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Easy proof for rational + irrational = irrational? watch

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    Ok so I have to prove a + b = irrational, where a is a rational number and b is an irrational number.

    But i get the opposite...

    a = \frac{m}{n}

    \frac{m}{n} + b = \frac{m+b(n)}{n}

    let, m + bn = h

    a + b = \frac{h}{n}

    So i get that it is rational...where am i going wrong?

    +REP
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    if b is irrational and any rational number is added then its always going to be irrational...x+1/y= xy-y+1/y xy/y=rational or irrational but the addition of a irrational i.e. 1/y its going to equal irrational...dont know if that helps
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    It looks to me like h (or m+bn) in your post above is irrational.
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    your assumption that h is rational lies on m+bn being rational, i.e a rational plus an irational. So you are assuming the disproof in your disproof!
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    (Original post by Mos Def)
    let, m + bn = h
    h has to be an integer by definition of rational number, which is in fact, false.
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    Ooohh right, so basically i don't assume the h bit, and leave it as , (m + bn)/n ...which is irrational...QED?
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    Basic proof by contradiction. Suppose a is rational, and write it as p/q, for some integers p and q. Suppose b is irrational.

    Now we want to prove that a + b is irrational. If it is not irrational, it is rational, and hence it can be written as: a + b = c/d, for some integers c/d; hence b = c/d - a = (cq-pd)/(dq). But then clearly b is rational, as cq-pd is an integer, as is dq. But this is a contradiction - hence a + b is irrational.
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    (Original post by GHOSH-5)
    Basic proof by contradiction. Suppose a is rational, and write it as p/q, for some integers p and q. Suppose b is irrational.

    Now we want to prove that a + b is irrational. If it is not irrational, it is rational, and hence it can be written as: a + b = c/d, for some integers c/d; hence b = c/d - a = (cq-pd)/(dq). But then clearly b is rational, as cq-pd is an integer, as is dq. But this is a contradiction - hence a + b is irrational.
    Oh right, this is similar to what I did, cheers!
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    found this on a wiki answers page, I hope it helps -

    say that 'a' is rational, and that 'b' is irrational.
    assume that a + b equals a rational number, called c.
    so a + b = c
    subtract a from both sides.
    you get b = c - a.
    but c - a is a rational number subtracted from a rational number, which should equal another rational number.
    However, b is an irrational number in our equation, so our assumption that a + b equals a rational number must be wrong.
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    (Original post by RVNmax)
    found this on a wiki answers page, I hope it helps -

    say that 'a' is rational, and that 'b' is irrational.
    assume that a + b equals a rational number, called c.
    so a + b = c
    subtract a from both sides.
    you get b = c - a.
    but c - a is a rational number subtracted from a rational number, which should equal another rational number.
    However, b is an irrational number in our equation, so our assumption that a + b equals a rational number must be wrong.
    I like that way, it's easy
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    (Original post by CHEM1STRY)
    I like that way, it's easy
    Yay I guess it helped then, have some rep for quoting me.
 
 
 
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