Talk:Dual number

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I added a division section, showing how dual numbers can be divided. I'll add other calculation sections later on. I want to add this exponentiation stuff to the page, but I can't find an independent source for it. Anyone know of one?

Exponentiation

{(a+b\varepsilon)^{c+d\varepsilon}}
= {a^{c+d\varepsilon}(1+{b\varepsilon \over a^{c+d\varepsilon}})^{c+d\varepsilon}} = {a^c a^{d\varepsilon}(1+{b\varepsilon \over a^c a^{d\varepsilon}})^{c+d\varepsilon}}
= {a^c(1 + \varepsilon ln(a)d)(1+{b\varepsilon \over a^c})^{c+d\varepsilon}} = {a^c(1 + \varepsilon ln(a)d)(1+{b\varepsilon \over a^c(1 + \varepsilon ln(a)d)})^{c+d\varepsilon}}
= {a^c(1 + \varepsilon ln(a)d)(1+{b\varepsilon \over a^c})^{c+d\varepsilon}}
= {a^c(1 + \varepsilon ln(a)d)({e^{b\varepsilon \over a^c}})^{c+d\varepsilon}} = {a^c(1 + \varepsilon ln(a)d)e^{{b\varepsilon \over a^c}(c+d\varepsilon)}} = {a^c(1 + \varepsilon ln(a)d)(e^{{b\varepsilon c \over a^c}}e^{b\varepsilon d\varepsilon \over a^c})}
= {a^c(1 + \varepsilon ln(a)d)((1 + {bc\varepsilon \over a^c})e^0)}
= {(1 + \varepsilon ln(a)d)(a^c + bc\varepsilon)} = {a^c + bc\varepsilon + \varepsilon ln(a)d a^c + \varepsilon ln(a)d bc\varepsilon}
= {a^c + \varepsilon(bc + ln(a)da^c)}

Which is definitely a dual number when a is greater than 0.

There's not really a need to explicitly write out the expressions for operations such as exponentiation. For any binary function on the reals, f, the natural extension to dual numbers is given by f(a+be,c+de) = f(a,c)+(b f1(a,c)+d f2(a,c))e, where f1 and f2 are the two partial derivatives of f with respect to its arguments. Sigfpe 01:02, 14 December 2006 (UTC)

I would like to know which Slavic languages in addition to Slovene use dual number.

I believe Slovenian and Sorbian are the two and only Slavic languages using the dual number. BT 18:07 23 Jun 2003 (UTC)

Contents

[edit] References

I have removed the reference to Clifford since the assertion given was not true when I checked the text of his article. The dual numbers are well-recognized by this name, but the origin of this convention has not been provided. There is a 1906 reference to Joseph Grunbaum given at Inversive ring geometry#Historical notes.Rgdboer 23:22, 17 August 2006 (UTC)

[edit] Category "Supernumber"?

I just saw the category entry "Supernumber", which is a term I've just now seen for the first time - can someone familiar with this concept elaborate or give reference(s)? I've recently rewritten and expanded the hypercomplex number article, which should include all algebraic systems with dimensionality that are commonly refered to as "numbers". If there's a distinct "supernumber" program that's not captured there, it should be added. Thanks, Jens Koeplinger 12:30, 25 August 2006 (UTC)

[edit] Images

It would be nice to have a picture or two to illustrate dual numbers (assuming that's even possible). As it's written now, I can't quite grasp what a dual number is from the first couple of paragraphs. (Actually, I can't grasp the concept at all, but I might just be having a slow day today.) — Loadmaster 21:27, 17 April 2007 (UTC)

Yes, graphics would be helpful. Its really the cartesian plane with the multiplicative embellishment. A figure might accent the "circle" x = -/+ 1, the slope y/x , and the product by examples. Problems enter when pushing to close to C with its Euclidean fixity. For instance, z z* = xx for dual numbers, so the "modulus" fails to separate points on lines x = const. Also, the product on the "circle" calls for adding slopes, a dangerous suggestion for beginners in analytic geometry to see. *** Graphics would also be a help at split-complex number. Rgdboer 22:13, 29 April 2007 (UTC)

[edit] Division

It seems like division isn't explained fully. In particular, it seems like there's a special case for when both real parts are zero:

\frac{0+b\epsilon}{0+d\epsilon} = \frac{b}{d}.

Is that right? —Ben FrantzDale 03:37, 5 May 2007 (UTC)

No, if w and z are dual numbers, the quotient w /z means w z −1 , and the inverse of z exists only when its real part is non-zero.Rgdboer 22:09, 7 May 2007 (UTC)

[edit] Silly Question

So, being that both of them can be expressed as 2x2 matrices, can one do math on both complex numbers and dual numbers together? Maybe even split-complex numbers...

I know they are different number systems altogether pretty much, but was just wondering if the math would work out or if there was any way to combine complex/dual numbers or how they would work with eachother.

If I'm right that the 2x2 matrices are not compatible with eachother, then would a 2x2 matrix with each of the four numbers being a 2x2 matrix themselves work? If so, would it be a matrix representing a complex number with four dual numbers inside or vice versa? 71.120.201.39 19:56, 11 May 2007 (UTC)

Given the nature of matrix multiplication (that you can apply the same algorithm to blocks of the matrix), this should work. That is,
\begin{bmatrix} a+b i & c+d i \\ 0 & a+ bi\end{bmatrix}
should be equivalent to
\begin{bmatrix} a & -b & c & -d \\ b & a & d & c \\ 0 & 0 & a & -b \\ 0 & 0 & b & a \end{bmatrix}.
So yes. —Ben FrantzDale 20:35, 11 May 2007 (UTC)
Refer to Real matrices (2 x 2) for the breakdown of the general 2 x 2 real matrix into one of the three types of complex number: ordinary, split, or dual.Rgdboer 23:07, 12 May 2007 (UTC)

[edit] Diagonal Matrix Elements

Should the diagonals of the matrix representation be (a / sqrt(2)) rather than a? The way it's written we would have (a + 0e)^2 == 2 * a^2 for pure real numbers.

[edit] Moebius transformations and parabolic rotations

I would propose to add to Geometric section (after "multiplication" rotations were described as shears) the following passage:

Less trivial and more "parabolic" rotations of dual numbers can be obtained by a usage of Moebius transformation, see arXiv:0707.4024

By the way, the cited paper contains some pictures related to dual numbers, let me know if you would like to use them in the Wiki-article. V.V.Kisil 20:44, 20 August 2007 (UTC)

The extra parabolic rotations are developed from the concept of "Galilean angle" in Isaak Yaglom's book you cite in the bibliography of "Inventing a wheel, the parabolic one". I agree that the article would be improved by expanding on this idea. However, the Galilean invariance article has a more physical bent, and these rotations are likened to parabolic particle trajectories in Yaglom's text. The idea is at home in both places, I'd just been thinking of the other first. Your "Inventing a wheel" article is clear enough (at the outset) for beginning students, a real credit to your composition. Rgdboer 23:17, 22 August 2007 (UTC)