Hello, I'm reading Georgi Shilov's Linear Algebra and it states:
"The numbers 1, 1+1=2, 2+1=3, etc. are said to be natural; it is assumed that none of these numbers is zero. Given two elements N and E, say, we can construct a field by the rules N+N=N, N+E=E, E+E=N, N*N=N, N*E=N, E*E=E. Then, in keeping with our notation, we should write N=0, E=1 and hence 2=1+1=0. To exclude such number systems, we require that all natural field elements be nonzero."
I am wondering why this is necessary. (I'm new to abstract algebra.) Why do we want to exclude such number systems? Is it just for convenience or is it because they actually create a contradiction?