steps of calculations. Square roots of negative numbers. Quantities that, when squared, became negative – quantities that, in the worldview of the time, "should not exist."

Strangely, if you accept this quantity in the calculation process and continue the operation, you will eventually arrive at the correct real solution. It is convenient, but difficult to accept that it truly exists for a long time. This is "Imaginary numberwas treated as

Humanity once tried to make this amount of quantity "non-existent."

The structural integrity did not allow for exclusion.

Orthogonal Independent Axes: The Birth of the Complex Plane

However, the structural integrity did not allow this amount to be excluded.

In the late 18th century, three mathematicians independently performed an operation. They broke the habit of depicting real numbers as a single line, and on that line Orthogonal independent axes I have newly drawn one. The horizontal axis is the real number, and the vertical axis is the imaginary number. Both are combined and drawn as a two-dimensional plane. This is called the "complex plane."

At this moment, the imaginary number, which had been deemed "that which should not exist," finally gained a structural place.

Gauss later wrote:

The designation "imaginary" is incorrect. These are completely real quantities; they simply do not manifest as observable phenomena.

Not being observed and not existing are not the same thing.。

In other words, the existence of imaginary numbers is not based on observation. Structural integrity was logically necessitated.

Before the axis is established and after it is established

It wasn't discovery that changed how we see the world, but instruments.

What Gauss and the others did wasn't "discovering" new numbers. It was adding another dimension to the world of numbers, which had previously been one-dimensional. Orthogonal independent axes We established a new [something].

Before the axis was established, imaginary numbers were considered "non-existent." After the axis was established, they became "existent." The phenomenon has not changed. What has changed is the device used to coordinate it.

Events with the same structure have occurred many times throughout history. Gambling has existed since ancient times, but uncertainty was not recognized as a quantifiable independent axis in the cognitive space until Pascal and Fermat established "probability" in the 17th century.

The moment the axis stands, your view of the world changes.

An adaptation of Extended Imaginary Theory

To the realm of existence description—a coordinate of the imaginary dimension

And now, this operation from the realm of numbers Domain of existential descriptions is being adapted.

That is the framework I named "Extended Imaginary Theory." It describes an object as a dual structure, Z = D + iD. Existence is re-envisioned as a superposition of the real dimension D and the imaginary dimension iD.

Gauss's intuition that "imaginary numbers are real" should not be confined to the realm of numbers.

To set coordinates for the unseen.。

That is the thinking device humanity needs next.

↓Murakumo's first thesis "Extended Imaginary Theory" is here↓

↓Murakami's Second Paper, "A Structural Theory of Origins," is here↓

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