could be this a general (and useful) rule to operate with ordinal? working on some exercises i found this rule that seems could be very useful to operate with ordinal. is correct?
$(\omega * m + n)*(\omega * p + q) = \omega ^2 * p + \omega * m * n + n$ with $n,m,p,q < \omega$.  * is the ordinary multiplication between ordinals
Why can't I find it in books?
 A: Ordinal multiplication and addition are associative but distribute only on the left, so we have, to begin with
$$ (\omega\cdot m+n)\cdot(\omega\cdot p+q) = (\omega\cdot m+n)\cdot\omega\cdot p + (\omega\cdot m+n)\cdot q $$
In the first of these terms, because $\omega\cdot p$ is a limit ordinal every $n$ is subsumed into the next copy of $\omega\cdot m$ (assuming that $m, p\ne 0$), but in the second one there's a last $n$ that ends up at the end. So we have
$$ \omega\cdot m\cdot\omega\cdot p + \omega\cdot m\cdot q + n $$
However, $m\cdot\omega=\omega$, so this is the same as
$$ \omega^2\cdot p + \omega\cdot mq + n $$
which is not quite what you wrote.
If $m$ and/or $p$ are $0$ we get special results that are not a special cases of this.

Why aren't such rules found in books? Because it is actually really rare that you need to make such concrete computations on even small transfinite ordinals like these. It wouldn't really be worth the trouble to memorize rules for it (and if textbooks included such rules, a certain kind of students would start wasting their time on memorizing them).
