Fortunately for computers the solution to multiplying irrational numbers is forcibly simplified by the number of bits you have to work with. Thereby, you apply the same reduction to addition that I explained to an approximation of said irrational numbers i.e. to a limited number of decimal (or binary in this case) places - as though they were rational numbers. To deal with (non-integer) numbers that have decimal points, variables of “float” or “double” type are used to represent numbers in an alternative way to allow more decimal places to be dealt with, even though the same number of binary places are being used - at the cost of accuracy. “Float” is short for floating decimal point, which hints at how the alternative representation manipulates bits to this end, although that’s not the only trick used. Still though, the limited number of bits nevertheless results in irrational numbers being dealt with as though they were rational - and the problem posed to mathematicians of irrational numbers is circumvented artificially.
Even outside of the world of computers, the same constraints are forced by practicality. Most vulgarly, for engineers and others who use irrational numbers in calculations that apply to everyday scales, the use of more decimal places quickly becomes redundant, meaning they too use irrational numbers like they were rational. But even for quantum physicists performing experiments at the quantum level, the use of more decimal places also becomes eventually redundant beyond a certain only slightly further threshold and the same truncation is resorted to - and everything still boils down to a twist on addition.
But what about for theoretical physicists and pure mathematicians? Their solution is even simpler: algebra. Got an irrational number? Use a letter to denote it (in exactly the same way that numbers denote integer quantities). This preserves the implied infinity of decimal places throughout all calculations, sometimes being cancelled out, sometimes being eliminated by certain properties of irrational numbers when used in, for example geometry. Consider that famous identity by Euler: “e^(pi)i = -1”. This equation is most simply shown on an 2D graph of a (complex) unit circle that plots its imaginary component against its real component, where pi is in radians and e is the base of natural logs as standard. Here we have 3 non-real numbers that when related to one another in a specific way amount to a real number, an integer no less.
Technically, “i” is just another kind of anomaly like irrational numbers that presents itself when building up maths from addition upwards - and to deal with that, in answer to your bonus question, the same solution of algebra is used. Mathematicians have to bring everything back into the realm of “boils down to addition” in order to deal with it - even the apparent exceptions that emerge from building maths in such a way. The whole point of maths is that it boils down to the simplest possible concepts, such that it is robust and consistent throughout.
In short: the practical solution is to truncate to a rational approximation, the theoretical solution is to use algebra.
Thank you for asking the first not-profoundly-stupid question in this thread for far too long.