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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

${\displaystyle {(a+b\varepsilon )^{c+d\varepsilon }}}$

${\displaystyle ={a^{c+d\varepsilon }(1+{b\varepsilon \over a^{c+d\varepsilon }})^{c+d\varepsilon }}={a^{c}a^{d\varepsilon }(1+{b\varepsilon \over a^{c}a^{d\varepsilon }})^{c+d\varepsilon }}}$

${\displaystyle ={a^{c}(1+\varepsilon ln(a)d)(1+{b\varepsilon \over a^{c}})^{c+d\varepsilon }}={a^{c}(1+\varepsilon ln(a)d)(1+{b\varepsilon \over a^{c}(1+\varepsilon ln(a)d)})^{c+d\varepsilon }}}$

${\displaystyle ={a^{c}(1+\varepsilon ln(a)d)(1+{b\varepsilon \over a^{c}})^{c+d\varepsilon }}}$

${\displaystyle ={a^{c}(1+\varepsilon ln(a)d)({e^{b\varepsilon \over a^{c}}})^{c+d\varepsilon }}={a^{c}(1+\varepsilon ln(a)d)e^{{b\varepsilon \over a^{c}}(c+d\varepsilon )}}={a^{c}(1+\varepsilon ln(a)d)(e^{b\varepsilon c \over a^{c}}e^{b\varepsilon d\varepsilon \over a^{c}})}}$

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

${\displaystyle ={(1+\varepsilon ln(a)d)(a^{c}+bc\varepsilon )}={a^{c}+bc\varepsilon +\varepsilon ln(a)da^{c}+\varepsilon ln(a)dbc\varepsilon }}$

${\displaystyle ={a^{c}+\varepsilon (bc+ln(a)da^{c})}}$

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) reply

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

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

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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) reply

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) reply

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) reply

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) reply

ε is kind of like a tiny real number. Having (2+3ε)(4-5ε) = 8+2ε is like having 2.0003×3.9995=8.000199985, but you only care about precision to the 0.0001 place, so you round it to 8.0002. DanBishop ( talk) 07:19, 23 May 2009 (UTC) reply

The epsilon (ε) in this article has nothing to do with "a tiny real number" such as found in the article epsilon-delta argument. Instead this epsilon represents a unit vector perpendicular to the real line at the origin. Therefore it generates a plane out of the line and ε, points in the plane being represented by dual numbers x + y ε. An old phrase for this generation of a new direction from the real line is "an imaginary number". Just as the imaginary number i generates the ordinary complex plane with its algebra based on ii = − 1, so in this article the "imaginary number" ε generates the dual number plane with its algebra based on ε² = 0. So there is nothing tiny about ε except its square, which is less than tiny. Rgdboer ( talk) 21:07, 24 May 2009 (UTC) reply

"a unit vector...": this should be "the unit distance along a line orthogonal to the complex number plane" (or to the quaternion skew field, etc.). ε is not an imaginary number, it is an abstract number. It might be easier to think of rε, the product of real r and dual ε, as a collapsing sphere with radius proportional to r. Then x+ε is a collapsing unit sphere surrounding x. — Preceding unsigned comment added by 2601:C:A780:961:2C3:51FF:FE74:7AF ( talk) 21:30, 15 February 2014 (UTC) reply

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:

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

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

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) reply

I just corrected the section because id contained a clear mistake: Division is not defined if the equation

${\displaystyle ax=b}$

has no solution x but also if it has no unique solution. Otherwise, zero could be divided by itself because 0·x = b "has a solution" (rather infinitely many, in fact the solution space is entire given algebraic structure) if b=0.-- Slow Phil ( talk) 14:01, 10 January 2011 (UTC) reply

I don't agree with your change or with your explanation. Take two dual numbers a and b with:

${\displaystyle a:=0+rd,}$

${\displaystyle b:=0+sd,}$

${\displaystyle r,s\in \mathbb {R} ,r\neq 0,}$

${\displaystyle dd=0.}$

Then for

${\displaystyle ax=b}$

there exists a unique solution:

${\displaystyle x=s/r}$.

