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Mathematical Fallacies

Mathematical Fallacies. by Robert J. Nemiroff Michigan Technological University. Physics X: About This Course. Officially "Extraordinary Concepts in Physics" Being taught for credit at Michigan Tech Light on math, heavy on concepts Anyone anywhere is welcome No textbook required

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Mathematical Fallacies

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  1. Mathematical Fallacies by Robert J. Nemiroff Michigan Technological University

  2. Physics X: About This Course • Officially "Extraordinary Concepts in Physics" • Being taught for credit at Michigan Tech • Light on math, heavy on concepts • Anyone anywhere is welcome • No textbook required • Wikipedia, web links, and lectures only • Find all the lectures with Google at: • "Starship Asterisk" then "Physics X"  • http://bb.nightskylive.net/asterisk/viewforum.php?f=39

  3. Proof That 1=2 a=b=1a2=aba2-b2=ab-b2(a+b)(a-b)=b(a-b)(a+b)=b2=1 What's wrong with this?

  4. Proof That 1=2 What's wrong with this? The step where (a-b) was eliminated was not mathematically logical, since (a-b)=0 and dividing by zero is not generally mathematically defined. Moral: Be careful with your algebra!

  5. Proof that 2π=0 x=2πsin(x)=0x=arcsin(0)x=0 Therefore2π=0 Why doesn't this work?

  6. Proof that 2π=0 Why doesn't this work? The arcsin function is multivalued so the arcsin(0) is not necessarily 0. Moral: Be careful with sinusoidal functions that you don't "jump" quadrants or phases!

  7. Variable Ambiguity x=1d/dx (x) = d/dx (1)1 = 0 The problem here is that x is not a true variable.  It is actually a constant, so that its derivative should also be zero. Moral: Know your variables from your constants!

  8. Negative Roots 1=11=sqrt(1)1=sqrt( (-1) * (-1) )1=sqrt(-1) * sqrt(-1)1=i * i1=-1 What went wrong?

  9. Negative Roots What went wrong? This is a tougher one!  The problem is that for sqrt(x*y) to be equal to sqrt(x)*sqrt(y), one or both numbers must be positive.  In this case, that wasn't true.

  10. Infinite Series: Associativity 0 = 0 + 0 + 0 + 0 + ... 0 = (1-1) + (1-1) + (1-1) + ... 0 = 1 + (-1+1) + (-1+1) + (-1+1) + ... 0 = 1 + 0 + 0 + 0 + ... 0 = 1 Problem: the associative law cannot be applied to infinite series that are not "absolutely convergent", meaning the the sum of the the absolute value of each term itself converges.

  11. Complex Roots x2 + x + 1 = 0 x2 = -x -1 x=-1 - 1/x ; Substitute x into the initial equation x2 + (-1 - 1/x) + 1 = 0 x2 - 1/x = 0 x2=1/x x3 =1 x=1; Substitute this x into the initial equation 12 + 1 + 1 = 0 3 = 0 ; What went wrong?

  12. Complex Roots 3 = 0 ; What went wrong? The problem is that x3=1 really has three roots, and the one chosen, x=1, was an extraneous solution given the previous mathematical context. Moral: Make sure your solution is a real mathematical and physical solution.

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