The post on two easy to grasp explanations on Gödel's theorem and the Banach-Tarski paradox made me think of other explanations that I found easy or insightful and that I could share them as well.

1) Here is a nice proof of the Pythagorean theorem:

2) An easy and concise explanation of expected utility calculations by Luke Muehlhauser:

Decision theory is about choosing among possible actions based on how much you desire the possible outcomes of those actions.

How does this work? We can describe what you want with something called a utility function, which assigns a number that expresses how much you desire each possible outcome (or “description of an entire possible future”). Perhaps a single scoop of ice cream has 40 “utils” for you, the death of your daughter has -⁠274,000 utils for you, and so on. This numerical representation of everything you care about is your utility function.

We can combine your probabilistic beliefs and your utility function to calculate the expected utility for any action under consideration. The expected utility of an action is the average utility of the action’s possible outcomes, weighted by the probability that each outcome occurs.

Suppose you’re walking along a freeway with your young daughter. You see an ice cream stand across the freeway, but you recently injured your leg and wouldn’t be able to move quickly across the freeway. Given what you know, if you send your daughter across the freeway to get you some ice cream, there’s a 60% chance you’ll get some ice cream, a 5% your child will be killed by speeding cars, and other probabilities for other outcomes.

To calculate the expected utility of sending your daughter across the freeway for ice cream, we multiply the utility of the first outcome by its probability: 0.6 × 40 = 24. Then, we add to this the product of the next outcome’s utility and its probability: 24 + (0.05 × -⁠274,000) = -⁠13,676. And suppose the sum of the products of the utilities and probabilities for other possible outcomes was 0. The expected utility of sending your daughter across the freeway for ice cream is thus very low (as we would expect from common sense). You should probably take one of the other actions available to you, for example the action of not sending your daughter across the freeway for ice cream — or, some action with even higher expected utility.

A rational agent aims to maximize its expected utility, because an agent that does so will on average get the most possible of what it wants, given its beliefs and desires.

3) Micro- and macroevolution visualized.

4) Slopes of Perpendicular Lines.

5) Proof of Euler's formula using power series expansions.

6) Proof of the Chain Rule.

7) Multiplying Negatives Makes A Positive.

8) Completing the Square and Derivation of Quadratic Formula.

9) Quadratic factorization.

10) Remainder Theorem and Factor Theorem.

11) Combinations with repetitions.

12) Löb's theorem.