Let's Prove that \(3=0\)

Algebra Level 4

I found this proof on a Chinese Q&A forum:

  1. \(x^2+x+1=0\)
  2. \(x^2=-x-1\)
  3. \(x=-1-\frac{1}{x}\)
  4. \(x^2+\big(-1-\frac{1}{x}\big)+1=0\qquad \) (substituting the equation in step 3 into the equation in step 1)
  5. \(x^2-\frac{1}{x}=0\)
  6. \(x^2=\frac{1}{x}\)
  7. \(x^3=1\)
  8. \(x=1\)
  9. \(1^2+1+1=0\qquad \) (substituting the equation in step 8 into the equation in step 1)
  10. \(3=0.\)

In which of these steps did I first make a mistake by using flawed logic? Or did we just prove that mathematics is inconsistent?


If you enjoyed this problem and know some calculus, you may want to consider trying this problem.

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