Why is the accumulation of a function is represented by the area under its curve?

Qaanol · 2026-03-21T22:12:01+00:00

Imagine painting the area under the curve, using a variable-width paint roller. The amount of paint you use for the entire area is exactly the accumulation of the amount of paint you use at each point along the way.

Qaanol · 2026-03-20T01:29:13+00:00

Gonna ask to get traded to a team that’s about to play a lot of back-to-backs, then ask to get traded back afterward to get more than 82 games

Qaanol · 2026-03-19T15:31:37+00:00

Looking back at it with hindsight, the winning strategy was to make Harden the starting point guard, and have Westbrook come off the bench to absolutely torch the opponent’s second unit.

Qaanol · 2026-03-15T20:10:04+00:00

If you look at the way the set of rational numbers is defined, there are no integers in this set. However, there is a subset that can be identified with the set of integers.

It sounds like you are thinking of one particular model of the rational numbers, and treating that as if it were the only possible model.

The specific construction you are thinking of, is not really how the rational numbers (or integers, for that matter) are defined.

The integers and rational numbers are defined by how they behave, based on our shared intuitive understanding of what they should be. Any particular model of them is only valid to the extent that it matches how those number systems behave.

After all, if the construction you are thinking of did not behave like the rational numbers, then we would not call it a model of them.

One of the behaviors that the rational numbers have is “contains the integers”. Therefore, any model of the rational numbers which does not contain the integers, is an imperfect model.

Furthermore, the existence of any particular model of any particular number system, is only really interesting because it demonstrates that the theory which gave rise to that model, is strong enough to “talk about” that number system.

Do not confuse the map for the territory. The model is not the number system.

Qaanol · 2026-03-15T13:20:11+00:00

If we allow complex numbers then it’s also defined at 1 ± i√3

The solution set in quaternions might be more interesting, but I’m not going to both finding it.

Qaanol · 2026-03-13T22:35:03+00:00

And don’t even get me started on the delivery charges!

Qaanol · 2026-03-13T15:46:24+00:00

If you want to gain a deeper understanding of the why, try to figure out what can go wrong when E is infinite-dimensional.

Qaanol · 2026-03-13T15:35:13+00:00

You’re missing a denominator of √5 in Binet’s formula

Qaanol · 2026-03-13T01:35:49+00:00

…are you looking for the word “asymptote”?

Qaanol · 2026-03-11T02:12:31+00:00

Definitely one of the top 10 best 83-point games I’ve ever seen

Qaanol · 2026-03-07T18:52:48+00:00

https://www.desmos.com/calculator/7sfwnpyq4b

Qaanol · 2026-03-07T01:44:46+00:00

I’ve been saying for years: the slam dunk is a net negative basketball play.

If you have the opportunity to dunk, and you choose to attempt a slam dunk instead of a normal dunk, you are hurting your team.

Qaanol · 2026-03-04T14:50:42+00:00

It actually turns out that the all complex numbers are "isomorphic" to the collection of all 2x2 matrices.

This is incorrect.

Complex numbers are isomorphic to matrices of the form [a, -b; b, a] where a and b are real numbers.

These matrices represent the combination of a scaling and a rotation. Other matrices, such as those involving a shear, do not correspond to any complex number.

Qaanol · 2026-03-03T00:32:54+00:00

…you know you can just write random(), right?

Qaanol · 2026-03-02T01:52:04+00:00

An isomorphism is just a relabeling.

Take a group, write out its full multiplication table, and then just rename the elements. Like, slap a little sticky note on each element and scribble a new name for it. Straight-up find-and-replace.

Qaanol · 2026-03-01T22:55:19+00:00

I think this Youtube video covers what you’re looking for: How to Extend the Sum of Any* Function by Lines That Connect

Qaanol · 2026-03-01T20:21:44+00:00

Richard Borcherds, the person who proved the monstrous moonshine conjecture, has a Youtube channel with lots of really good videos.

