The term "zip" meaning zero originated from a specific slang usage among children.
According to available information, "zip" meaning “nothing” came from kids in the 1900s when they referred to getting a zero on a test. Now, if you have zip, you have nothing, and not just on a test. Essentially, receiving a "zip" on a test meant you scored zero. This usage then expanded beyond academic scores to signify a complete lack of something in general.
