SOLUTION: Show that 7 + √2 is not a rational number

SOLUTION: Show that 7 + √2 is not a rational number

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Question 189409: Show that 7 + √2 is not a rational number
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

I'll do you one better. Let's prove that the sum of any rational number and any irrational number is irrational.

Assume that there exists an irrational number x and a rational number

,

such that the sum:

is rational.

Then there exists integers c and d such that

is a rational number and,

But then

and since a, b, c, and d are integers, bd, bc, and ad are integers, and further bc - ad is an integer, hence

is rational by definition.

But that leads to a contradiction because x was originally assumed to be irrational.

Therefore, reductio ad absurdum,

is irrational.

John

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