2013년 10월 15일 화요일

Rational Multiplication is Closed

 

 
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Theorem

The operation of multiplication on the set of rational numbers Q is well-defined and closed:
x,yQ:x×yQ


Proof 1

Follows directly from the definition of rational numbers as the quotient field of the integral domain (Z,+,×) of integers.
So (Q,+,×) is a field, and therefore a priori × is well-defined and closed on Q .



Proof 2

From the definition of rational numbers, there exists four integers p , q , r , s , where:
q0
s0
p q =x
r s =y
We have that:
p×rZ
q×sZ
Since q0 and s0 , we have that q×s0 .
Therefore, by the definition of rational numbers:
x×y=p×r q×s Q .
Hence the result.

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