Rational Plus Irrational?

Explain why the sum of a rational number and irrational number MUST BE irrational?

4 Answers

  • Math
    Lv 7
    7 months ago
    Favorite Answer

    Let's assume that the sum of the rational number a/b (where a and b are integers) and irrational number x is also a rational number c/d (where c and d are some integer), i.e.

    (a/b) + x = c/d

    Then x = (c/d) - (a/b)

    RHS is clearly rational while LHS is irrational. This is a contradiction.

    Hence the sum must be irrational

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  • JOHN
    Lv 7
    7 months ago

    If u (rational) + β (irrational) = v (rational),

    then u - v (rational) = -β (irrational);

    i.e., rational = irrational. Contradiction.

    So u + β = γ (irrational),

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  • Pope
    Lv 7
    7 months ago

    Suppose it is not true. In that case, let there be a rational number r, and an irrational number x. Suppose their sum is rational, so this relation must be true:

    r + x = a/b, for some certain integers a and b

    Since r is rational, r = n/m, for some integers m and n.

    r + x = a/b

    m/n + x = a/b

    x = a/b - m/n

    x = (an - bm)/(bn)

    But a, b, m, and n are all integers. Therefore, so are (an - bm) and bn. That means x is the ratio of two integers, which cannot be so for an irrational x. By contradiction, we must reject the proposition that r + x can be rational. Since the sum cannot be rational, it must be irrational.

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  • Mark
    Lv 7
    7 months ago

    Because the irrational number can't be represented by a fraction, so for instance, 4 + pi will be irrational, something like 7.14159(continue on into infinity).

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