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).

--Bakey 12:35, 6 December 2012 (UTC)

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