Q: In Z7, give a reasonable interpretation for the expressions 1/2, - 2/3 , sqrt(-3), and -1/6.
A: We can rewrite each expression in Z7 as follows:
Let x belong to Z7 such that:
2^-1 = x (mod 7)
2 = 2*x (mod 7)
Since x=4 satisfies this equation we find that 2^-1 = 4 (mod 7)
Similarly:
-2/3 = x (mod 7)
-2 = 3*x (mod 7)
Since -2 = 5 (mod 7) we have that:
5 = 3*x (mod 7)
and thus x=4 satisfies this equation such that -2/3 = 4 (mod 7)
sqrt(-3) = x (mod 7)
-3 = x^2 (mod 7)
Since -3 = 4 (mod 7) we have that:
4 = x^2 (mod 7)
Thus by similar argument x=-2,2 or rather x=2,5. Thus sqrt(-3) = 2 or sqrt(-3) = 5 (mod 7).
-1/6 = x (mod 7)
-1 = 6*x (mod 7)
Since -1 = 6 (mod 7) we have that:
6 = 6*x (mod 7)
and thus x = 1 such that 1 = -1/6 (mod 7).
