## Wednesday, June 24, 2015

### Apparent Mathematical Paradoxes

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. $i^2 = \sqrt{-1} \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1 \neq -1$

#### 2 comments:

Anonymous said...

sqrt(ab) = sqrt(a) x sqrt(b) apparently does not apply to imaginary numbers.

Sachin Shanbhag said...

Spot on.