Apparently this is a thing. Consider the function $f(x) = \frac{1}{1-x}$. Maybe you know that you can express $f(x)$ as an infinite polynomial:

$$\frac{1}{1 - x} = 1 + x + x^2 + x^3 + x^4 + ...$$

Monday, September 5th, 2016

Apparently this is a thing. Consider the function $f(x) = \frac{1}{1-x}$. Maybe you know that you can express $f(x)$ as an infinite polynomial:

$$\frac{1}{1 - x} = 1 + x + x^2 + x^3 + x^4 + ...$$

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Tuesday, August 16th, 2016

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.

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Monday, April 11th, 2016

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.

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Thursday, March 31st, 2016

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").

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Thursday, March 10th, 2016

The complex number $z = a + bi$ can be drawn. Plot the imaginary part on the $y$-axis, and the real part on the $x$-axis.

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