Solution for Section 3.7 Question 1b

1b) True. To prove that 1 + sqrt8.jpg (686 bytes)is irrational, you might want to try a proof by contradiction.

Proof Suppose that  1 + sqrt8.jpg (686 bytes) is a rational number. Hence there exist integers a and b such that 1 + sqrt8.jpg (686 bytes)= a / b, where bnoteq.jpg (604 bytes)0.

1 + sqrt8.jpg (686 bytes) = a / b
sqrt8.jpg (686 bytes) = (a / b) - 1
2 ·sqrt2.jpg (659 bytes) = (a - b) / b
sqrt2.jpg (659 bytes) = (a - b) / 2b

Since a and b are both integers, (a - b)  and    2b will both the integers. Furthermore, since b noteq.jpg (604 bytes)0, we know that 2bnoteq.jpg (604 bytes)0. Therefore, if  1 + sqrt8.jpg (686 bytes) is a rational number, then sqrt2.jpg (659 bytes)is a rational number, which is not true. Hence, 1 + sqrt8.jpg (686 bytes)is irrational.

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