Complex Ancestors: How Imaginary Numbers Repair Broken Trees in Mathematics and Physics

From missing husbands and brick walls in genealogy to Krein spaces and quantum theory, a gentle tour of why i and complexification are too useful to be merely “imaginary.”

Mathematics likes to pretend it is inevitable. Once the definitions are written down and the theorems proved, everything appears as if it could not have been otherwise. But this is, at least in part, an illusion. Beneath the formal surface lies a quieter layer of choices, interpretations, and habits of thought—some so ingrained that we no longer notice them.

One of these habits concerns complex numbers.

We use them constantly. They appear in quantum mechanics, signal theory, spectral analysis, and in many corners of pure mathematics. They behave well, they simplify formulas, and they make difficult structures manageable. In short, they are extremely convenient. But convenience, as history repeatedly reminds us, is not the same thing as understanding. So it is natural to ask: what, exactly, is this imaginary unit that sits at the heart of so much mathematics?

To approach that question, let us leave mathematics for a moment and step into a more familiar territory: genealogy.

Imagine you are building a family tree and everything is going smoothly—names, dates, marriages, baptisms, wills, all nicely documented—until you hit a gap that refuses to be filled. The records insist that someone must be there. Property passes through them, DNA patterns point to them, surnames shift because of them. And yet: no birth certificate, no parish entry, no grave. Officially, they do not exist. Practically, they must.

Mathematicians face a similar situation.

There are equations—perfectly reasonable ones—that simply cannot be solved using ordinary real numbers. For example, the equation x²+1==0 asks for a number whose square is −1. But no real number has this property. The “records,” so to speak, come up empty. And yet the structure of mathematics strongly suggests that a solution ought to exist.

So mathematicians do something very familiar to a genealogist: they introduce a placeholder that is not directly observable but is logically necessary. This is the imaginary unit i, defined by the rule i²=−1. It is not found among the “living population” of real numbers, but once introduced, it restores consistency and allows the whole system to function again.

And, just as in genealogy, we do not keep this hypothetical ancestor in isolation.

You would place an inferred ancestor within the family structure, attached to spouses and children. Mathematics does the same: it does not use i alone, but combines real and imaginary parts into a single entity, a complex number of the form a+bi. You might think of this as “Grandma Mary (well-documented) together with her unknown but necessary partner.” It is one entry in the ledger, not two separate people, but internally it carries both the known and the inferred.

This combination turns out to be extraordinarily powerful. Equations that were previously unsolvable suddenly have solutions. Patterns that looked broken become coherent. In engineering, complex numbers describe oscillations, waves, and electrical circuits. In physics, they are essential in quantum theory, where probabilities themselves seem to carry a kind of hidden phase—something not directly observable, but crucial for making correct predictions.

There is even a geometric way to think about this. Real numbers form a line; complex numbers form a plane. Adding the imaginary part is like adding a new direction in which things can vary. What looked impossible in one dimension becomes natural in two.

From a genealogical point of view, the lesson is not so strange. Sometimes, to make sense of the data, you must allow for entities that are not directly recorded but are strongly implied. They are not arbitrary inventions—they are disciplined hypotheses that restore order and explanatory power. In genealogy, an inferred ancestor invites further research; it is a guide, not a final truth. In mathematics, imaginary numbers are more stable—they behave consistently, and their usefulness has been tested in countless applications. Still, their origin carries the same flavor: a response to a gap that could not simply be ignored.

And just as a single missing person can reshape an entire family tree, the introduction of i reshapes mathematics in a profound way. What began as a technical fix turns into a whole new landscape—one in which previously disconnected pieces suddenly fit together. Not bad for someone who, officially, does not exist.

So far, we have looked at complex numbers as a kind of logically necessary “missing ancestor.” But the story does not end there. There is also the question of how we interpret this strange object i.

In a thought-provoking paper published in the Journal of Humanistic Mathematics, Andrew Powell revisits the imaginary unit from a geometric perspective. He suggests that i is not an abstract algebraic curiosity, but something rooted in the structure of the plane itself: a rotation, a bivector, a geometric operation disguised as a number. In this view, complex numbers are not an artificial extension of reality but a natural language for describing it.

This idea resonates with the program developed by David Hestenes, who has long argued that the imaginary unit in quantum theory reflects an underlying Clifford algebra structure. According to this perspective, the mystery of i dissolves once we stop looking for it in the wrong place. It was never “imaginary”—only misidentified.

And yet, there is another story, equally compelling and not entirely compatible.

Moretti and Oppio approach the problem from the side of physics rather than geometry. They show that if one begins with real or even quaternionic Hilbert spaces and imposes the symmetry requirements of relativistic quantum theory, the complex structure emerges almost unavoidably. In this account, complex numbers are not fundamental ingredients but consequences—artifacts of symmetry rather than primitives of thought.

