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 $f : \mathbb{R} \to \mathbb{C}$ given by the rule $f(t) = e^{i2\pi t}$, also known as the phaser with frequency $2\pi$. We show that this function is in fact the constant function $1$:

$$\forall t \in \mathbb{R}: f(t) = e^{i2\pi t} = \left(e^{i2\pi}\right)^t = 1^t = 1.$$

$\square$

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