You’ve been calculating percentages all wrong! 😃
Why doesn’t a 50% gain followed by a 50% loss get you back to your starting point? Because the standard way we define percentage increases is needlessly complicated. This post introduces the “true increase”, a new definition based on exponentials that makes growth calculations intuitive: percentages simply add up. A 10% gain and a 20% gain is exactly a 30% gain.
You will see how this change of perspective makes compounding obsolete and simplifies everything from interest rates to the “rule of 70”.

Plancks law for black body radiation explains the color of a glowing hot object, the light from our sun, and even the energy radiated by the Earth. This post offers a complete derivation of that famous equation for black body radiation, starting from the foundational principles of thermodynamics. We build the formula from the ground up, relying on just a few clearly stated postulates. You will see how a profound physical result emerges from a surprisingly simple and direct thermodynamical approach.

The fundamental mathematical ideas behind thermodynamics

Some notes on the elementary properties of the exponential families

A proof of the divergence of the harmonic series with the Coq proof assistant.

A companion matrix is a matrix with a prescribed characteristic polynomial. I would like to show them from a broader perspective: companion matrices are the matrix version of a shift operator.

My goal is to derive Maxwell’s equations of electromagnetism with almost no effort at all. As often in mathematics, things look simpler when there is less structure. Here, as in mechanics, we do not assume any prior metric, so the geometry of the space at hand is very simple.

The variational equation is a fundamental tool in engineering. It is the description of the sensitivity with respect to the initial conditions. A remarkable fact, often presented only in \(\mathbf{R}^n\), is that this is in fact the solution of another differential equation, sometimes called the variational equation. I would like to show that this is valid in general manifolds.