This is the essence of what you just changed. No? Thanks, Jens Koeplinger ( talk) 22:12, 14 January 2011 (UTC) reply

Nevermind, I see what you mean. Thanks, Koeplinger ( talk) 15:54, 15 January 2011 (UTC) reply

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) reply

Given the nature of matrix multiplication (that you can apply the same algorithm to blocks of the matrix), this should work. That is,

${\displaystyle {\begin{bmatrix}a+bi&c+di\\0&a+bi\end{bmatrix}}}$

should be equivalent to

${\displaystyle {\begin{bmatrix}a&-b&c&-d\\b&a&d&c\\0&0&a&-b\\0&0&b&a\end{bmatrix}}}$.

So yes. —Ben FrantzDale 20:35, 11 May 2007 (UTC) reply

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) reply

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.

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) reply

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) reply

There are some primes missing. This is because the source markup contains primes quite with primes. I don't know how to escape primes within primes so maybe someone else could fix it. Sigfpe ( talk) 15:51, 25 July 2008 (UTC) reply

In algebraic geometry dual numbers over fields other than the reals are useful to define an infinitesimal deformation (essentially a flat family over the ring of dual numbers over the field of interest). 69.234.22.86 ( talk) 02:49, 10 August 2008 (UTC) reply

Complex numbers have a polar form that simplifies multiplication and division: (r cis θ) × (s cis φ) = rs cis (θ + φ). Analogously, If we define P(a, β) = a + aβε, then P(a, β) × P(c, δ) = (a + aβε)(c + cδε) = ac+ac(β+δ)ε = P(ac, β+δ). Is there a standard notation for this? DanBishop ( talk) 06:50, 23 May 2009 (UTC) reply

The cis notation, not uncommon but by no means standard, expresses the exponential function as in Euler's formula:

cis(θ) =${\displaystyle \ \exp(i\theta )=\cos \theta +i\sin \theta }$.

In dual numbers one also has

${\displaystyle \exp(\theta \epsilon )\ \exp(\phi \epsilon )=\exp((\theta +\phi )\epsilon )}$.

Extending the cis notation into other number rings beyond the ordinary complex plane is not advisable; the study of transformation groups and their "infinitesimal algebras" by means of Lie theory refers to the exponential function at every juncture. Rgdboer ( talk) 20:58, 24 May 2009 (UTC) reply

Slow Phil - you've done an that concerns me. Let me go line by line (rather than rudely reverting it).

In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring Rx] by the ideal generated by the polynomial x²,

Rx]/x².

The image of x in the quotient is the "imaginary" unit j.

The article here is about the dual numbers, I don't see why a statement about the split-complex numbers should be here. If it would relate to dual numbers then you would also have to clarify that ${\displaystyle x^{2}:=0}$ is your ideal; because the split-complex, complex, and dual numbers all can be understood as a quotient ring ${\displaystyle R[x]/x^{2}}$. However, the next section "Generalization" already states that, so I see no need to duplicate this information.

With this description, it is clear that the dual numbers form a commutative ring with characteristic 0.

I don't like wording "... it is clear that ...". To me, it is already clear from the lead-in that the algebra is commutative, that it is a ring, and that its characteristic is zero. I see that the article in the current form doesn't state the system's characteristic yet, so it would seem sufficient to add a simple note somewhere, "... with characteristic 0". The reader then can follow the link to find out more about what that means. This would be informative, and would not make a judgment whether this property should or should not be clear to the reader.

Moreover if we define scalar multiplication in the obvious manner, the dual numbers actually form a commutative and associative algebra over the reals of dimension two.

Since the dual numbers are defined on a vector space over the reals, this includes scalar multiplication already. Again, I don't like to judge whether something should be "obvious" to the reader. Instead of adding this paragraph, I suggest adding (and referencing) vectors space, for example in the lead in: Change "The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers." into "The dual numbers are a two dimensional vector space over the reals, equipped with a commutative, associative, and unital vector multiplication." Something alone these lines. Maybe the entire lead-in needs to be straightened out.

The algebra is not a division algebra or field since the imaginary elements are not invertible. In fact, all of the nonzero imaginary elements are zero divisors (also see the section "Division").

