# 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

• 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.

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