Question

Consider the conjecture that the sum of a rational number and an irrational number is ALWAYS irrational. To try to prove this conjecture, Alfred begins with the assumption that the sum is rational.
Let a = a rational number.
Let b = an irrational number.
Assume a + b = c and c is rational (attempting to disprove the conjecture).

a + b = c
b = c - a

Explain how Alfred's argument contradicts his initial assumption proving that the sum cannot be rational


A) If c is rational, either b or a would have to be rational as well.
B) If a is rational, either b or c would have to be rational as well.
C) If b is irrational, the difference of two rational numbers could never be equal to it.
D) If a + b is rational, then a and a could be either rational or irrational.

Answer 1

Answer:

Remember that b = c - a. If b is irrational, the difference of two rational numbers could never be equal to it

Answer 2

Answer:

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

The premise that c is rational is false

Step-by-step explanation:

The assumption that c is rational means that the difference c-a must be rational. That difference is b, so b=c-a would mean that b must be rational. But b is defined to be irrational. Since an irrational number cannot equal a rational number, a contradiction arises and the assumption that c is rational must be false. Therefore the sum (c) must be irrational. (B)