My suggestion here would be to maybe add to the lead-in: "The dual numbers are not a division algebra, since multiplication is generally not invertible" or similar. Then, the existing section "Division" in the article should probably be moved up, either before or after "Geometry". Personally I would put it before "Geometry", but others may feel differently. On a sidenote, I suggest against using emphasis (here: "not").

Overall, it looks as if the entire article could use a bit of smoothing out; but not by adding another section with mostly duplicate information. Thanks, Jens Koeplinger ( talk) 19:46, 29 January 2011 (UTC) reply

PS: I've just read over the German version of this article, and see where your edit is coming from. Thank you for working on merging material across the languages! Koeplinger ( talk) 20:01, 29 January 2011 (UTC) reply

In addition to Koeplinger's observations, the indeterminate of a polynomial ring is usually denoted with a capital X or Y. Fixed this and some other slips. Rgdboer ( talk) 00:50, 30 January 2011 (UTC) reply

I note that another name for dual numbers, parabolic numbers, has been added. I fail to readily find a mention thereof with a Google search. Is this name notable enough to mention? — Quondum 05:03, 5 November 2012 (UTC) reply

Perhaps, if we can find a source. Tkuvho ( talk) 16:35, 3 December 2012 (UTC) reply

Here is one. Double sharp ( talk) 15:31, 17 March 2016 (UTC) reply

Notable enough to mention? I dont even understand what that inquiry means. If indeed you have found an alternative nomenclature then yes, of course it should be mentioned. This is an encyclopedia, after all, and people are coming here from a multitude of sources and dialects. Why would you deliberately remove an alternative phrasing some might use? 50.125.86.70 ( talk) 20:35, 20 July 2019 (UTC) reply

The following lines have been removed as they require a demonstration or reference:

Exponentiation of dual numbers follows the general rule:

${\displaystyle (a+b\varepsilon )^{c+d\varepsilon }=a^{c}+\varepsilon (b(ca^{c-1})+d(a^{c}\ln a))}$

The subject can be discussed in this Talk space. Rgdboer ( talk) 03:12, 23 December 2013 (UTC) reply

I haven't gone through the detail, but since the exponential mapping has such a simple form, the inverse mapping (the logarithm) seems straightforward to define, and presumably from this exponentiation (only for a > 0). So a demonstration should be easy, if it is considered worthwhile. However, I don't feel a burning need for this to be in the article. — Quondum 04:35, 23 December 2013 (UTC) reply

While the behavior of a nilpotent element might suggest infinitesimals, the analogy collapses when there is more than one nilpotent element, as in Nilpotent#Nilpotency in physics, where there may be multiple anti-commuting nilpotent elements.

Shmuel (Seymour J.) Metz Username:Chatul ( talk) 22:43, 19 December 2014 (UTC) reply

What's wrong with noncommuting infinitesimals? Alain Connes loves them. Tkuvho ( talk) 20:31, 20 December 2014 (UTC) reply

A dual-imaginary number is a number of the form a+bi+(c+di)ε, or equivalently a+bi+cε+dεi. 99.185.0.100 ( talk) 16:31, 10 May 2015 (UTC) reply

The dual numbers can be complexified to what one might call dual complex numbers or complex dual numbers (not dual-imaginary numbers). The question is whether this is notable and should be mentioned in this article, and whether it can be sourced. There are many such algebras that are not notable in their own right, but that are easily found. — Quondum 17:10, 10 May 2015 (UTC) reply

See Eduard Study#Hypercomplex numbers for mention of this system. Rgdboer ( talk) 20:49, 11 May 2015 (UTC) reply

I guess there are different possibilities that fit the description. The semiquaternions (or Study's quaternions) you linked to has iε = −εi, so they are not commutative. Their geometric use is interesting. There is also the commutative version, which would be the complexified dual numbers that I referred to. — Quondum 21:31, 11 May 2015 (UTC) reply

Why are you correcting someones use of terminology when you admit you have no sources and no clue about its usefulness or noteworthiness. Who are you do decide what a concept yet to be published about should be called? 50.125.86.70 ( talk) 20:31, 20 July 2019 (UTC) reply

