Question

Consider the statement: "The sum of a rational number and an irrational number is always rational."



Choose ,begin emphasis,all,end emphasis, of the expressions that show that this statement is false.

Answer options with 6 options
A.
StartRoot 25 EndRoot plus pi

B.
0.56 plus pi

C.
seven-eighths plus StartRoot 13 EndRoot

D.
pi plus StartRoot 17 EndRoot

E.
0.45 plus 0.96

F.
StartRoot 18 EndRoot plus StartRoot 21 EndRoot

Answer 1

A, B, C

Reason: In each of A, B, C one addend is rational and the other is irrational:
- A: sqrt(25)=5 (rational) and π (irrational) → 5+π is irrational.
- B: 0.56 = 14/25 (rational) and π (irrational) → 0.56+π is irrational.
- C: 7/8 (rational) and sqrt(13) (irrational) → 7/8+√13 is irrational.

If r is rational and x is irrational, r+x cannot be rational (otherwise x=(r+x)−r would be rational), so these are counterexamples.