The following is a “proof” that one equals zero.

Consider two non-zero 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 (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.

**How to Cite this Page:**.

Su, Francis E., et al. “One Equals Zero!.” *Math Fun Facts*.

**References:**

R. Vakil, *A Mathematical Mosaic*, 1996. p. 199.

**Fun Fact suggested by: **

James Baglama