[hervature]

Others have handled the appropriate construction of irrational numbers. The claim here is not that every student needs to know irrational numbers are a limiting sequence but this is exactly how mathematicians think of it. So it is strange to hear another mathematician claim that repeated addition is somehow lying. At some point, people kind of just accept that it is just something you punch into your calculator and don't even think about it anymore.

Your example is quite bad because sqrt(2)*sqrt(2) = sqrt(2*2) = sqrt(4) = 2. So repeated addition works fine. Let's focus instead on pi*pi. The way calculators do this is precisely as some type of limiting sequence depending on how much precision you want. Because, one cannot "calculate" pi*pi exactly because it is irrational. So, you have 3*3, then 3.1*3.1, then 3.14*3.14, etc. which are all repeated additions with some division (e.g. 314*314/(100*100)). In reality, when multiplying two irrational numbers, we just use enough decimal points for floating point precision and then chop off any potentially erroneous digits after the multiplication.*