Let a be a rational number and b be an irrational number.

Question

a + b = c, assume c is rational

a + b - a = c - a, subtract a from both sides

b = c - a, which means that b is rational. But this contradicts the initial assumption.

The above proof shows that the sum of an irrational and rational number is always ________. Explain.
Responses
A rational. Since an irrational number cannot equal a rational number.rational. Since an irrational number cannot equal a rational number.
B irrational. Since an irrational number cannot equal a rational number.irrational. Since an irrational number cannot equal a rational number.
C rational. Since you can write it as the subtraction of two rational numbers.rational. Since you can write it as the subtraction of two rational numbers.
D irrational. Since you can write it as the subtraction of two rational numbers.

Answer 1

The correct response is:

B: irrational. Since an irrational number cannot equal a rational number.

Explanation: In your proof, you started with the assumption that the sum of a rational number \(a\) and an irrational number \(b\) equals a rational number \(c\). You then manipulated the equation to arrive at the conclusion that \(b\) is rational, which contradicts the assumption that \(b\) is irrational. This demonstrates that the assumption that \(a + b\) is rational must be false. Therefore, the sum of a rational number and an irrational number must always be irrational.