Results 1 to 3 of 3

Math Help - x and y are irrational, can x+y be irrational

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    36

    x and y are irrational, can x+y be irrational

    the answer is yes, but i need to show an example, would i just do it like pi+ square root 2 or would i need to do a proof by contradiciton, could i please have a worked solution please.

    also if n is any positive integer then sqrt(2)/n is irrational. notice that this means we can find irrational numbers as small as we like simply by increasing n.
    use the fact that sqrt(2)/n can be made as small as we desire to show that between every pair of distinct rational numbers there is an irrational number. that is suppose x Є Q and y Є Q and that x < y show that there is an irrational number z with x < z < y
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,396
    Thanks
    1846
    If the question is simply can "irrational plus irrational be irrational", then you only need to show an example to answer "yes": \sqrt{2}+ \sqrt{2}= 2\sqrt{2}. If the question is must "irrational plus irrational be irrational", a counter example shows that the answer is "no": \sqrt{2}+ (1-\sqrt{2})= 1.

    To show that \sqrt{2}/n is irrational for all n, use a proof "by contradiction": If \sqrt{2}/n were rational, we would have \sqrt{2}/n= a/b for integers a and b. Then \sqrt{2}= (an)/b showing that \sqrt{2} is rational, a contradiction.

    Since 1< \sqrt{2}< 2, \sqrt{2}/n< 1 for n> 2. If x and y are rational, x< y, y- x is a positive number and (y- x)\sqrt{2}/n is positive and less than y- x. Finally, x+ (y-x)\sqrt{2}/n is larger than x but less than y.
    Last edited by HallsofIvy; March 12th 2009 at 05:21 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Like a stone-audioslave ADARSH's Avatar
    Joined
    Aug 2008
    From
    India
    Posts
    726
    Thanks
    2
    Quote Originally Posted by b0mb3rz View Post
    the answer is yes, but i need to show an example, would i just do it like pi+ square root 2 or would i need to do a proof by contradiciton, could i please have a worked solution please.

    also if n is any positive integer then sqrt(2)/n is irrational. notice that this means we can find irrational numbers as small as we like simply by increasing n.
    use the fact that sqrt(2)/n can be made as small as we desire to show that between every pair of distinct rational numbers there is an irrational number. that is suppose x Є Q and y Є Q and that x < y show that there is an irrational number z with x < z < y

    A simpler example if you want ( It has nothing to do with square root (2)
    or pi
    x=0.01001000100001.....

    This is not recurring & has infinite digits in a sequence

    y=0.02002000200002....

    This is also not recurring & is same

    x+y= 0.03003000300003....
    neither of these is rational
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: July 19th 2010, 06:04 PM
  2. example of irrational + irrational = rational
    Posted in the Algebra Forum
    Replies: 3
    Last Post: April 4th 2010, 04:44 AM
  3. Replies: 1
    Last Post: March 23rd 2010, 01:55 PM
  4. Replies: 2
    Last Post: January 31st 2010, 06:40 AM
  5. Replies: 7
    Last Post: January 29th 2009, 04:26 AM

Search Tags


/mathhelpforum @mathhelpforum