The set of integers is denoted with the letter Z (rather than I) because Zahl (pl. Zahlen) is the German word for a number. Probably quite useful, as otherwise we'd be struggling for a letter to use for imaginary numbers. There's suggestion that the reason Z stuck for the integers is because two of the most influential number theorists, Euler and Gauss, were both German and both used Z to represent the set of integers in their mathematical writing.
Incidentally, the word integer comes from the root verb tangere (to touch), and with the prefix means intact or whole - hence, the integers are the set of whole numbers. We also get the word integrity, which has the meaning of being whole or undivided.
That's your lot for tonight - I'm now going to veg in front of the telly for a couple of hours before sleeping for, like, a week.
(Image credit: By Joxemai - Own work, CC BY-SA 3.0, commons.wikimedia.org/w/index....)