Find a mistake type of math problems

Question

I am interested in the problems where the formulation of the problem has some kind of mistake in it and as a consequence gives unexpected answer.

Can't explain it better than this example:

For example: $4^2 = 4 \cdot 4 = 4+ 4 + 4+4$ (sum $4$ times) Similarly: $$\frac{d}{dx}x^2=\frac{d}{dx}(x\cdot x)=\frac{d}{dx}(x+x+...+x) = 1 + 1 + ... + 1 = x$$

Since the $1$'s are summed $x$ times. I hope you see what went wrong :) If you have encountered problems of this type, please share.

EDIT: I am aware that my proposed example has a mathematically inconsistent step, this type of expansion is only allowed for natural numbers making the function non-differentiable. However this is the kind of inconsistencies I find amusing.

Answer 1

In $\frac{d}{dx}(x+x+...+x) $, you have $x$ terms, but $x$ isn't an integer.

Answer 2

I discovered this one on my own. AFAIK, no one else has. $$1 = {\rm abs}\,(-1) = ~|~ {-1} ~|~ = \det\, [-1] = -1.$$ "Proof by ambiguous notation."

Answer 3

Your equation does not make sense for many possible values of $x$. The equation

\begin{align} x \cdot x &= \underbrace {x + x + \dots + x}_{x \text{ times}}, \end{align}

only holds when $x$ is a nonnegative integer. But $x$ is an indeterminant which can take on real values as well. So you can't write $x^2$ as the sum of $x$ copies of $x$ when $x$ is not an integer. It simply does not make sense.

Moreover, you cannot differentiate the identity, as it only holds for the integers. You at least need the identity to hold about a neighborhood of a point $x$ in the real numbers, since the derivative is a local property.

Answer 4

This is one of the problems you are interested in: $$1=\sqrt 1=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=i\cdot i=-1$$ What is wrong?

Answer 5

$(-32)^{\frac{1}{5}}\neq(-32)^{\frac{2}{10}}$:

• $(-32)^{\frac{1}{5}}=\sqrt[5]{(-32)^1}=\sqrt[5]{-32}=-2$

• $(-32)^{\frac{2}{10}}=\sqrt[10]{(-32)^2}=\sqrt[10]{1024}=\pm2$