See if you can find what's wrong with the well-known "proof" that 2 = 1, outlined below:
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Given.
a = b
Multiply both sides by a.
a² = ab
Add a² on both sides.
a² + a² = a² + ab
Simplify the left side.
2a² = a² + ab
Subtract 2ab from both sides.
2a² - 2ab = a² + ab - 2ab
Simplify right side.
2a² - 2ab = a² - ab
Factor 2 out of each term on left side.
2(a² - ab) = a² - ab
Divide both sides by a² - ab.
2(a² - ab)/(a² - ab) = (a² - ab)/(a² - ab)
Which "proves:"
2 = 1
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Can you find the error?
Come on ... don't look ahead until you at least give it a try!
All right, here's the mistake:
Look at the eighth step. If a = b (given), then a² - ab = 0. Division by zero is impossible, and therefore not allowed. Breaking the "never divide by zero" rule breaks the proof.
Multiply both sides by a.
a² = ab
Add a² on both sides.
a² + a² = a² + ab
Simplify the left side.
2a² = a² + ab
Subtract 2ab from both sides.
2a² - 2ab = a² + ab - 2ab
Simplify right side.
2a² - 2ab = a² - ab
Factor 2 out of each term on left side.
2(a² - ab) = a² - ab
Divide both sides by a² - ab.
2(a² - ab)/(a² - ab) = (a² - ab)/(a² - ab)
Which "proves:"
2 = 1
---
Can you find the error?
Come on ... don't look ahead until you at least give it a try!
All right, here's the mistake:
Look at the eighth step. If a = b (given), then a² - ab = 0. Division by zero is impossible, and therefore not allowed. Breaking the "never divide by zero" rule breaks the proof.
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