This YouTube video describes the construction of the triplex numbers, a sort of three-dimensional analogue of the complex numbers.
One starts by defining new units \(i\) and \(j\), and imposing the relations
$$i^2 = j$$
$$j^2 = i$$
$$ij = ji = 1$$
Below is an example of multiplication with triplex numbers. Arithmetic is carried out in the usual fashion, keeping in mind to follow the new rules to “collapse” the answer into a triplex number:
$$(1+j)(i+j)=i+j+ij+j^{2}$$
$$=1+2i+j$$
With any algebraic construction, there is always the question of applications. When I first saw triplex numbers, I wondered what kind of problems they might facilitate solving. The complex numbers, for example, are invaluable simply by virtue of being the prototypical algebraically closed field, and the quaternions are also of immense value, though their applications are more specific in nature.
As a starting point, one can think about solutions of polynomial equations. Do the triplex numbers facilitate solving certain kinds of polynomial equations, perhaps resulting in easier formulas? Do the triplex numbers solve “more” polynomial equations than the real numbers? In other words, do there exist polynomials lacking real roots which turn out to have roots in the triplex numbers?
It may help to consider a concrete example. Let us look at the polynomial
$$f(x)=x^{7}+x^{6}-6x^{5}-x^{2}-x+6$$
This polynomial has three real roots, and four strictly complex roots. What about roots in the triplex numbers? Are there any strictly triplex roots? How many strictly triplex roots are there? None? Four? Fourteen? Four hundred? Infinitely many?
It turns out that there are eighteen strictly triplex roots of the polynomial above. In fact, you can easily compute the exact number of triplex roots of any polynomial if the number of real and complex roots are known. Say that \(n\) is the number of real roots, and \(m\) the number of strictly complex roots of a polynomial, then the number of strictly triplex roots is given by the formula \(n(n+m) – n\).
This formula implies, perhaps disappointingly, that if a polynomial has no real roots, then it has no triplex roots, either. So the triplex numbers are not any “better” than the real numbers when it comes to solving polynomial equations.
These facts are not so obvious at first. However, one can show that the triplex numbers are actually structurally identical to another number system which is much easier to work with, namely \(\mathbb{R} \times \mathbb{C}\). Counting triplex roots of polynomials thus amounts to counting the possible ways that one can arrange real and complex roots in an ordered pair.
In this paper, I describe the map between \(\mathbb{T}\) and \(\mathbb{R} \times \mathbb{C}\) explicitly, and prove some properties of the triplex numbers, including the formula for counting triplex roots of polynomials mentioned above.
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