Imaginary numbers are not imaginary at all. The truth is, they have had far more impact on our lives than anything truly imaginary ever could. Without imaginary numbers, and the vital role they played in putting electricity into homes, factories, and internet server-farms, the modern world would not existMichael Brooks, “Imaginary Numbers Are Reality” at Nautilus (February 9, 2022)
Imaginary numbers, are we recall from school, are the square roots of minus numbers. Two plus numbers, multiplied, result in a plus number. But so do two minus numbers. The square roots of minus numbers must exist but they exist in an “imaginary” sense. And yet they matter:
Students who might complain to their math teacher that there’s no point in anyone learning how to use imaginary numbers would have to put down their phone, turn off their music, and pull the wires out of their broadband router.Michael Brooks, “Imaginary Numbers Are Reality” at Nautilus (February 9, 2022)
While acknowledging that “A full mathematical description of nature” requires imaginary numbers to exist, Brooks considers the term “imaginary” numbers unhelpful because he considers all numbers to be imaginary. Toward the end of his essay, he asserts, “So what we’re discovering here is not some deep mystery about the universe, but a clear and useful set of relationships that are a consequence of defining numbers in various different ways.”
But what does his claim that the numbers are “not some deep mystery about the universe” leave us? Recent studies have shown that imaginary numbers — which we can’t really represent by objects, the way we can represent natural numbers by objects — are needed to describe reality. Quantum mechanics pioneers did not like them and worked out ways around them:
In fact, even the founders of quantum mechanics themselves thought that the implications of having complex numbers in their equations was disquieting. In a letter to his friend Hendrik Lorentz, physicist Erwin Schrödinger — the first person to introduce complex numbers into quantum theory, with his quantum wave function (ψ) — wrote, “What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. Ψ is surely fundamentally a real function.”Ben Turner, “Imaginary numbers could be needed to describe reality, new studies find” at LiveScience (December 10, 2021)
If you are willing to alter quantum mechanics, so that it becomes even more non-local than it already is, then you can still create the necessary entanglement with real valued numbers.
Why is it controversial? Well, if you belong to the shut-up and calculate camp, then this finding is entirely irrelevant. Because there’s nothing wrong with complex numbers in the first place. So that’s why you have half of the people saying “what’s the point” or “why all the fuss about it”. If you, on the other hand, are in the camp of people who think there’s something wrong with quantum mechanics because it uses complex numbers that we can never measure, then you are now caught between a rock and a hard place. Either embrace complex numbers, or accept that nature is even more non-local than quantum mechanics.Sabine Hossenfelder, “Do complex numbers exist?” at BackRe(Action) (March 6, 2021)
These are not good times for simple materialism.
You may also wish to read:
Why the unknowable number exists but is uncomputable. Sensing that a computer program is “elegant” requires discernment. Proving mathematically that it is elegant is, Chaitin shows, impossible. Gregory Chaitin walks readers through his proof of unknowability, which is based on the Law of Non-contradiction.
Most real numbers are not real, or not in the way you think. Most real numbers contain an encoding of all of the books in the US Library of Congress. The infinite only exists as an idea in our minds. Therefore, curiously, most real numbers are not real. (Robert J. Marks)
Can we add new numbers to mathematics? We can work with hyperreal numbers using conventional methods. Surprisingly, yes. It began when the guy who discovered irrational numbers was—we are told—tossed into the sea. (Jonathan Bartlett)