Dual number division

Question

I was reading the Wikipedia article on dual numbers and I read the first part. Then on my own I tried to find how to divide two dual numbers. I came up with: $$\frac{a+b\epsilon}{c+d\epsilon}=\frac{a+b\epsilon}{c+d\epsilon}*\frac{\epsilon}{\epsilon}=\frac {a\epsilon+b*0}{c\epsilon+d*0}=\frac {a\epsilon}{c\epsilon}=\frac {a}{c}$$ but the wiki page says that it's something else. Could somebody please explain why I am wrong?

Answer 1

You want to find $x+y\varepsilon$ such that $$ a+b\varepsilon=(c+d\varepsilon)(x+y\varepsilon) $$ that reduces to $$ \begin{cases} cx=a\\[6px] dx+cy=b \end{cases} $$ Since $$ \det\begin{bmatrix}c & 0 \\ d & c\end{bmatrix}=c^2 $$ you know that $c+d\varepsilon$ is invertible if and only if $c\ne0$ (assuming we're over a field) and, easily, $$ x=ac^{-1},\quad y=(bc-ad)c^{-2} $$

If you try $ac^{-1}(c+d\varepsilon)$ you don't find $a+b\varepsilon$, do you?

Where's the error? You're doing $0/0$. If you could multiply the numerator and denominator by $\varepsilon$, you could as well multiply them by $\varepsilon^2$, just do the same twice.