= Generalized Cantor normal form
{c}
{title2=$\alpha=\sum_{i<m}\rho^{\beta_i}\xi_i$}
For every <ordinal> base $\rho>1$, a nonzero ordinal has a unique finite <ordinal addition> expansion with strictly decreasing exponents $\beta_i$ and coefficients $0<\xi_i<\rho$. Choose the largest power at most the remaining ordinal and apply the <ordinal division algorithm>. The next remainder has smaller leading exponent, so <well-foundedness> forces termination. Each lower tail is smaller than its preceding power; uniqueness of division consequently fixes the exponent, coefficient and tail successively. For $\rho=\omega$ this is <Cantor normal form>.
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