$$\begin{array}{rcl} z^{-i} &=& \exp(-i\ln z) \\ &=& \exp(-i\ln(re^{i\varphi})) \\ &=& \exp(-i[\ln_\Bbb R (r) + i\varphi + 2ni\pi]) \\ &=& \exp(-i\ln_\Bbb R (r) + \varphi - 2n\pi) \\ &=& \exp(\varphi - 2n\pi) [\cos(\ln r)-i\sin(\ln r)] \\ \end{array}$$ For an understanding, we must view number...

Ernst Mayr in his last book titled "What Makes Biology Unique?" argues that many of the theories in biology do not need any mathematical support. He says that much of biology is only conceptual and cannot be describe by mathematical formulations. In the meantime, he also argues that these propert...

What we as humans can understand is automatically limited by our processing apparatus. If part of the world were inherently contradictory, we would find some way to pretend that it wasn't, simply because inherent contradiction is beyond the bounds of what we can reasonably abide. We might give ...

In complex algebra we use square roots of unity. Is there any algebra where instead we use cube roots of unity? For example, properties like, as in usual complex algebra $z = x+iy$, we say $|z| = \sqrt{x^2+y^2}$, where in the new algebra $z = x+\omega y$, we say $|z| = (x^3+y^3)^{1/3}$, where $\o...

Subtraction between infinite cardinals cannot be well-defined. This is a good example why. We know that $|\Bbb N|$ and $|\Bbb Z|$ are both of the same cardinality, but so is $|\Bbb{Z\setminus N}|$. On the other hand, $|\Bbb N\setminus\{k\in\Bbb N\mid k>2\}|=3$ (zero is a natural number here). S...

Naive solution for some intuition (with division of zero involved): $$\begin{array}{rcl} x^2 y' - y &=& 0 \\ x^2 y' &=& y \\ \dfrac{y'}{y} &=& \dfrac1{x^2} \\ \displaystyle \int \dfrac{\mathrm dy}{y} &=& \displaystyle \int \dfrac1{x^2} \ \mathrm dx \\ \ln y &=& C-\dfrac1x \\ y &=& Ae^{-1/x} \end...