My new friend showed me with an example: the number 2019 is 2(6!)+4(5!)+4(4!)+2!+1. Or more formally, it’s 2(6!)+4(5!)+4(4!)+0(3!)+1(2!)+1(1!)+0(0!). (That last term is a little silly, but we’ll want it later.) Whew, that’s a lot of digits and punctuation marks! The exclamation point is of course the factorial symbol, not an expression of excitement about numbers. For a positive integer n, n! is the product of the integers 1 through n.
To write a number factoradically (or in the factorial number system, as some people call it), you express it as the sum of multiples of factorials with the rule that you can’t use a coefficient larger than n for the n! term. That is, you are not allowed to write the number 6 as 3(2!). You’d have to write it as 1(3!) or just 3!.