Not OP, but I'll give it a shot. At a low level pretty much all standard integer math is mod arithmetic, typically mod 2^32 or 2^64 depending on if a computer is 32 or 64 bit. This means that when a number goes above this limit it "overflows" and only the part left after dividing by the mod size is left. Additionally, negative numbers are stored by essentially counting down from the maximum number that can be stored + 1, in scholarly terms this is the twos complement. For instance on a 32 bit system -7 as a signed integer would be stored identically to the unsigned integer (2^32)-7. For anyone new to programming, unsigned meaning an integer variable that can only be positive and signed being an integer variable that could be positive or negative.

Now all of this combines to make addition, subtraction, and even multiplication essentially identical for signed and unsigned numbers. On an 8 bit system (so integers can be between 0 to 255 inclusive) you could be trying to add the signed number 40 and the unsigned number 200. Naturally, this is 200+40 or 240. Now inside the computer, 240 is stored in the exact same way that -16 is so it is important that the software knows the correct way to handle the variable. This is equivalent to subtracting 56 from 40, and the signed integer -56 is represented in the exact same way as the unsigned integer 200.

Now this is where it starts to get crazy. What if you add the signed integer -5 (represented by 251 in the unsigned world) and the unsigned integer 100? Well it is equivalent to 251 + 100 = 351. However, now we encounter modular arithmetic, because this is an 8 bit system and the maximum value for an integer is 255 we calculate everythihg mod 256, so ar left with 351 % 256 = 95. You can mentally view this as the calculation occurring with 1s and 0s, where everytime a 1 is added to a 1, there is a single bit carried and the last bit is dropped, which is exactly how full and half adders work in a processor!

This also works for multiplication! If we multiply -5 by 5, we get -25 easily. But for computers we do 251 * 5 = 1255. The modulo math starts to get tricky for us puny humans but for a computer it is trivial, 1255 % 256 = 231. As a signed integer that is equal to -25, as it is equivalent to (255+1-25).

For division, everything is just crazy. 250/5 is way different than -5/5, no amount if mod arithmetic will save it. Unsigned 50 and signed -1 have completely different representations, and this is even an example without fractions and rounding.

I may have went long winded, but hopefully this helps someone!

You probably do this all the time with time without realising it.

If it’s 11 o’clock and we agree to meet in two hours, you instinctively know I’m talking about 1 o’clock. But what you’ve just done is add two numbers mod(ulus) 12.

In effect your brain added 11+2 to get 13, then took away 12 to make it fit, and came up with the modulus (or remainder) of 1.

Modulus arithmetic is therefore sometimes known as clock arithmetic.

You can think of it as applying a mod operator after each operation.

Say we wanted to work with mod 5 Then 6 + 8 = 14 = 4 mod 5