Remarks
- We have to verify that the definitions of addition and
multiplication are consistent. That is, if
x = x' mod n and y = y' mod n,
then
x + y = x' + y' mod n and xy = x'y' mod n. For then, the outcome of an addition or multiplication is independent of the chosen representatives.
Well, x = x' mod n (respectively, y = y' mod n) means that there exists an integer a (respectively, b) such that x - x' = na (respectively, y - y' = nb). This implies(x + y) - (x' + y') = (x - x') + (y - y') = na + nb = n (a + b). Hence x + y = x' + y' mod n.
Similarly we find
xy - x'y' = x(y - y') + (x - x')y' = nbx + nay' = n(bx + ay'). Hence xy = x'y' mod n.
-
In computations mod n the following properties of the
two operations addition and multiplication are often tacitly used. They look quite straightforward and are easy to use in practice. But since we have constructed a new arithmetical structure, they do require proofs.
Here is a list of the properties we mean (similar properties will reoccur in different contexts throughout Algebra Interactive; in fact arithmetic in the integers satisfies similar properties). For all a, b, c in Z/nZ
the following properties hold.
- a + b = b + a (commutativity of addition);
- a · b = b · a (commutativity of multiplication);
- (a + b) + c = a + (b + c) (associativity of addition);
- (a · b) · c = a · (b · c) (associativity of multiplication);
- a(b + c) = ab + ac (distributivity of multiplication over addition);
-
If
a
Z/nZ
then its opposite -a is the unique element b such that
a + b = 0.
- a + 0 = 0 + a = a;
- a · 1 = 1 · a = a.
To prove these properties, one computes left-hand and right-hand sides in each case using the definitions. These are easy checks.