imaginary numbers are not the same thing as irrational numbers
An ultrafinitist is still allowed to call that 'i'.
E.g. when you calculate the area of a plot of land do you take into account the curvature of the Earth? You have to make a bunch of compromises in the first place to even talk about what the area of a plot land means.
Math is a bunch of useful systems that we humans have devised. We tend to gravitate towards the ones that help us describe and predict things in the real world.
But there is plenty of math which doesn't do either. It's just as real as the math that does.
https://plato.stanford.edu/entries/church-turing/decision-pr...
The short answer is that they deal with such things symbolically.
foo=3.14159265...
Where after 5 is some continuing sequence of decimals.
The series of functions is literally just:
foo(0) = 3 foo(1) = 3.1 foo(2) = 3.14...
And to be clear, it's not just like, an algorithm that estimates pi, it's literally just a list of return values that is infinitely long that return more and more digits of whatever the number is. That is actually how he defines pi.
https://youtu.be/lcIbCZR0HbU?si=3YxcHfPlCFrlr5h3&t=2080
pi _happens_ to be computable, and there are more efficient functions that will produce those numbers, but you could do the same thing with an incomputable number, you just need a definition for the number which is infinitely long.
To be clear, I don't think any of this is a good idea, just pointing out that if he's going to allow that kind of definition of pi (ie, admit a definition that is just an infinite list of decimal representations), you can just do the same thing with any real number you like. He of course will say that he's _not_ allowing any _infinite list_, only an arbitrary long one.
All the numbers you get this way are going to be rational, and if you require them to be finite, you can't even identify them with any irrational numbers. At least with the computable numbers you get an infinite set of irrational numbers along with the rationals, while still never touching the vast majority of all numbers (the remaining, incomputable irrationals).
I don't know how you'd do electrical engineering with the rational complex field, because electrical engineering and physics in general involves a lot of irrational quantities and calculus, and the standard foundations of these concepts use real numbers.
It's really up to finitists to show that there are problems with these methods and that they have a better way of doing things, because so far the standard way seems to work very well.