A variable with a^2=0

Question

I want to do something simple.

I want to define a variable such that a^2=0, so that say:

Exp(a) = 1+a

And any other function automatically f(a)=1+f'(0)a.

Is it possible to do that in Mathematica?

Answer 1

I guess what you need is

1 + Exp[a] + f[Exp[b]] /. Exp[x_] -> 1 + x

2+a+f[1+b]

Edit:

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for your comment:

a^3 + a^2 /. a^3 -> a^2 + 1

works as you wish.

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f[a] /. a -> 0

Answer 2

Another using [Full]Simplify method

a^2 + 1 // Simplify[#, a^2 == 0]&

Answer 3

I suppose you could do:

Power[a, n_] /; n >= 2 ^:= 0

(using UpSetDelayed)

a
(* a *)
a^2
(* 0 *)
Series[Exp[a], {a, 0, 10}] // Normal
(* 1 + a *)