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)

[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)

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