二個數學的証明 要用英文喔 二十點~~~~
question1:show that 根號2 is irrational.hint:suppose that 根號2=p/q.......
question2:(a)if n is odd,n平方 is odd
(b)if n平方is odd,n is odd 注意:這一題要反證回去 和上一題不同
(c)n is odd if and only if n平方 is odd 這是要證明a和b可以互通
- ddLv 62 0 年前最佳解答
Q1suppose that 根號2 is rationalthen there exist integers p & q s.t. 根號2=p/q where gcd(p,q)=1so, we have 2=p2/q2 i.e. p2=2q2=> 2 | p2but 2 is a primeso, 2 | plet p=2ktherefore, (2k)2=2q2 i.e. q2=2k2so, 2 | q (as p)then 2 | gcd(p,q)=1 (contradiction)Q2(a)if n is odd, we can write n=2k+1 for some integer kthen, n2=(2k+1)2=2(2k2+2k)+1 is also odd(b)suppose that n is eventhen, we can write n=2k for some integer kso, we have n2=(2k)2=2(2k2) is also even (contradiction)(c)by (a) and (b)we have n is odd iff n2 is odd