It is a curious situation. Geometry seems to suggest that complex numbers are deeply natural, while physics suggests they are, in some sense, imposed. One might be tempted to resolve the tension, but perhaps that temptation should be resisted. Mathematics has a long tradition of supporting multiple explanations for the same phenomenon, each illuminating a different aspect of it, none fully canceling the others.

Perhaps the role of complex numbers is not a single story but a convergence of several: geometric, algebraic, physical, and even informational. Or perhaps we are still missing the simplest explanation, hidden in plain sight, waiting for a different point of view. In the meantime, we proceed pragmatically. Complex structures work. They organize theories, simplify computations, and reveal patterns that would otherwise remain obscured. Whether they are discovered or invented, fundamental or emergent, they have earned their place at the center of modern mathematics and physics.

Now, if we return to our genealogical analogy and push it a little further, we arrive at a more subtle situation—one that is closer to the kind of structures that appear in functional analysis and quantum theory.

In an ordinary family tree, relationships are “positive” in a straightforward sense: parenthood, descent, inheritance. But suppose you are dealing with more complicated records—conflicting testimonies, adoptions, disputed inheritances, or lines of descent that carry different kinds of weight or credibility. Some connections reinforce each other; others partially cancel or cast doubt. The structure is no longer purely additive in a simple sense. It carries an internal tension.

This is, in spirit, what happens when we move from Hilbert spaces to Krein spaces.

A Hilbert space is the comfortable world where everything behaves nicely: lengths are always positive, angles make sense, and orthogonality behaves as expected. It is like a perfectly documented family tree with consistent and reliable records. A Krein space, by contrast, allows for an indefinite inner product. Some directions have “positive length,” others have “negative length,” and the geometry reflects a balance between these opposing contributions. It is a space with built-in ambiguity—structured, but no longer entirely reassuring.

Such spaces are not exotic curiosities. They appear naturally in areas of physics where symmetry and causality play a role, particularly in relativistic settings. In a way, they encode the idea that not all contributions to a system are of the same type or sign.

Now comes the key step.

Just as real numbers can be extended to complex numbers to restore completeness and flexibility, real Krein spaces can be “complexified.” One takes a structure defined over the real numbers and enlarges it so that complex scalars are allowed. At first glance, this may look like a purely technical move—another formal extension, another layer of abstraction.

But if we keep our analogy in mind, it is something more familiar.

It is like taking a partially reconstructed family tree—with its uncertainties, its mixed evidence, its positive and negative indications—and embedding it into a richer framework where hidden relationships become easier to express and manipulate. The complexification does not erase the original structure; it reveals additional degrees of freedom that were already, in some sense, implicit.

And just as before, one may ask: are we discovering something that was always there, or are we choosing a language that makes the situation manageable?

This question becomes particularly sharp in the context of Krein spaces. When we pass from a real indefinite structure to a complex one, are we uncovering a deeper geometric or physical reality? Or are we introducing a tool—powerful, elegant, and perhaps indispensable—that helps us organize what would otherwise remain intractable?

The mathematics itself does not insist on a single answer. What it does provide is a framework in which both viewpoints can coexist: the real structure, with its tensions and ambiguities, and its complex extension, where those tensions can be handled with greater flexibility. Much like in genealogy, where a well-chosen hypothesis can illuminate an entire network of relationships, complexification often turns a difficult structure into one that can be understood, analyzed, and—eventually—used.

This brings us to the present discussion: the complexification of real Krein spaces. At first glance, this may appear as a technical construction within functional analysis. But it is also another instance of the same underlying question: when we pass from real to complex structures, are we uncovering something essential, or are we choosing a particularly effective language?

It would be satisfying to settle this once and for all.

Mathematics, however, is not always in the business of providing satisfaction. Sometimes it offers something more durable: a framework in which the question can continue to live.

The full mathematics—with the scary-looking formulas—is on my math blog. Here, we stay with the ideas, the analogies, and the questions that refuse to disappear quite as easily as an imaginary ancestor.

References:
Hestenes, D., “Clifford Algebra and the Interpretation of Quantum Me chanics”, in Clifford Algebras and their Applications in Mathematical Physics, edited by J. S. R. Chisholm and A. K. Common, Reidel, 1986.

Moretti, V., Oppio, M., “Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincare symmetry”, Reviews in Mathematical

Moretti, V., Oppio, M., “Quantum theory in quaternionic Hilbert space: How Poincare symmetry reduces the theory to the standard complex one”, Reviews in Mathematical Physics 31, 1950013 (2019).

Powell, A.W., ``What is an Imaginary Number? The Plane and Beyond’‘, Journal of Humanistic Mathematics} 14(2) (2024), 264-285.