问一个证明题,能用英文回答更好证明√2+√3是无理数,用反证法.

Question

问一个证明题,能用英文回答更好
证明√2+√3是无理数,用反证法.

Answer 1

1 Assume that the sum of√2 and√3 is a rational number.
假设√2+√3是一个有理数
2 So (√2+√3) could be express as p/q that both of p and q are integers,p/q is a fraction in lowest terms.
那么√2+√3可以被表示为一个最简分数p/q,p,q均为整数
3 Square both side.
So 2+3+2√6 = p^2/q^2
两边平方
则5+2√6=p^2/q^2
4 5+2√6 is a irrational number ,so p^2/q^2 must be a irrational number.
因为5+2√6是一个无理数,那么 p^2/q^2 也是一个无理数
5 Because of both p and q are integers,so both p^2 and q^2 are integers and p/q is a fraction in lowest terms,so p^2/q^2 is also a fraction in lowest terms.But a irrational number can not be express as a fraction in lowest terms,it's contrary with 4.
由于p和q都是整数,所以p^2和q^2也都是整数,且p/q是一个最简分数,所以p^2/q^2是一个最简分数,但是一个无理数不可能被表达为一个最简分数,这和结论(4)相矛盾
6 So √2+√3 is not a rational number.
It's a irrational number.
所以√2+√3不是一个有理数
是一个无理数
全部手写..英语可能有少量错误..
同学.这东西真的很费劲哎..- -|||
想必你也是某个出国班的吧..
不过能帮上你的忙就好..