Question

设集合M={x|x=a+b√5,a,b∈Z},若x∈M,y∈M,试判断xy,x/y是否属于集合M.

Answer 1

x=a+b√5,y=m+n√5

a,b,m,n∈Z

xy=(a+b√5)(m+n√5）=(am+5bn)+(an+bm)√5

因为a,b,m,n∈Z,所以am+5bn∈Z,an+bm∈Z,所以xy∈M

x/y=(a+b√5)/(m+n√5)=[(a+b√5)(m-n√5]/[（m+n√5）(m-n√5)]

=【(am-5bn)+（mb-an)√5】/(m^2-5n^2)

=（am-5bn）/(m^2-5n^2) + (mb-an)/(m^2-5n^2) *√5

因为（am-5bn）/(m^2-5n^2)不一定属于整数.(mb-an)/(m^2-5n^2) 也不一定属于整数,所以x/y不属于M