查看完整版本: MATH expert come in

MATH expert come in

<P>Need help in number theory:</P>
<P>(1) If a is rational, b is irrational, prove that a+b is irrational.</P>
<P>(2) if a is rational, b is irrational, prove that ab is irrational, for a != 0.</P>
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<P>This is like common sense... How to prove it??</P>
<P><IMG alt="" src="http://www.umasscssa.org/forum/images/smilies/default/00000247.gif" border=0 smilieid="48">&nbsp;<IMG alt="" src="http://www.umasscssa.org/forum/images/smilies/default/00000247.gif" border=0 smilieid="48"> </P>

<P>hehe，就是因为是common sense，才能体现出数学家孜孜不倦的钻研精神啊～～～～</P>
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<P>用定义直接推就好了</P>

Assume (a+b) is rational, so (a+b)=c/d, where gcd(c,d)=1.
=> (a+b)d=c => c|d
Since gcd(c,d)=1, c!|d
Hence, there is a contradiction in the assumption.
=> (a+b) cannot be expressed in the form c/d. i.e. (a+b) is irrational.


Is my proof correct??

<P>。。。</P>
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<P>没那么复杂了，我贴了第一题的答案，第二题照着做吧</P>

[[i] 本帖最后由 k4knight 于 2008-4-10 07:29 PM 编辑 [/i]]

where??

<P>[quote]原帖由 <I>Scoriger</I> 于 2008-4-10 07:28 PM 发表 <A href="http://www.umasscssa.org/forum/redirect.php?goto=findpost&amp;pid=129004&amp;ptid=10536" target=_blank><IMG alt="" src="http://www.umasscssa.org/forum/images/common/back.gif" border=0></A> where?? [/quote]</P>
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<P>汗，我果然BC</P>

我开始认真复习复习复习::inspired

<P>[quote]原帖由 <I>k4knight</I> 于 2008-4-10 06:31 PM 发表 <A href="http://www.umasscssa.org/forum/redirect.php?goto=findpost&amp;pid=129005&amp;ptid=10536" target=_blank><IMG alt="" src="http://www.umasscssa.org/forum/images/common/back.gif" border=0></A> &nbsp; 汗，我果然BC [/quote]</P>
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<P>Thank you!!! you possess great niubility!!!</P>
<P><IMG alt="" src="http://www.umasscssa.org/forum/images/smilies/default/more/00000243.gif" border=0 smilieid="56">&nbsp;<IMG alt="" src="http://www.umasscssa.org/forum/images/smilies/default/more/00000243.gif" border=0 smilieid="56"> <IMG alt="" src="http://www.umasscssa.org/forum/images/smilies/default/more/00000243.gif" border=0 smilieid="56"> </P>

<P>[quote]原帖由 <I>marthanovera</I> 于 2008-4-10 06:40 PM 发表 <A href="http://www.umasscssa.org/forum/redirect.php?goto=findpost&amp;pid=129008&amp;ptid=10536" target=_blank><IMG alt="" src="http://www.umasscssa.org/forum/images/common/back.gif" border=0></A> 我开始认真复习复习复习<IMG src="http://www.umasscssa.org/forum/images/smilies/yangcongtou/inspired.gif" border=0 smilieid="93"> [/quote]</P>
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<P>Don't worry! this homework is due on monday...</P>
<P>i have not started my revision for tmr's exam! haha...</P>

fan zheng
a/b - c/d
(a/b)/(c/d)