Essentially, this defines a homomorphic pairing function[1] for the integers. A Gaussian integer can be seen as the pair (a, b) where a and b are integers. This assigns a unique natural number to each one.<p>One can use this to show that pairs of integers have the same cardinality as natural numbers.<p>Further, since one can sum the binary representations of two numbers and get the representation of the result, this is homomorphic under addition. I have never before seen a pairing function with that property (not that I have seen many).<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Pairing_function#Cantor_pairing_function" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Pairing_function#Cantor_pairin...</a>