The section on differentiation does not seem like that much of a surprise, at least to my mind (which is admittedly rather new to this stuff). It seems like ε is acting as simply a sort of first-order approximation to an infinitesimal dx: a "proper" infinitesimal system would have orders of magnitude corresponding to all (dx)ⁿ (integer n), and the dual numbers approximate it with ε² = 0. Thus, messing around with expressions of the form f(x) + ε f'(x) is a good way to recall all those differentiation rules in calculus. (For example, you can get to the product rule by multiplying two such expressions, or the chain rule by considering what happens if you apply a function g on such an expression. Although the latter is not a real proof at all, because it is not really clear to me if there is a single legitimate way to extend a general, not necessarily smooth function from the reals to the dual numbers.) If you defined "k-polydual numbers" which adjoin elements ε_k defined by ε^k
_k = 0, so that a general k-polydual number would be a₀ + Σ^{k − 1}
₁ a
_iεⁱ
_k, you would get similar effects with multiple derivatives, and you'd go further up the Taylor series. If you let k go towards infinity, you'd recover what I think are the infinitesimals of the hyperreal numbers, right? So, if this is really as simple as I think it is, then shouldn't the accent be on why this method of differentiation is useful, instead of a detailed exposé of its properties? Double sharp ( talk) 15:53, 17 March 2016 (UTC) reply

P.S. Even then, the dual numbers don't seem as useful as "real" infinitesimals, because ε is not invertible. I wouldn't be surprised too if the exact nilpotence of ε (as opposed to squaring to some smaller infinitesimal) creates problems with analysis, but I don't yet know enough to say what it might be. So isn't this more to do with infinitesimals than dual numbers, which are relevant in this context solely for being a crude simulacrum of the infinitesimals? Double sharp ( talk) 15:58, 17 March 2016 (UTC) reply

P.P.S. Further, it seems to me that the algebraic derivation of the extension of polynomials to dual numbers is also not that useful. After the algebraic manipulations, you then have to recognise for yourself independently that the terms involving ε add up to form bP'(a)ε. You're not going to get the derivative from this without knowing it beforehand for some other source. And these are polynomials! Why do we need this way of computing derivatives when the power rule is so easy? (If this is an actual application, there must be a reason.) Double sharp ( talk) 16:17, 17 March 2016 (UTC) reply

I would guess that you are being far too narrow and limited in your thinking. The article itself gives many different generalizations, most or all of which have nothing at all to do with differentiation. One example: Galois theory, where you take quotients of the ring of polynomials by some fixed polynomial, rather than just x^2, as in this article. Another, much much more complex example, is the ideas of Grassmanians, which consider not just one such epsilon, but any number of them, anti-commuting, and forming the foundation of superspace. And even so, this is just scratching the surface. Don't get yoked, blinded into thinking that everything is about real numbers and differentiable functions, and then a whole new world can open up for you. 67.198.37.16 ( talk) 02:44, 7 May 2016 (UTC) reply

Oh, and for your continuing amusement and amazement: you might perhaps see practical utility in this: the definition of a tangent bundle can be taken as the sheaf (mathematics) of morphisms of the locally ringed space of differentiable functions into the ring of dual numbers. This is an example of a very abstract statement that does talk about differentiability, and uses it to define the tangent space, which is the general, generic concept of the "first derivative" on a manifold. What you call "k-polydual numbers" up above are called jet (mathematics). 67.198.37.16 ( talk) 03:57, 7 May 2016 (UTC) reply

(Fair enough: I was fairly annoyed IRL at something else when I wrote that, and I guess it spilled over. ^_^) I do indeed find the generalisations you mention a lot more interesting than the way the article currently talks about differentiation, which seems to just take differentiable functions over R as the basis. (So perhaps the article should actually mention all this, because I'm not seeing all these delights there...) Double sharp ( talk) 04:30, 7 May 2016 (UTC) reply

"what links here": The article on superspace links this article, as a "trivial example", the article on differentiable manifold mentions the sheaf-tangent-space construction. I don't really want to start a section of "all the places where dual numbers show up", that risks getting off-track... 67.198.37.16 ( talk) 07:28, 7 May 2016 (UTC) reply

What is ${\displaystyle |\varepsilon |}$ ? Im presuming it should be a value of 1, as taken from context epsilon should be unital. 50.125.86.70 ( talk) 20:38, 20 July 2019 (UTC) reply

