I recently finished discrete math and am interested in some of the more theoretical math concepts. We didn't really cover number theory at all in my class though. Ran across this statement online:
If $a_1 + a_2 + · · ·+a_j = n$, where $a_1, a_2, . . . , a_j$ are positive integers with $a_1 ≥ a_2 ≥ · · · ≥ a_j$ , then we say that $a_1, a_2, . . . , a_j$ is a partition of the positive integer n into j positive integers.
I understand what is happening here, but why it is useful to do this? This seems to be simply a fancier way to say break the number apart for easier arithmetic/etc as taught in grade school. Clearly there is much more to it but I'm not clear why it would be useful from a math standpoint, which is where the insight really is.