SETI greeting featuring two Humans, a strand of DNA and binary language
Binary language was a term used to describe a basic form of computer code used by many cultures throughout the Milky Way Galaxy. Binary language consists of ones and zeros, typically in groups of eight or sixteen characters.
Humankind's understanding of binary language was incorporated into their standard SETI greeting during the 20th century. (VOY: "Future's End")
The Bynars of Bynaus in the Beta Magellan system had a culture completely based on binary code. Their minds were supplemented by high-speed computation modules, dependent to a central computer. Despite the dependence, the interconnectedness of their society allowed a tremendous amount of communications and information transfer.
Another aspect of their binary thinking was that, for them, there were only two choices: "one" or "zero" – "yes" or "no". (TNG: "11001001")
Nanites communicating in binary code
When nanites gained sentience aboard the USS Enterprise-D they tried to communicate with the crew in binary code. (TNG: "Evolution")
Federation computers used an even more advanced system, using the measurements quads, kiloquads and gigaquads to measure data. The systems were based in binary, yet also accommodating the use of trinary code. (VOY: "Hope and Fear")
Star Trek computers are frequently referred to operating in binary capacities, such as "11001001," VOY: "Favorite Son", and VOY: "Distant Origin". However, references to trinary code, and the possible obsolescence of binary, have been made in modern Trek productions.