Show that squareroot 2 cannot be a rational number Proceed b

Show that squareroot 2 cannot be a rational number. Proceed by proof by contradiction: assume that squareroot 2 is a fraction k/l with integers k and l notequalto 0. On squaring both sides we get 2 = k^2/l^2, or equivalently 2l^2 = k^2. We may assume that any common 2 factors of k and l have been canceled. Can you now argue that 2l^2 has a different number of 2 factors from k^2? Why would that be a contradiction and to what? You need not to translate the proof to propositional formulas. Simply write down your reasoning.

Solution

Let us k/l is not reducible to any lower fraction.

So there are only 3 possibilities for k and l :

1) k is odd and l is even

2xlxl = kxk

This cannot be the case as LHS would be even and RHS would be odd.

2) k is odd and l is odd

2xlxl = kxk

This too cannot be the case as LHS would be even and RHS would be odd.

3) k is even and l is odd

2xlxl = kxk

This cannot be the case as RHS would be a multiple of 4 but LHS would not be a multiple of 4.

Also k and l cannot both be even, because then the fraction would be further reducible.