[Epistemic status: providing suggestions for how to think about something (or how to justify how you already think about something) based on my own experience.]

When deciding between various options under uncertainty, one attractive framework is to calculate the expected value of each option, then choose the option with the highest expected value.

[Epistemic status: clarifying language in the hope of helping people who were confused like me, not stating anything about what is or isn't the case in the world.]

When you imagine how the world should be, imagine also what work needs to be done to get it there. But do not yet imagine yourself doing that work. To do so at this stage would be to invite excess complication. First imagine what someone will need to do. Then, in place of "someone" put "myself".

This is an experiment where I narrate my thought process as I solve a math problem. My goal: writeups of the solutions to math problems usually present a polished, streamlined version of the solver's thought process that omits errors, wrong turns, and heuristics. I wondered what it would be like to represent the thought process "warts and all."

Most programming languages have a way to represent "no value" or "nothing". Python has None, Ruby has nil, Java and friends have null. Javascript has two ways to represent this concept -- undefined and null.

I bought a bottle of cranberry juice which said "100% juice" on the label. But, later, I found that the label said it included apple juice. I became confused and suspicious about fruit juice labels, a state which lasted many years.

But it turns out that fruit juice labels are pretty comprehensible. I learned this from user rumtscho's lovely post on this topic at cooking.stackexchange.com. The following is just a restatement of that post.

One way to convey digital information across distances is through copper wire (Ethernet cable). Here we just vary the voltage in the wire between two states A and B. When we are at A, we are sending a 0, and when we are at B we are sending a 1.

What is voltage? Voltage is the delta between two points of an electrical field.

These states are called symbols.

Number of symbols / seconds is a unit called baud. If your symbol rate is 1 symbol per second, you are sending information at 1 baud.

(I'm just writing this to help myself remember. YMMV.)

Question: we can write down a general formula for the roots of a quadratic, cubic, or quartic polynomial in terms of the coefficients. Why can't we do it for a quintic polynomial?

How easy is it to do a lot of good? Imagine that opportunities to improve the world were tangible and visible. Say they look like purple jelly beans. When you pick one up, bam! Someone's life is a little better.

In world A, jelly beans are plentiful. Maybe they rain down from the sky every week. Everywhere in the country, this is true. The sidewalks are covered in jelly beans. Everyone picks up a few just going to work every day.

The Feynman Lectures on Physics are calmly beautiful and I am reading them right now instead of going to bed, but what the hell is with this paragraph in the middle of chapter 3?

One of the most impressive discoveries was the origin of the energy of the stars, that makes them continue to burn. One of the men who discovered this was out with his girlfriend the night after he realized that nuclear reactions must be going on in the stars in order to make them shine. She said “Look at how pretty the stars shine!” He said “Yes, and right now I am the only man in the world who knows why they shine.” She merely laughed at him. She was not impressed with being out with the only man who, at that moment, knew why stars shine. Well, it is sad to be alone, but that is the way it is in this world.

Some subgroups of a group $G$ are normal, and some are not. It's not easy to get an intuition for what that means. A week or two after encountering the concept in my abstract algebra class, I knew the following:

Normal subgroups $N$ of $G$ are the only ones you can quotient by; i.e. you can write $G/N$ and it's a valid group.

They are the kernel of some homomorphism $G \rightarrow H$, where $H$ is any other group. Kernel means every element gets mapped to the identity (the subgroup is "killed").