The Greeks considered 1 to be a prime number. In the 1800s, the definition was changed to be "A number greater than one that has no divisors except 1 and itself." One of the arbitrary laws of arithmetic that has yet to be resolved is rounding. The usual way to do rounding is "point 5 and above go to the next higher number." However "point 5" is exactly half way between two numbers and always going to the higher number is an arbitrary choice. Some people prefer to round to the even number.
Algebra has several arbitrary laws. Some of them are merely conventions -- like using a, b and c for constants and x, y and z for variables. Others, like using juxtaposition to represent multiplication -- are much more pervasive. The juxtaposition law and omitting the exponent when it is 1 seem to be laws just to make polynomials tidy. Without these arbitrary simplifying laws, the expression 3x + 5 would be written 3 * x^1 + 5. Without the arbitrary law that multiplication takes precedence over addition, it would be necessary to write the expression (3 * x^1) + 5.
The most arbitrary laws in geometry involve angles. We call a right angle 90 degrees and a full circle 360 degrees for no other reason than that is what the Babylonians did, although we long ago rejected everything else about Babylonian mathematics. Somehow, this archaic arbitrary measuring system escaped the almost universal metric revolution -- time is the only other system to escape the metric makeover. Geometry is older than other branches and some of the conventions of geometry continue into other branches of mathematics -- like the way we refer to 2nd and 3rd power exponents as squares and cubes to reflect their role in calculating volumes.
0! is defined to be 1, although this seems counterintuitive. The reason for the arbitrary law is that it makes some formulas work out. For example, the formula for the number of ways n things can be arranges in groups of k is n!/k!(n-k)! and if we define 0! = 1, the formula makes sense when n = k. A more egregious example is 0^0. It might seem that 0 to any power would be 0, but there is a law that says that anything to the zero power is 1 -- to make several formulas work out -- so we often have 0^0 arbitrarly defined to be 1.