On the Subject of Complexity

It's imaginary, but in the worst way imaginable.

This module is from ██████-i[8+8i], and it has the following displays from top to bottom and then from right to left: operator, submit button, backspace, button for negatives, right screen, digit appender, left screen and a digit selector.

To solve the module, perform simple math problems in the complex base x+yi.

There are 3 stages, each of which picks a random operator from the set +-×÷. A strike will reset the stage.

All numbers are given in base x+yi, where x and y are both within a range of 2 to 8 inclusive. The symbols on the module must be interpreted as shown in the table.

The right screen shows the two numbers to apply the operator on (top operator bottom). The left screen shows your current input. The input must be entered from most significant digit to least significant digit.

To find the base you will be using for the module, Look at the highest digit the real and imaginary component can be and add 1 to both. The value derived from the real component is represented as x and the other value as y. The base would be labeled as base x+yi.

In this base you should interpret real and imaginary as separate. The real component would be set up in base x and the imaginary component in base y. Then afterwards the digits simply get merged. When the two numbers do not have equal lengths, 0's will be prepended to the one with less digits.

Normal base systems (with base b) work as follows: starting from the rightmost digit every step left multiplies the initial multiplier of 1 by b. Add all the values in the number together after having each multiplied by the value of the multiplier when it was at that position. All available digits are within 0 to b-1 inclusive.

Buttons

The digit selector has a left and right side to select. Clicking the left side will add to the real component, where the right side adds to the imaginary component. When no more digit is available for either side, it will fall back to 0 or 0i respectively.

The digit appender appends the selected digit to the number.

The negator has a left and right side that negates the real and imaginary part of the inputted number respectively

The backspace button removes latest appended digit.

The submit button submits the current inputted number. An empty input will be seen as 0+0i.

ValueSymbolValueSymbolValueSymbol
0Aa0iAa++
1Ba1iAb--
2Ca2iAc**
3Da3iAd//
4Ea4iAe-Re<
5Fa5iAf-Im >
6Ga6iAg-n<>
7Ha7iAh==

The numbers are read from top to bottom. The real and imaginary numbers merge as left half and right half respectively.

Useful Formulae

i2=-1

(a+b)(c+d)=ac+ad+bc+bd

Note: The module may or may not be a bit broken due to having traveled through environments it was not protected to. The negatives are all wired up to the same network, so if that button is broken, they shall be discarded in your answer.