TITLE: Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?
QUESTION [37 upvotes]: I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an example of a nonstandard differentiation:

The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity... the typical method from Newton through the 19th century would have been simply to discard the $dx^2$ term.

I've never heard anything like this before, and really find it fascinating that Newton's method was to define the relation $dx^2 = 0$. If we actually formalize the above structure by taking $\mathbb{R}$ and adjoining an element $dx^2 = 0$ to it, we get the "dual numbers," isomorphic to the quotient ring $\mathbb{R}[x]/x^2$. I'd seen some things about how this algebra plays into automated differentiation algorithms for some computer software systems, but I've never heard anything about Newton directly working in this algebra. So I have a few questions:

Does anyone have more historical information on the way that Newton performed differentiation, and its relation to the dual numbers?
Does anyone know how effectively real analysis can be formalized with the dual numbers? Does the resulting system play nice enough to develop all of the important modern results?
If we start with $\mathbb{C}[x]/x^2$ instead, can we likewise develop complex analysis?

Since this idea is so simple, I'm very curious how powerful it is. I'm also curious if it has any major drawbacks too, since I'm not sure why anyone would mess with the foundational baggage involved in defining the hyperreals if this simple 2-dimensional real algebra could really do the trick.

REPLY [4 votes]: Duals Numbers, attributed to Eduard Study, are already 
practically used, for example here:
https://github.com/JuliaDiff/DualNumbers.jl
From my point of view division seems not so much a problem, since it fails when ordinary division would also fail. Here is a set of arithmetic operations defined for duals, I am writing (x,y) instead of x+εy:
$$-(x,y) = (-x,-y)$$
$$(x,y)+(z,t) = (x+z,y+t)$$
$$(x,y)*(z,t) = (x*z,x*t+y*z)$$
$$\frac{(x,y)}{(z,t)} = (\frac{x}{z}, \frac{y*z-x*t}{z^2})$$
I guess the claim that duals cannot be used to define derivative, stems from a confusion with Jerome Keislers standard part. He writes translated to dual equations the following, and division is exactly the problem:
f((x,h)) - f((x,0))
------------------- = (f'(x), e) /* doesn't work */
      (0,h)

But if we use the hypothesis:
f((x,y)) = (f(x), f'(x)*y)

We then find the following by using this hypothesis and the
aforementioned arithmetic operations:
f((x,h)) - f((x,0))
------------------- = (0, f'(x))  /* works */
      (h,0)

And if this isn't convincing enough, we can also use the
hypothesis to show, that duals reflect the chain rule:
f(g( (x,1) )) = f( (g(x), g'(x)) )

              = ( f(g(x)), f'(g(x))*g'(x) )

Hyperduals are an extension of duals where second or higher 
order derivatives can be also calculated. 
But currently I rather would wish for duals that can compute 
f(x+) and f(x-) for me, i.e. left and right derivative. 
Currently experimenting.