The made-up word "nodulus" is closely related in meaning to the conventional "modulus" of mathematics. So I'll start by describing that, for anyone who hated math so much they kicked it out of their memory as soon as they could possibly get away with it.

The simplest definition of "modulus" is "the
integer remainder after a division problem is done". The word is frequently used with the word "base", which is the divisor in that same division problem.

Therefore, if we take a number like 11 and divide it by, say, 8, the remainder of that division (the part that can't be wholly divided by 8) is 3, and so "3 is the modulus of 11 with respect to the base of 8".

That's all you need to know about a modulus to understand a "nodulus". A nodulus is always associated with the SQUARE of the base. Therefore, in the above example, 11 would be a nodulus with respect to the base of 8, because the square of 8 is 64, and obviously 64 cannot wholly divide into 11.

However, a larger number, say 77, would have a nodulus of 13 with respect to the base of 8, since 77-64=13.

Now, where might such a thing as a nodulus actually get used in mathematics? See the link for an example!

Great word! Except... when I started reading this one, I thought a nodulus would describe that resulting head bob people do (however slight) after they know they've solved a problem. This would have applications outside mathematics...

Love it. There is some connection between the sequence of integers and their squares, in that if you add up the values of the sequence of odd numbers {1, 3, 5, 7, 9, 11, 13 ...} - you get the sequence of squares {1, 4, 9, 16, 25, 36, 49 ...} since one is just the aggregation of the other, it's interesting to think how this would play out in circular (modular) arithmetic. But, rather than thinking of modular math as being something that you can map non-modular math onto, it should be able to consider it as a discrete set of mathematics itself, in which squares and square-roots have a meaning without resorting to going outside of modular arithmetic. I'm not sure what that would be, but I'm sure it would be preferable.

Going back to the sequences, if the sequence {1, 3, 5, 7, 9, 11, 13, 15 ...} were put into a modulo 8 system, what would it look like?
{1, 3, 5, 7, 1, 3, 5, 7} ?
And the resulting sequence of additions would be:
{1, 4, 1, 0, 1, 4, 1, 0}
I'm not sure how that helps, but it should be consistent.

Sorry, I don't see a global use for this concept at all, and would much rather see words invented to denote (x/x+1) and 1/(1+y), which are extremely commonly occurring expressions which are cumbersome to write.

[phundug], I specified that the word has "occasional" use. If you want a different word that means something else, which might find more common usage, feel free to invent it!