What are Imaginary Numbers? How "Non-existent Numbers" Change the Way We See the World [Real Numberism]

In junior high and high school, we all learn about "numbers that become negative when squared." But many people move on without ever knowing where such numbers exist in reality. Imaginary numbers – as the name suggests, they are "imaginary." However, these numbers are not fictional tales. Rather, they quietly expand our very perspective on what we call "reality."

What are imaginary numbers in the first place?──Square it−1numbers that become

When you square a regular number (a real number), it always becomes zero or greater. 2 x 2 = 4, (-3) x (-3) = 9. No matter what, it never becomes negative. However, if you consider "a number that becomes -1 when squared," there is no answer within the realm of real numbers.

So mathematics gave a new symbol to that answer. That is the imaginary unit i. The definition is just one line, i² = -1. From this single agreement, a new world of numbers opens up.

Let a and b be real numbers. A number that can be written as z = a + bi is called a complex number. 'a' is the real part, and 'bi' is the imaginary part. Real numbers are just a special case where b = 0. In other words, the real numbers R, which we are familiar with, are a subset of the broader set of complex numbers C.

imaginaryimaginarybut it's not "fictional."

The name "imaginary number" is almost a historical mistake. In the 16th century, while solving cubic equations, the "square root of a negative number" inevitably appeared. Mathematicians at the time struggled with this, calling it an "impossible number" or an "imaginary number." The naming reflects this confusion.

However, later it was shown by Gauss and Euler that complex numbers can be represented as points on a plane (the complex plane). Imaginary numbers became mathematical realities on equal footing with real numbers. Today, imaginary numbers are indispensable in cutting-edge fields dealing with reality, such as electrical engineering, quantum mechanics, and signal processing.

The wavefunction in quantum mechanics takes on complex values. Far from being a "useless imaginary," the imaginary unit is an indispensable tool for describing reality. In essence, the imaginary number is imaginary but not fictional.

Real numbers are merely the "real part" of complex numbers.Realism

Here's an important structure that mathematics reveals. With just the real numbers R, we cannot answer questions like "what number squares to -1?". The system doesn't close. So, mathematics did neither belittle the real numbers nor leave an empty space. It minimally added a new axis, i, orthogonal to the real numbers R. Then, the system cleanly closes.

i is outside of R, but not imaginary. Only by being united with R does it complete the world of numbers. Real numbers were merely a part of this vast complex plane, the horizontal axis.

From here to philosophy──Isn't the "measurable world" also just the real part?

Extended imaginary theory borrows this mathematical framework as a model for how we perceive the world. We tend to consider only what can be measured and described numerically as "real." However, no matter how precisely we describe an object, there will always be a surplus that has not yet taken shape as meaning.

The description only captures the result once it has already taken shape. What precedes the formation of that shape is not recorded in any numerical value.

This composition is strikingly similar to the system of numbers that could not be closed with just real numbers. Therefore, the Extended Imaginary Theory writes the existence Z of an object as a superposition of real dimensions D that are fully described and imaginary dimensions iD that remain in front of them.

Z = D + ID

Just as i was an orthogonal uniaxial in mathematics to R, iD is an irreducible independent uniaxial to D. The "measurable reality" is merely the real part of a broader existence. However, the refusal to acknowledge that iD and continuing to define and treat this world with only D is called the "real number system."

Seeing the world through imaginary numbers

When you learn about imaginary numbers, a new dimension is added to the world. Instead of discarding the "impossible" concept of the square root of a negative number, you accept it by adding another axis. At that moment, problems that couldn't be solved become solvable, and a line expands into a plane.

Can't the same be done for how we see reality? Before dismissing what cannot be measured as "imagination," why not ask if there isn't another axis of reality present?

What Shiza Plus calls "structural vision" is the ability to see this one axis.

Realism is not wrong, it is narrow.

Just as real numbers are merely the real part of complex numbers, depicted reality is also merely the real part of a broader existence. Imaginary numbers are the closest gateway that first teaches us about that "breadth."

↓The fourth paper, "The Far North of Realism," is here↓

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