Math grad student here to offer some mathematical perspective (hopefully). Initially 'i' was invented in order to be able to calculate the roots (zeros) of a polynomial of degree 3 (wikipedia will tell you as much if you look for Gerolamo Cardano and Niccolo Tartaglia). Using 'i' abstractly (not as some sort of number, but just a symbol) turned out to allow simple arithmatic and still render sensible results. This inspired mathematicians to further examine, and adding i to the set of real numbers turned out to render a field (as a fellow commentor explained). However, the field of COMPLEX numbers turned out to allow a much much much more extensive theory with loads of great results, the most basic one being that ANY polynomial of degree n with complex (or real) coefficients has EXACTLY n roots, think of x²⁺¹ (this is called 'being algebraically closed' and is an extremely useful property in algebra) However, the field of complex numbers has several other brilliant properties that can be used to derive properties for ordinary real problems as well. There exists a book 'Complex proofs of real problems' that expands on this. The solving of these real problems in its turn gave rise to important consequences for practical engineering issues (many of which were mentioned by commentors before me), physical issues, etcetera. The full history of complex calculus is probably impossible to explain to a five-year old but I hope this is something at least. If you have further questions or want some more wikipedia to browse, I'd be glad to help. TL;DR: the complex numbers have far better properties than the real ones and these properties can be used to find invaluable 'real' results applicable to engineering, physics, informatics, etc.