Applied Mathematics

February 1, 2024

Science is simple!

Multiplication aligns, division relates.

Ratio-nality? Good things on top, bad things below. The product grows when the desired happens.

Numbers and words are both rational - they structure tangible perceptions on top of abstract units.

Changes, when they take place, are related to each other just as the things they’re measuring do. More often then not, you’ll want to try log x, exp x, x log x, and x^k (with k being 2 here most often), only to speed up or slow down impacts.

One compared thing can be ranks - that can be useful where principal places on the list might be more crucial.

Many things - money, performance, connections - follow exponential distributions (read about Zipf’s law, it’s amazing!). A few have a lot and most have little. Taking the logarithm turns the exponential inequality into a rational relationship - it balances out the extremes. For money, taking the logarithm shows how to fairly spread influence across payers.

The rate of change often matches the structural relationships between things.

For instance, we have gravity taking two squares of masses divided by the square of distance because mass strengthens gravity at the same rate that mass itself grows, and we want both masses’ impact related to the same distance along one line for two directions.

It often helps to think of lines of measument as of axes. Dimensions correspond to independent directions of possible change.

The core idea is that math gives us tools to logically relate, compare, and order measurements in a simple way. Focusing on the core operations and functions reveals the straightforward patterns behind complex systems. These principles are applied to any practical framework.

(to be continued)