# Anti-dual numbers and what are their properties?

• A
• Anixx
In summary, the conversation discusses a proposed system of numbers called "anti-dual numbers" that is based on a unit curve defined by a reciprocal function. The system defines addition and multiplication in a way that makes them commutative and associative, and it also introduces the concept of divisors of infinity. However, it is unclear what algebraic and analytic properties this system has, and whether it is free from zero divisors. These questions should be investigated before considering the system as a viable option.
Anixx
In [this post][1] user William Ryman asked what would happen if we try to build "complex numbers" with shapes other than circle or hyperbola in the role of a "unit circle".

[Here][2] I proposed three shapes that could work. The common principle behind them being
that if the unit curve is defined as ##r=r(\phi)##, an arbitrary point, corresponding to a 2-dimensional number on the plane ##z=(a,b)## is characterized by angle ##\alpha(z)=\text{atan2}(b,a)##, magnitude ##M(z)=\frac{\sqrt{a^2+b^2}}{r(\alpha(z))}## and argument ##\operatorname{arg}(z)=\int_0^{\alpha(z)} r(\phi)^2 d\phi##, twice the area of a sector between the radius-vector and ##x## axis.

The addition of numbers is defined element-wise as ##(a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2)##.

The multiplication is defined in such a way that the arguments are added and magnitudes are multiplied: ##\operatorname{arg}(uv)=\operatorname{arg}(u)+\operatorname{arg}(v)## and ##M(uv)=M(u)M(v)##.

These definitions make addition and multiplication commutative and associative.

So, I decided to consider the number system based on the following equation for unit curve: ##r=|\cos\phi|##. This function is reciprocal to the function defining dual numbers, so I called the system "anti-dual numbers".

The expressions for modulus and argument of a number ##z=(a,b)## thus would be:

##M(z)=\frac{a^2+b^2}{a}##

##\arg z=\frac{1}{2} \left(\frac{a b}{a^2+b^2}+\arctan \left(b/a\right)\right)##

These expressions are valid for the first quarter of the plane, in other quarters we should account that negative modulus corresponds to a shift of argument by ##\pi/2## (not by ##\pi## as in complex numbers!), that's why we have to add the functions arg and mod which are intended to represent the canonical form.

The expression for the angle of direction of radius-vector as a function of argument is [from this post by Tyma Gaidash][5]:

##\phi (z)=\arcsin\sqrt{I_{\frac{4 \arg z}{\pi }}^{-1}\left(\frac{1}{2},\frac{3}{2}\right)}##

This expression involves [inverse beta regularized][6] function.

The code below for Mathematica system provides functions for determining argument and modulus of a number ##(a,b)##, determining Cartesian coordinates based on modulus and argument as well as a function that multiplies two numbers given in Cartesian coordinates.

 ar[a_, b_] := 1/2 ((a b)/(a^2 + b^2) + ArcTan[b/a]) m[a_, b_] := (a^2 + b^2)/a arg[a_, b_] := ar[a, b] + Pi/2 Sign[b] HeavisideTheta[-m[a, b]] mod[a_, b_] := Abs[m[a, b]] \[Phi][A_] := ArcSin[Sqrt[InverseBetaRegularized[4 A/Pi, 1/2, 3/2]]] // FullSimplify angle[A_] := Piecewise[{{\[Phi][A], 0 <= A < Pi/4}, {\[Phi][A - Pi/4] + Pi/2, Pi/4 < A <= Pi/2}, {-\[Phi][-A], -Pi/4 < A < 0}, {-\[Phi][-A + Pi/4] - Pi/2, -Pi/2 < A < Pi/4}}] X[m_, A_] := m Cos[angle[A]] Abs[Cos[angle[A]]] Y[m_, A_] := m Sin[angle[A]] Abs[Cos[angle[A]]] Multiply[{a1_, b1_}, {a2_, b2_}] := {X[m[a1, b1] m[a2, b2], ar[a1, b1] + ar[a2, b2]], Y[m[a1, b1] m[a2, b2], ar[a1, b1] + ar[a2, b2]]}

Example:

a := -1; b := -1
arg[a, b]
mod[a, b]

Output:

1/2 (1/2 + Pi/4) - Pi/2
2

Multiplication:

Multiply[{1, 1}, {1, 1}] // N

Output:

{-1.10363, 1.78788}

----------------------

That said, I wonder, what algebraic and analytic properties this system has? It seems to be a 2-dimensional hypercomplex commutative numbering system that is not isomorphic to complex, split-complex and dual numbers.

One interesting feature of this system is existence of divisors of infinity because ##(0,1)(0,1)=\infty## (multiplication by divisors of infinity cannot be handled by the provided code though). This makes the system not closed under multiplication unless an improper element ##\infty## is attached.

What else can be said about the system? [1]: https://math.stackexchange.com/q/4459901/2513
[2]: https://mathoverflow.net/questions/423657/lemniscate-numbers-and-others-what-would-be-the-properties
[3]: https://i.stack.imgur.com/R3dRX.png
[4]: https://en.wikipedia.org/wiki/Atan2
[5]: https://math.stackexchange.com/a/4390291/2513
[6]: https://mathworld.wolfram.com/RegularizedBetaFunction.html

Anixx said:
That said, I wonder, what algebraic and analytic properties this system has?
It is you who should check this before you propose something new. Something which by the way is prohibited to discuss on PF per our rules.

My first thought was:
There is more than one irreducible quadratic real polynomial, e.g. ##x^2+x+1.##
and my second was:
Is that thing free from zero divisors, i.e. does it contain ##\mathbb{R}##?

Anyway. We do not discuss personal speculations, even less if they are intended to shift the workload from the inventor to the reader.

## 1. What are anti-dual numbers?

Anti-dual numbers are a type of hypercomplex number that extends the real numbers with a dual part, which is a number that is the square root of a negative real number. They are denoted as a + bε, where a and b are real numbers and ε is the dual unit.

## 2. What are the properties of anti-dual numbers?

Anti-dual numbers have several properties, including the distributive, associative, and commutative properties for addition and multiplication. They also have a zero element (0 + 0ε) and a multiplicative identity (1 + 0ε). Additionally, they follow the rule of distributivity with respect to multiplication by real numbers.

## 3. How are anti-dual numbers different from dual numbers?

While both anti-dual numbers and dual numbers have a dual part, they differ in the sign of the dual unit. In anti-dual numbers, the dual unit is negative, while in dual numbers, it is positive. This results in different algebraic properties and applications for each type of number.

## 4. What are some applications of anti-dual numbers?

Anti-dual numbers have various applications in fields such as physics, engineering, and robotics. They can be used to represent quantities with both real and imaginary components, such as velocity and acceleration. They are also useful in solving differential equations and in computer graphics for representing rotations and translations.

## 5. How are anti-dual numbers related to other hypercomplex numbers?

Anti-dual numbers are a subset of hypercomplex numbers, which also include complex numbers, dual numbers, and split-complex numbers. They share some properties with these other types of numbers, but also have distinct algebraic properties and applications. For example, anti-dual numbers are commutative under multiplication, while split-complex numbers are not.

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