可利用反證法,要用到有理數(shù)和無理數(shù)的定義.
整數(shù)和分數(shù)統(tǒng)稱有理數(shù),也就是說對一個有理數(shù)必可表為a/b其中a、b是某個整數(shù),反之不能這樣表示的就是無理數(shù).
Proof:Assume x is a rational number and y is a irrational number,
then there exist two integers a,b that x=a/b.
The product of x,y is z=xy=ay/b.(1)
If z is a rational number,then there exist two integer c,d that z=c/d(2)
from(1)(2) we get ay/b=c/d ,that is y=bc/ad.
As we know,a,b,c,d are all integers ,which make y must be a rational number,that is a contravention.
Thus,z must be a irrational number.