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We call i a dual number if it satisfies $$i^2=0$$

In matrix representation, it's $$ i^2=\begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix}^2=0 $$

Now I have a number satisfying $$ j^3=0 $$ where $$ j \ne j^2 \ne j^3 $$ In matrix representation, it's

$$ j^3=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix}^3=0 $$

But what do we call for this number j? If I recall correctly, it's not a higher dimensional dual number as its basis is only univariate. Also, I have seen its application in automatic differentiation.

Thanks!

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    If an element $a^n=0$ for some natural $n$, then $a$ is said to be nilpotent. I'm not sure of a specific case when $n=3$. – 2017-02-23
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    Just something come up in my mind: How about the name for the case i^n=1 and i^n=-1? – 2017-02-23

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A square matrix $A$ is clled nilpotent if $A^n=0$ for some positive integer $n$

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    Yes nilpotent is indeed the correct name. Sorry for being greedy, do you know the name for the case i^n=1 and i^n=-1 too? – 2017-02-23