# One Equals Zero!

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.

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 x-y=0. Notice that halfway through our “proof” we divided by (x-y).

For a more subtle “proof” of this kind, see One Equals Zero: Integral Form.

Su, Francis E., et al. “One Equals Zero!.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
R. Vakil, A Mathematical Mosaic, 1996. p. 199.

Fun Fact suggested by:
James Baglama

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