2.docx - 2 is irrational From the definition of irrational...

• 1

This preview shows page 1 out of 1 page.

2 is irrational. From the definition of irrational , this theorem is interpreted as saying: “for all integers m and n , m/n =6 √2”. We prove this equivalent formulation as follows: Proof: Let m and n be arbitrary integers with n = 0 (6 as m/n is undefined if n = 0). Suppose that . By dividing through by any common factors greater than 1, we obtain where m and n have no common factors. Then ( m ) 2 = 2( n ) 2 . Thus ( m ) 2 is even, and so m must also be even (the square of an odd integer is odd since (2 k + 1) 2 = 4 k 2 + 4 k + 1 = 2(2 k 2 + 2 k ) + 1). Let