Omni, Take Two
Did you know that 2 equals 1? Here is the proof:
(1) X = Y ; Given (2) X^2 = XY ; Multiply both sides by X (3) X^2-Y^2 = XY-Y^2 ; Subtract Y^2 from both sides (4) (X+Y)(X-Y) = Y(X-Y) ; Factor (5) X+Y = Y ; Cancel out (X-Y) term (6) 2Y = Y ; Substitute X for Y, by equation 1 (7) 2 = 1 ; Divide both sides by Y -- "Omni", proof that 2 equals 1
If you think this is too easy, here is another neat proof spotted by my friend Avital Steinitz while teaching a Signals and Systems course.
Consider the function given by the rule , also known as the phaser with frequency . We show that this function is in fact the constant function :