怎樣證明 √2 + ∛3 是無理數 ?

In the following proof, we will use the so called rational root theorem which can be found in any Pure Mathematics textbook or http://en.wikipedia.org/wiki/Rational_root_theorem

Proof:

Assume that √2 + ∛3 is a rational number, called it x

Then x = √2 + ∛3 or x - √2 = ∛3

Triple on both sides

x^3 - 3√2x^2 + 6x - 2√2 = 3

x^3 + 6x - 3 = 3√2x^2 + 2√2

Square on both sides

x^6 + 12x^4 - 6x^3 + 36x^2 - 36x + 9 = 18x^4 + 24x^2 + 8

x^6 - 6x^4 - 6x^3 + 12x^2 - 36x + 1 = 0

We see that if x has a rational root, then its form should be either 1 or -1, which is impossible by direct substitution. We conclude that √2 + ∛3 is an irrational number.