Paradoxes involving summations of divergent series have invaded my Facebook stream.These include, for example claims like, \[1 + 2 + 3 + ... = -1/12,\] \[1 -1/2 + 1/3 -1/4 + ... = 0,\] which involve reasoning that jumps from "infinity = infinity + something" - which is fine, to conclude that therefore, "something = 0" - which is faulty.
Such problems are great at teaching subtleties that can usually be shoved under the rug.
Here is yet another example that a student (Nathan C.) recently brought up. We know that \(i^2 = -1\). Consider the following steps and identify the misstep. \[i^2 = \sqrt{-1} \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1 \neq -1\]
Such problems are great at teaching subtleties that can usually be shoved under the rug.
Here is yet another example that a student (Nathan C.) recently brought up. We know that \(i^2 = -1\). Consider the following steps and identify the misstep. \[i^2 = \sqrt{-1} \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1 \neq -1\]
2 comments:
sqrt(ab) = sqrt(a) x sqrt(b) apparently does not apply to imaginary numbers.
Spot on.
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