2 = 1 Proof: Where Is the Mistake?

The question, solved step by step

Start with x = y. Show that 2 = 1.

Step 1: start with x = y.
Step 2: multiply both sides by x to get x² = xy.
Step 3: subtract y² from both sides: x² - y² = xy - y².
Step 4: factor both sides: (x + y)(x - y) = y(x - y).
Step 5: divide both sides by (x - y). This is the mistake, because x - y = 0.
The "result" x + y = y, so 2y = y and 2 = 1, is false.

Answer

The mistake is step 5: dividing by x - y, which equals 0.

The common mistake

Blaming an earlier step such as the factoring. Steps 1 to 4 are all valid. The error is dividing both sides by x - y when x = y makes it zero.

Parent tip

Try the same steps with numbers: let x = y = 1. Then x - y = 0, so step 5 would be dividing by 0. Seeing a real zero helps students remember the rule.

Video transcript

0:00Two equals one? Watch, then find the mistake!

0:03Start with x equals y. Times x on both sides.

0:07Take away y squared. Then factor.

0:10Divide by x minus y.

0:12Since x equals y, two y equals y. So two equals one!

0:17Which step is wrong?

0:22Step five! X minus y is zero, and you can never divide by zero!

0:27Did you spot it? Comment the step!

Questions parents ask

Where is the mistake in the proof that 2 = 1?

The mistake is dividing both sides by x - y in step 5. Because the proof starts with x = y, x - y is 0, and dividing by zero is not allowed. All the steps before it are correct algebra.

Why can you not divide by zero?

Division asks "how many times does the divisor fit into this number?" For zero there is no sensible answer. If 5 ÷ 0 had an answer, multiplying it back by 0 would have to give 5, but anything times 0 is 0.

Is the proof valid if x is not equal to y?

The proof starts with x = y, so it is not meant to work otherwise. If x and y were different, the step x² = xy would be false. So the proof is built on x = y, which is exactly what makes x - y equal to zero.

What grade learns about dividing by zero?

Children first learn that division by zero is undefined in elementary school, and it comes back in Algebra 1 and Algebra 2 when solving equations. Factoring and dividing both sides, as in this proof, are typical Grade 8 and 9 skills.

2 = 1 proof: find the mistake