The following is a "proof" that one equals zero.
Consider two nonzero numbers x and y such that
x = y.
Then x^{2} = xy.
Subtract the same thing from both sides:
x^{2}  y^{2} = xy  y^{2}.
Dividing by (xy), 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.
What's wrong with this "proof"?
Presentation Suggestions:
This Fun Fact is a reminder for students to always
check when they are dividing by unknown variables
for cases where the denominator might be zero.
The Math Behind the Fact:
The problem with this "proof"
is that if x=y, then xy=0. Notice that halfway
through our "proof" we divided by (xy).
For a more subtle "proof" of this kind,
see One Equals Zero: Integral Form.
How to Cite this Page:
Su, Francis E., et al. "One Equals Zero!."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
