Dual Numbers......help!

[BoxAngle]

I consider myself not too bad at maths but I cannot grasp dual numbers. In wikipedia it says they are expressions of the form a+bε (ε is greek Epsilon), a and b are real numbers and ε is a symbol taken to satisfy ε²=0. Now surely the square root of 0 is 0 so ε must be 0 and that leaves a+b! I don't get it! Is there a mathmetician on here who could explain this in easy to understand terms? Thanks, here's the wikipedia page:

https://en.wikipedia.org/wiki/Dual_number

 

[BoxAngle]

PS I hope this is the correct section I could not see one for maths.

[Prudie]

I'm not bad at maths but I've never heard of dual numbers. The main thing, and it says it in Wiki, this ε must be an imaginary number but it says it cannot be 0. You can't actually square zero or divide by it for that matter so no it's not 0x0 =O.

 

It's a bit like the more well known imaginary number i, that's the square root of minus 1 and that can't actually exist in our number system either but it's still used as the base for complex numbers etc.  

[BoxAngle]

Thanks, yes I know about complex numbers i being √-1 etc but they talk about hyper complex numbers and the explanation for them is even stranger!. Can someone explain this in a way that mere mortals can comprehend?

[Raving-Mod]

Afternoon BoxEdge, welcome to AnswerBank!

There are people on here with all sorts of interests and coincidentally, there is a member of Mensa who is interested in the same area but unfortunately he's temporarily unavailable today.

I don't know anything about "dual" in the context you're asking about but given it can mean having more than one of something at the same time, I'm sure he could tell you all about it.

[BoxAngle]

Thanks, it's BoxAngle.

[Raving-Mod]

Apologies BoxAngle, I'm Too Tired Today.

[beso]

Prudie is not quite correct. While it is not possible to divide by zero, it certainly can be squared.

The epsilon component in these numbers is anoher type of imaginary number that is defined as square root of zero that is not zero itself.

The concept was introduced by William Clifford who was a very important pioneering mathematician. He did a lot of abstract work in the fields non-Euclidean geomery, higher order dimensions and founded geometric algebra. He built on the work of other mathematical geniuses like Bernhard Riemann.

Not much of what he did had immediate practical applications for everyday life at the time but contributed to the groundwork of mathematics that subsequetly became extremely important, including the calculations involved in Relativity.

Remarkably, in 1876, forty years before Einstein fully described gravitation in General Relativity, Clifford suggested that gravity was due to a curvature of space and that matter is a manifestation of curvature in a space-time manifold.

Unfortunately he died aged just 33, in 1879, the same year as James Clerk Maxwell who first described Electromagnetic Radiation. Both men had laid important foundations for Einstein's work but unfortunately did not live to witness its fruition.

https://en.wikipedia.org/wiki/William_Kingdon_Clifford

[BoxAngle]

Thanks beso that helps. So ε repesents √0 in a similar way to i represents √-1, got it.

[ClareTG0ld]

I am not sure that beso is quite correct either, although I much prefer that answer to prudie's. 

First of all, it's unhelpful to use the word "imaginary", although it's altogether too late to do anything about that. The name has stuck, but it's horribly misleading. Imaginary numbers are no less meaningful than any other number, and merely represent a different concept. A number i such that i^2 = -1 very much can exist: we just declare that it exists, and then it does, on the same footing as 1 or 2 or 3.14159...

Secondly, it's arguably unhelpful to focus on the "square root" definition of such numbers. In particular, for dual numbers, ε²=0, it follows almost instantly that 0 has infinitely many "square roots" - because (2ε)² = 4ε² = 0, etc etc., while on the other hand numbers like -1 still don't have any "square roots" over the dual numbers, so you get this weird situation where the amount of "square roots" a number has is either 0,2, or infinity. 

A (hopefully) far more helpful way to think about all this is geometrically. I'll expand on what I mean later, but a clue comes from the alternative names of circular, hyperbolic and parabolic numbers for what are usually known as complex, split-complex/double, and dual numbers respectively.