I rewrote the section of "algebra representations". While doing so, I got a bit stuck on why 1 has to be represented by the identity matrix. I have to admit that I don't know representation theory, and the article on representation theory only talks about groups. But I think that a homomorphism of unital algebras should always map the identity to the identity. So if our "representation" is understood as a unital algebra homorphism of the dual numbers into M_2(R) which is furthermore injective, then I think we can conclude that 1 must be mapped to the identity matrix. [edit] There is also an article on algebra representations which claims in the introduction that a unit is always represented by the identity map (or identity matrix, if one uses matrices instead of linear operators). -- Svennik ( talk) 12:41, 14 March 2021 (UTC) reply

The dual numbers do indeed provide a baby example of a phenomenon which separates modern algebraic geometry from classical algebraic geometry. The spectrum of the dual numbers provides a baby example of a scheme which is not simultaneously an algebraic variety. It's also got interesting links to differential geometry. The reason the aforementioned user gives for deleting this content is that the explicit term "dual numbers" isn't used in algebraic geometry. Even if true, I don't think this is a good reason. The user should explain why he keeps trying to remove the content, and we can debate his reasons here. -- Svennik ( talk) 22:26, 21 December 2021 (UTC) reply

It is a basic policy that WP:original research is strictly forbidden in Wikipedia. The content that you have restored is original research, since none of the treatises on scheme theory that I know mention explicitely dual numbers. Even if some mention them as the simplest example of a non-reduced ring, they do not discuss them specifically. If you know a standard (modern) textbook that discusses dual numbers specifically and support that content, be free to add it again with explicit references. Without such sources, I will follow WP:OR, and remove again this content, until you provide (if you can) WP:reliable sources. D.Lazard ( talk) 16:57, 22 December 2021 (UTC) reply

Found relevant material. Done. -- Svennik ( talk) 09:01, 27 December 2023 (UTC) reply

I've spoken to algebraic geometers, and the connections between the dual numbers and schemes, the square-zero construction in ring/module theory, etc, are well known to them, and therefore belong in this article. I think somebody can find a more explicit references in the work of Lawvere in Synthetic Differential Geometry.

I'd like to arbitrate this dispute. I suppose my decision to revert Lazard's revert (many months later) counts as edit warring? [edit] See /info/en/?search=Wikipedia:Dispute_resolution_noticeboard#Dual_numbers Svennik ( talk) 19:51, 26 April 2023 (UTC) reply

My feeling is that if there are citations then this information belongs in the article. But if this is a case of 'everybody knows' and is composed from synthesizing several sources then it doesn't belong here. This information is not going to disappear. If there is a real, significant link then there will be a peer reviewed article. At that time we can update this article. OrewaTel ( talk) 22:21, 28 April 2023 (UTC) reply

There is a lot of elementary material that is not currently discussed here, which is probably more useful to most readers than anything about schemes or module theory. – jacobolus  (t) 22:38, 28 April 2023 (UTC) reply

In the Differentiation section is the sentence, "Consider the real dual numbers above." However, there are no numbers above, and this sentence seems to have been added by an IP editor ( 63.82.98.175 on 22:02, 2005 April 28 (UTC)) also without any numbers above. Seems the sentence has lasted all these years since then. I considered removing it myself, but was concerned that maybe I am missing something that everyone else is seeing (thus explaining why it has lasted all these years). Can this sentence be removed? — al-Shimoni ( talk) 02:35, 1 May 2023 (UTC) reply

I believe by "above", the author means the definition given earlier in the article. But this sentence can be safely removed. I tried rewriting those few sentences for clarity. – jacobolus  (t) 18:17, 1 May 2023 (UTC) reply

Since GitHub is a user-generated site it cannot be taken as a reliable source according to WP:UGC. Accordingly, "Beyond Complex Numbers | M.E. Irizarry-Gelpí". meirizarrygelpi.github.io. Retrieved 2024-01-26. has been removed. The term Nilplex number is unique to that author, not likely to be adopted for usage. Rgdboer ( talk) 01:40, 27 January 2024 (UTC) reply