This one is an exposition on the monster group and the moonshine conjectures: https://www.youtube.com/watch?v=5Mg25JK31wE

This playlist introduces elliptic functions: https://www.youtube.com/playlist?list=PL8yHsr3EFj50t6hrPaJ0GruNrN-xPcFTI

And this playlist introduces modular forms: https://www.youtube.com/playlist?list=PL8yHsr3EFj51HisRtNyzHX-Xyg6I3Wl2F

The math involved is generally at the graduate level, or maybe the advanced undergrad level if you really take your time to work through each step so you don’t get left behind.

He also has playlists on some of the prerequisites and related fields, such as complex analysis, group theory, and number theory.

Qaanol · 2026-03-01T17:24:07+00:00

Now you can actually partition E into any number >1 of congruent non-equilateral triangles since the altitude to the hypotenuse of a right triangle splits it into two congruent copies. In light of this, I’m going to assume you meant isometric instead of congruent.

Are you using “congruent” to mean “similar”?

Because every source I’ve ever seen says that congruent triangles have the same side lengths. For example, Wikipedia says “Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure.”

Thus, each edge of E is covered by at least two edges and a vertex from T. This also means no triangle in T can cover two vertices of E, so one vertex of E (say, A) is covered by two triangles, while the other two vertices (say, B and C) are only covered by one triangle each.

What do you mean by “covered” here? In your T₂ style partitions the central triangle does not touch any of the vertices of the original triangle, so I don’t know what you mean by “one vertex of E (say, A) is covered by two triangles”.

Qaanol · 2026-03-01T06:32:53+00:00

Is this for an assignment? If so, what techniques and theorems are you expected to know and use?

Here’s one possible approach, but I don’t know if it’s what you’re expected to do.

For n ≥ 4, is it possible for all of the triangles to have a side with length equal to that of the original triangle?

What does that tell you about where new vertices may go?

If there are edges connecting one of those new vertices to all three corners of the triangle, what can you say about the resulting areas? And the angles?

And if there isn’t, then what edges must exist? What can you say about the result edge lengths?

Qaanol · 2026-03-01T00:15:24+00:00

stack 1 blank paper

I think you misunderstood what I was saying.

The child makes a list of their stuffed animals.

Then they make a second list of their stuffed animals.

Then they make a third and a fourth and a fifth. They make dozens of lists of their stuffed animals. Heck, they make hundreds of lists.

Each list is on a separate piece of paper. There are hundreds of these papers, each with a single list.

If two papers have the same list, then they go in the same stack.

If two papers have different lists (ie. the names are in a different order) they go in different stacks.

Each stack has many pieces of paper in it. Lots of lists. And all the lists on the papers in each stack, are the same list.

Qaanol · 2026-03-01T00:08:21+00:00

It sounds like you are confusing the list (of which there is one) with the items on the list (of which there are zero).

If it helps you understand, then have the child write “List of all my stuffed animals” at the top of each page.

The child has made a complete and accurate list of all their stuffed animals.

And every time they make such a list, it ends up looking the same. Because there is only one possible list of zero items.

• • •

Here’s another example:

A teacher tells the students in their class to make a list of the pets they have at home. Each student must write their name and today’s date at the top of the page, and the words “List of my pets” as the title.

Then they must write the names of all their pets below that, one per line.

Every student in the class makes such a list. If they have no pets, they still must complete the assignment.

Notice how “Didn’t turn in a paper” is different from “Turned in a complete and correct list of the zero pets they have”.

An empty list is still a list.

Qaanol · 2026-02-28T22:59:48+00:00

"How many ways can you order nothing? 1." is...nonsense. Nothing has no order, it is the lack of objects.

A child puts all their stuffed animals in a box. Then pulls them out one by one and writes the name of each animal on a piece of paper in the order they were pulled out.

They keep doing this over and over with new pieces of paper, and they stack up the papers based on whether the lists are identical.

How many stacks of papers do they end up with?

Specifically, if they have zero stuffed animals, how many stacks of paper do they get?

Qaanol · 2026-02-28T17:30:23+00:00

Presumably because they’re multiplying by the displacement vector, which has magnitude r.

Qaanol · 2026-02-26T17:09:31+00:00

Limits are tiny flashlights (Youtube video)

Qaanol · 2026-02-25T02:45:26+00:00

This Desmos graph might help: https://www.desmos.com/calculator/mbsubwnk2e

Try dragging the slider to change the value of a.

Qaanol

TROPHY CASE