# Why/when is it useful to partition an integer into a set of smaller integers?

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.

Thanks.

• Well for one thing, their analysis, for example counting how many partitions an integer has, predicting the general growth rate, etc... tends to be some of the deepest math, fusing multiple areas of complex analysis, number theory, algorithm design, etc... Aug 6, 2016 at 5:22
• Seconding the comment by @frog. To get some idea of how much neat math is connected with partitions, you could start with en.wikipedia.org/wiki/… Aug 6, 2016 at 6:57