by – Chip Whaley
When we hear √2, most of us think “that’s imaginary”—and it is! But what if we really did try to use this number, β=i√2, as a base for counting? Even more bizarre: This single “imaginary” base can uniquely encode every point in a 2D plane into a linear string of digits. Let’s explore how that works and why it’s so interesting.
The image on the left shows the correspondence between each integer on the x axis and its unique correspondence in base -2 on the y axis between 0 and 31. You can see the fractal nature of this system, which would repeat if you go larger (see graphics later in this post). The image of the right is the same thing but in base i√2, with the imaginary part expanding the structure into the z axis. The fractilization on the left is of rectangles at 45degree angles to the plane. The one on the right is composed of rectangular solids at a more complex orientation, but also perfectly tiling the plane.
1. The Setup: Complex Bases
Most of us learned to count in base 10 (decimal). Some are familiar with base 2 (binary) or perhaps base 16 (hex). These are all real bases, bigger than 1 in absolute value. But mathematically, there’s no law saying the base can’t be a complex number. The only real requirement is:
∣ β ∣ > 1
If ∣β∣ is greater than 1, then we can build expansions “digit by digit” that converge to any complex value. For β=i√2, we note that:
| i√2 | ≈ 1.4142
That’s bigger than 1, so it’s fair game to try and build expansions in that base.
2. How Does a Complex Base Work?
Suppose we want to represent some complex number zzz. We choose digits d0,d1,d2,… in some finite digit set—often something like {0,1,…,m−1}. Then we write
where β=i√2. Each power β^k is also a complex number. Because ∣β∣>1, the terms β^k grow in magnitude as k increases (or as k becomes more negative, if we allow digits after a “decimal” point). This is analogous to how base-10 expansions work, except everything now lives in the plane, not on a single line.
But wait, how do we choose digits so that every complex number can be encoded in a systematic way—ideally with a unique representation? That’s the crux of the theory of complex β-expansions. The short version is that we pick a digit set that forms a complete residue system in the 2D lattice that β generates. For certain algebraic integers like i√2, this process is well-studied in mathematics.
3. Mapping Two Dimensions to One Dimension
Here’s one of the mind-blowing parts: When β is complex, you might expect we need two separate expansions (one for the real part, one for the imaginary part). But actually, the single string of digits
(d0,d1,d2,… )
already captures both the real and imaginary parts of the final sum
z=d0+d1xβ+d2xβ^2+⋯
We’re basically mapping the entire plane C onto an infinite set of digit sequences. In topological terms, the set of infinite digit sequences is a “one-dimensional” object (like a line or a ray) if you think of it just as sequences of digits. But because each digit sequence corresponds to one specific point in the plane—and vice versa—the mapping is a one-to-one correspondence!
Put differently: If you imagine all possible strings of digits, you can think of them as points on a “line” (one dimension, the dimension of the index k). Meanwhile, each such string “unfolds” into a unique point in 2D space. This is what we mean when we say it “maps a 2D plane one-to-one into a single ray (or single side of an axis).” The “axis” in this analogy is the axis of digit sequences.
4. Why i√2 in Particular?
There’s a special charm about β=i√2:
- It’s an Algebraic Integer. It satisfies a simple polynomial equation x2+2=0x^2 + 2 = 0x2+2=0. Having β satisfy a low-degree polynomial with integer coefficients typically ensures certain “nice” properties in the representation system.
- Moderate Magnitude. √2≈1.4142 isn’t too large. That means expansions in this base converge at a decent rate (each digit we add refines the point more and more).
- Symmetry and Geometry. Powers of i√2 rotate and scale the plane. Specifically,
- β1=i√2 is a 90° rotation from √2 on the real axis,
- β2=(i√2)^2=−2, purely real and negative,
- β3=(i√2)^3==−2i√2, and so on.
So each successive power rotates by 90° (a quarter turn) and scales by √2. This interplay of rotation + scaling leads to intriguing patterns in how digits combine to represent points.
- The fractal structure generated is of perfect 3d rectangular solids in a fractal structure that perfectly tiles the plane.
The first image above is in base negative 2 for 0 through 127, but the remaining 3 are 3d graphs of base i√2 from 0 to 127. The three 3d graphs are all the same dataset, but rotated to give you a better feel for the image.
5. Highlights and Quirks
- No Separate “Real/Imag” Splitting. In ordinary usage, you might write a complex number z as x+iyx + i yx+iy. In base β=i√2, we do it all in one digit expansion. That’s mathematically elegant and can be surprising at first.
- Fractal Boundaries. If we restrict digit strings (e.g., only a certain finite set of digit combinations), the set of all possible sums can form fractal regions in the plane. This is analogous to how certain real base expansions with constraints (like “only 0s and 1s” in decimal) create Cantor sets. In the complex realm, these shapes can be elaborate 2D fractals.
- Practical or Not? In day-to-day computing, nobody’s using base i√2 to store or display data, because the overhead is quite large compared to simpler integer bases. Still, for theoretical computer science and mathematics, this reveals that positional representation is not limited to real, integer bases. The concept is far more general!
- One-to-One from 2D to 1D. The biggest mind-bender: we typically think that a line (1D) can’t “cover” a plane (2D). But because each position in the digit string adds more detail in the plane, we effectively get a space-filling capacity. In fact, it’s a well-known phenomenon that even for real bases, a single infinite sequence can specify an uncountable continuum of points. With complex bases, it just so happens that the dimension is now 2, so we’re describing a 2D point with a 1D digit string.
In Summary
Using i√2 as a base might look like a bizarre parlor trick, but it opens a fascinating window into the world of complex number systems. Every point on the 2D complex plane can be uniquely labeled by a single string of “digits” in this base, creating a one-to-one mapping between the plane and what is essentially a linear, one-dimensional axis of digit sequences.
This underscores how flexible positional number systems can be—once you realize you can let go of the assumption that the base has to be a positive real integer. We can rotate, flip, and scale the plane in discrete steps, using imaginary powers of β to navigate every possible point in 2D.
If you ever grow tired of binary, decimal, or even hexadecimal, consider taking a leap into the complex plane and exploring expansions in base i√2. It’s a wild ride, but it reminds us that mathematics has no shortage of creative surprises hiding in plain sight.
Further Reading
- Donald Knuth, The Art of Computer Programming, Vol. 2
Look for the section on complex bases; while it may not discuss i√2 in particular, it lays out the fundamentals for how any complex base representation works. - Research on Gaussian (Complex) Number Systems
Academics have studied expansions in complex quadratic fields. Search for terms like “complex β\betaβ-expansions,” “Gaussian integers,” or “complex bases in numeration systems” for more details. - The Fractal Side
Check out articles or lectures on “fractal tilings” in complex bases. You’ll see pictures reminiscent of Penrose tilings and other geometric wonders, except they’re generated by powers of β\betaβ and digit sets.
Happy exploring in base i√2
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