Algebra Sleuth: Proof that 1 = 2?

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What You Need:

  • Proof that 1 = 2 (see below)

What You Do:

  1. Show your teen the proof.
  2. Ask her to tell you which step is invalid. She should determine both which number is wrong, and why.
  3. Help her keep going until she understands the answer.

The Proof that 2 = 1

1) a = b 1) Given

2) a2 = ab 2) Multiply both sides by a

3) a2-b2 = ab-b2 3) Subtract b2 from both sides

4) (a+b)(a-b) = b(a-b) 4) Factor both sides

5) (a+b) = b 5) Divide both sides by (a-b)

6) a+a = a 6) Substitute a for b

7) 2a = a 7) Addition

8) 2 = 1 8) Divide both sides by a


Part one: Step five is wrong. The rules of mathematics do not allow us to divide by zero.

Since a and b are equal, (a-b) = 0. Therefore, we cannot divide by (a-b)!

Note: To explain why you can't divide something by zero, ask your student how she would divide a pizza into 0 pieces. Impossible! The fewest number of pieces she could make would be one piece—the whole pizza!

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