!seany

The following is a "proof" that one equals zero.

Consider two non-zero numbers x and y such that

x = y.
Then x² = xy.
Subtract the same thing from both sides:
x² - y² = xy - y².
Dividing by (x-y), obtain
x + y = y.
Since x = y, we see that
2 y = y.
Thus 2 = 1, since we started with y nonzero.
Subtracting 1 from both sides,
1 = 0.

This equation seems to be flawless, doesn't it?

Also..

Consider the following integral:

INTEGRAL (1/x) dx
Perform integration by parts: let
u = 1/x , dv = dx
du = -1/x² dx , v = x
Then obtain:
INTEGRAL (1/x) dx = (1/x)*x - INTEGRAL x (-1/x²) dx
= 1 + INTEGRAL (1/x) dx
which implies that 0 = 1.

This also seems flawless right?

However, for the first "proving",

The problem with this "proof" is that if x=y, then x-y=0. Notice that halfway through our "proof" we divided by (x-y).

and as for the second one, it is a common mistake in integration by parts. Students often forget about the constant of integration for indefinite integrals. In this case, the constants on both sides will differ by 1.

acknoledgemtns : Su, Francis E., et al. "One Equals Zero!." Mudd Math Fun Facts.

have a nice day