The most familiar group is our number system; we have a bunch of numbers, and we have this thing called "$+$" which can take two numbers and churn out another number. A group is just a set of objects, and you have one "operation" (usually called "$\ast$" or "$+$", but those are just names) that tells you how to combine objects to produce other objects in that set, subject to a few rules. Why devise such a "strange chasing game" in a set of objects? That's because such systems are omnipresent in a stupendous array of phenomena:

- Most
**number systems**(whole numbers, fractions, real numbers, complex numbers...) **"Weird" number systems**that can have strange rules like $1 + 1 = 0$ are used in binary computation, and are crucial in encrypting information for secrecy. Similarly, the "*clock algebra*" has $2359 + 0001 = 0000$.