I’ve always tended to like math more than the average person. I was less a fan of *arithmetic* — the labour intensive process of adding, subtracting, multiplying, or dividing our way to an answer to a question that a calculator could figure out instantly. More so, I’ve tended to find myself intrigued by the tools that math gives us to help us understand the world better. Sometimes that involves making use of arithmetic, but often it can simply mean applying *ideas* from the world of math in order to make better sense of the world, or to help us make better decisions.

Here are a few especially useful tools and intuitions from the world of math, which I found myself using at least once this past month.

# 1. Fermi Estimations

**Fermi Estimations** are a really cool way to get a surprisingly close guess to answer questions about the world, even if it seems like you currently have NO applicable information about what you’re guessing about.

Think of it as a method of *systematic guestimation*.

My first exposure to the idea was in Randall Munroe’s XKCD What If? post where he explains the idea and applies it to a bizzare question. It’s a fairly short post, and worth reading through!

Kyle Hill also explains the idea in his video below, where he works through some more example problems.

Using a **Fermi Estimation** can be a fantastic way to get a pretty decent ballpark guess for any piece of information you might be curious about, but using a fraction of the time and effort you would need to actually calculate a precise answer.

# 2. The Pareto Principle & Zipf’s Law

The **Pareto Principle**, also sometimes known as the **80/20 Rule**, can be found in an insane amount of places. In the productivity sphere, it’s often expressed as the idea that *“80% of the value of one’s work results from just 20% of the effort”*. Which already is a really useful piece of information; but, the applications and lessons Pareto can offer go way beyond simple productivity insights.

V-Sauce has a great video that digs into some of the weirdnesses of the **Pareto Principle**, and specifically a different way of looking at the principle, called “Zipf’s Law”:

This rule has so many useful applications. Hank Green uses it to decide when is the best time to “ship” his creative work. You can use it to help you *learn a new language* faster, as Benny Lewis explains. Pareto originally had his insight while studying the *productivity* of his garden.

In his first book, Tim Ferriss notably suggests doing “80/20 Analyses” on any part of your life you want to see improvement on — from business, to relationships, to fitness, and beyond.

# 3. Bayes’ Theorem

This one is new to me this last couple years, but I’ve found it super helpful.

Basically, **Bayes’ Theorem** is a *systematic way to update your understanding of a situation based on new information.*

3Blue1Brown does a great introduction to the idea.

The area I’ve found **Bayes’ Theorem** most helpful has to be in the area of understanding medical tests.

Take the example of testing positive for COVID with a rapid test: If you don’t have any symptoms of COVID, and you haven’t knowingly been exposed to someone with the sickness, what do you do with that information?

There’s actually a decent math-based intuition for helping deal with that situation.

The video below, again by **3Blue1Brown**, was great for giving a paradigm for how to process that information, and also better internalize the meaning and application of **Bayes’ Theorem**.