Enderw24, I think you’re getting too hung up on the symbols we use to represent numbers, as opposed to the numbers themselves. This whole issue of different base systems, for example, is pretty meaningless: hexadecimal, duodecimal, binary, and so on are just different names. What matters isn’t what symbol we use to represent sixteen, be it “sixteen”, “16”, “10”, “14”, “10000”, or even “XVI”, what matters is that sixteen is what I get if I add one to itself, then add the result to itself, and then do that two more times. Even The Ryan’s five-by-five matrices are just different names. What matters is how the numbers behave. No matter what kind of system of arithmetic you might try to invent, either it will have an object I can call “one”, and another that I can call “two”, and a notion of “plus” such that “one plus one” equals “two”…or else it won’t describe the way real-world objects behave.
Your questions of “Why must 1 be 1?” and “Why must 1 be a whole number?”…well, I’m not sure the first question is even meaningful, and the second is like asking “Why must apples be fruit?”. The answer is, “We don’t have a choice in the matter”.