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I encountered the concepts of split-complex and dual numbers through these videos, and am struggling to understand the reason/motivation for defining them. The argument presented in these videos is that they are similar to introducing imaginary numbers, but this doesn't seem very convincing to me; since imaginary numbers allow solutions to $x^2=-1$, but $x^2=1$ and $x^2=0$ already have solutions in real numbers, so what's the point of introducing new non-real numbers that satisfy these equations?

Integreek
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  • There's plenty of information about this on Wikipedia: https://en.wikipedia.org/wiki/Split-complex_number, https://en.wikipedia.org/wiki/Dual_number – Qiaochu Yuan Jul 02 '25 at 03:46
  • @QiaochuYuan yes, I have gone through those pages, but still the existence of split-complex and dual numbers seem very counter-intuitive to me. Like I understand imaginary numbers help us to solve $x^2=-1$ as it has no real roots, but I can’t understand why someone would want solutions to $x^2=1$ other than $x=\pm1$ and $x^2=0$ other than $x=0$? I could have never thought of this by myself. – Integreek Jul 02 '25 at 03:49
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    For dual numbers see https://math.stackexchange.com/questions/2894248/what-is-the-motivation-for-the-development-of-dual-numbers

    Automatic differentiation is a very practical application.

     – Cheerful Parsnip Jul 02 '25 at 04:16
  • You do not need a motivation for mathematical facts to exist. Are you asking what are their uses or what? Dual numbers are used in automatic differenciation, for instance, and split-complex in interval arithmetic. – Anixx Jul 02 '25 at 05:47
  • @Anixx I was asking what was the thought process behind defining them. – Integreek Jul 02 '25 at 06:06
  • @Integreek complex numbers, dual numbers and split-complex numbers are the only commutative and associative 2-dimensional algebras over reals. Is this enough? – Anixx Jul 02 '25 at 06:08
  • @Anixx sorry I don’t understand your point, is it possible to put it in simpler words(I am a high school graduate)? – Integreek Jul 02 '25 at 06:09
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    @Integreek in simple words, there are 3 2-dimensional systems like this, that's why they are studied. – Anixx Jul 02 '25 at 06:12
  • Note that split-complex numbers represent algebraically a 2D pseudo-Euclidian complex plane, while dual numbers permit to deal with nilpotent objects in algebraic terms. – Abezhiko Jul 02 '25 at 07:15

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Complex numbers developed as an essential instrument for solving polynomial equations with no real solutions. An illustrative example is $x^2 + 1 = 0$. Since squaring a real number cannot produce a negative result, the introduction of $i^2 = -1$ becomes necessary. Thus, a complex number is $a+i b$ with $i^2=-1$. In contrast, dual numbers $a + \varepsilon b$ with $\varepsilon^2 = 0$ and double numbers $a + \jmath b$ with $\jmath^2 = 1$ were developed from a geometric perspective, and their purposes bear no relation to solving polynomials. As you have noted, they are in no way necessary for finding solutions to polynomial equations. I suggest you read "On the three types of complex number and planar transformations" paper by J Rooney.

Feel free to define whatever you like and create your own algebraic structure. If it proves useful, others will adopt it; if not, it might just end up in academic papers. Dual number algebra has numerous practical applications in robotics and computer graphics, particularly in the context of dual quaternion algebra.

CroCo
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