maths

if x and y are irrational,which of the following expression must be irrational?

1.2x

2.x+2y

c.2xy

d,all of the above.

4 個解答

評分
  • 6 年前
    最愛解答

    Question

    If x and y are irrational, which of the following expression must be irrational?

    A. 2x

    B. x + 2y

    C. 2xy

    D. All of the above.

    Answer

    A.

    Explanation

    Counterexample for B.

    Let x = 2√3, y = -√3, both x and y are irrational,

    then x + 2y = 2√3 + 2(-√3) = 0 is rational.

    Counterexample for C.

    Let x = y = √2, both x and y are irrational,

    then 2xy = 2√2√2 = 4 is rational.

    Then, clearly D is also wrong.

    By elimination, B, C, D are wrong, the answer is A.

    With more explanation:

    Suppose 2x is rational, then 2x = p/q where p and q are integers.

    Then, x = p/(2q).

    Here p and 2q are both integers, so x should be rational.

    We have "2x is rational" ⇒ "x is rational".

    Therefore, by proof of contrapositive,

    we have "x is irrational" ⇒ "2x is irrational".

    Therefore, A is correct (that is, must be irrational).

  • 6 年前

    must be irrational: b, c ,d

    -------------------------------------------------a is correct -------------------------------------

  • 6 年前

    i think that the answer may be D

  • 6 年前

    i think is D

    maybe

    資料來源: me
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