Is there any special reason behind the formula of radix complements? My concern is with whether memorizing the formula is the only way to remember it
If you think of say a bike odometer or a customer counter with for exampne 4 digits, you know it will count from 0000 to 9999 and then rolls over back to 0000, so they show the same count every 10000 counts because there is only four digits.
As these will only count upwards, and it happens to read a count of 5555, and you want to subtract 1, you need to go upwards 9999 counts before the reading is 5554.
Thus, -1 is same as +9999, which equals the highest number that can be displayed. Extending that, -2 equals +9998.
The same applies in binary, for a four bit value, -1 is same as +1111 (15 in decimal), or largest representable 4-bit number. Extending that, -2 is +1110 in binary or 14 in decimal.
That is what your formulas describe in a generalized mathematical fashion.