Have I Discovered a New Hypercomplex Number System?
The following article summarizes a possible discovery in mathematics that might turn out to be important. In 2023, I constructed a new set of hypercomplex numbers: a 10-element system I call the decenions. This system extends the familiar family of quaternions, octonions, and related structures, though it falls outside the standard 2ⁿ progression seen in hypercomplex algebras. I’ve found it mathematically intriguing and worth sharing for further exploration.
Mathematics relies on imaginary numbers, starting with i, the square root of –1, to complete algebraic operations. Building on that, hypercomplex numbers extend the number system into higher dimensions. The first such system, discovered in 1843 by William Rowan Hamilton, was the quaternion set: four elements (1, i, j, k), where each imaginary numbers squares to –1 and multiplication is anticommutative (ij = k, but ji = –k). Quaternions are like different “flavors” of number. Each one can also be multiplied by any positive or negative number, so, for example, 3j × 4k = 12i. These numbers can also be added and subtracted but keep their identities intact because they are different in “kind.” Hence, we can have a compound hypercomplex expression such as 7 + 2i – 5k, which is irreducible.
Octonions expanded this structure to eight elements, with seven imaginaries and one real, using similar rules but with more complex relationships. Beyond octonions came sedenions (15 imaginaries), which introduce additional complications such as the loss of associativity and the emergence of zero divisors. Generally, it is thought that valid hypercomplex systems grow in powers of two: 4, 8, 16, 32, and so on.
The multiplication table for decenions is shown below. The decenion table is generated through an array-based method that differs from the Cayley-Dickson process, which doubles the size of hypercomplex sets. In contrast, this approach yields a 10-element system using a 3×3 array. The method is too advanced and possibly speculative in its further extensions to describe here. T he entire set includes the real identity element (“one”), called e₀. One times anything keeps it the same, so e₀ is left out of the table. The nine imaginary elements are labeled e₁ through e₉. Any imaginary element times itself equals –1. The other imaginaries multiply such that ab = c, where all three are different elements with indicated signs. As with other hypercomplex systems, multiplication is anticommutative: ab = –ba. For example, e₁e₂ = e₃, but e₂e₁ = –e₃. In the table below, multiplication is read as left column times top row. For those familiar with product “triads,” we can derive them from the table as: 123, 147, 159, 168, 249, 258, 267, 348, 357, 369, 456, 789.
What’s important is that the table is fully consistent. It has no contradictions such as ab = c yet also ad = c, and no “loose ends” such as elements that can’t be rightly matched up with others. However, it is fine to have more ways to get the same product, e.g., ab = pq = c. It is thereby fully analogous to the tables for octonions, sedenions, etc.
I don’t yet know how fully the decenions align with other hypercomplex sets. They appear to behave similarly at a basic level, but deeper properties remain to be explored. In particular, it will be important to examine associativity, norm behavior, and the presence or absence of zero divisors. (For example, in larger hypercomplex systems, it is possible to form a product where (a + b) (c + d) = 0, even though none of the individual components are zero — an unusual phenomenon for those unfamiliar with this subject.
Of personal and sociological interest: Although I have worked in science and technical fields, I am a 69-year-old amateur or hobbyist (some coursework but no degree in math). Can amateurs make important discoveries? Sure. In 2022, hobbyist David Smith solved the einstein tile problem. Here, “einstein” comes from “one stone,” not the man. Smith found a 13-sided shape (“tile”) that fits to fill space using ever more tiles, but the pattern never repeats. Articles had titles such as “UK Hobbyist Stuns Math World With ‘Amazing’ New Shapes” (phys.org, June 10, 2023).
Whether or not this structure ultimately proves useful or fully consistent, I believe that exploring such variants can broaden our understanding of hypercomplex mathematics. If this topic interests you, I encourage you to read more about the history of hypercomplex numbers. If you are studying mathematics or know someone who is, I would be curious to hear your thoughts on the decenion structure. As with any mathematical idea, it will take many eyes and perspectives to understand where this might lead. In a letter to the editor in the July issue, I expressed interest in seeing a math-focused SIG within Mensa; perhaps discussions like this could help spark one. If you’re interested, email me at paradoxer@tni.net.
Acknowledgment: Thanks to Dr. Tevian Dray of Oregon State University for help checking my rules and construction of decenions.
A brief, similar exposition of decenions, “What Happens If I Discovered New Sets of Hypercomplex Numbers?,” appeared in Tidewater Mensa’s October 2023 M-Tides, Editor Denise Ross.
Neil Bates
NEIL BATES worked in science and technical fields for many years, including with NASA Langley and Jefferson Lab. He’s now an independent hobbyist trying to extend frontiers in science, math, and philosophy.
Tidewater Mensa | Joined 1985