Wau: The Most Amazing, Ancient, and Singular Number
In this video Vi Hart talks about a number that was discovered in East Asia and in many other ancient societies. She describes the different mathematical relationships, usually infinite fractions, that included Wau (some that just included Wau and others were equal to Wau). She related the special number to nature and other fields of study like physics.
https://www.youtube.com/watch?v=wsOXvQn3JuE
Why Every Proof that .999… = 1 is Wrong
In this video Vi Hart goes through some of the many proofs that are used to show that .9 repeating is equal to one. She goes on to prove or attempt to prove these wrong to say that the two are different numbers that are independent of each other. She invalidates these proofs and then briefly talks about why mathematicians believe it is so.
https://www.youtube.com/watch?v=CfJzrmS9UfY
Doodling in Math Class: Stars
In this video Vi Hart talks about drawing stars and connects this activity to math. She talks about stars with different numbers of points and how the number of points relates to the number of sides in each of the polygons that make up the star. She then makes this star drawing into a “game”.
I was a little confused, especially during Hart’s “Doodling in Math Class: Stars” video not because I did not understand what she was saying just because I felt that she was going a little to quick while speaking and the video recording was also sped up so while trying to process what she had just said she was already on to the next part of her speaking and all throughout her hand was moving very quickly on the paper. She did show what she was saying at the same time that she was saying it. For this reason I had to pause the video multiple times to reassess what she said. There was not anything in the video that particularly bewildered me or made me say wow except for when Hart started drawing the much larger stars with many points (when she began using the protractor): I thought that the stars that she made were beautiful and they also require a lot of talent to make. I learned that a star with p points can be drawn with 2 p/2- gons; this was a mathematical relationship that I had not been aware of. In fact I have not ever looked into the math behind stars before. I also learned that even with a set number of points in the star and a set number of polygons formed within the star, there are still multiple ways of drawing the star even with the restrictions. Although I did have to pause the video ever so often so that I did not fall behind on what Hart was saying, I for the most part understand everything that she said. So the only question I have after watching the video, is not directly about the math or the drawing in the video, it is: how does Hart discover the relationships between things like the sides of the stars and the number of polygons in the star; or rather: how is she so successful at finding things like the “game” in this video, that still relate to math, but that a majority other people would not be able to discover?
I think the video could really help me with my own math teaching, even though I did not really enjoy the fast pace of it. What I can learn from Hart is that I need to implement creativity in the classroom at least every once in a while especially in order to keep the students engaged. As Hart emphasizes in the beginning of the video while talking about how she was bored during her math class while learning about factoring, classes that are only lecture can get boring and can cause students to become distracted. So rather than have my students lose focus like Hart did from what the class was actually about: I will give my students an activity to do in class that directly relates to the topic of the day. This way they have something fun to look forward to and are then more inclined to pay attention during class so that they can preform well during the activity. This way they get the fun that they want in class, without losing focus from the actual class and making fun for themselves. Hart’s video really helped me come up with this idea for my own classroom because she expresses that class can get boring, which is true for any student at one point or another and that there are fun and thought-provoking activities to do during class that can still be mathematical and related to class. Kids have a hard time focusing on school work especially for note-taking and problem solving but they are often all for interactive activities, and so I have come to realize after watching this video that combining the two ways of learning can help to take away some of this lack of focus.
So recently I've been thinking about the incompleteness theorems, specifically the first one. One often sees a proof of the first incompleteness theorem using the diagonal lemma and a "provability predicate". So what I mean is let's say $T$ is some formal system of arithmetic (like $PA$, $Q$ or any other that is strong enough […]
You arrive on the island of Knights (who always tell the truth) and Knaves (who always lie) and encounter two inhabitants, A and B. (a) A states “If I am a knave, then B is a knight!”. What can you conclude? (b) B replies “I am a knight or 2 + 3 = 5”. What […]
(a) Let the domain consist of all people in a pub and P (x) denote the statement that x is drinking. Briefly argue whether this statement is always true or not. ∃x (P (x) → ∀y P (y)) The statement means there is someone in the pub such that, if he or she is drinking, […]
I'm trying to understand the statement of Gödel's second incompleteness theorem. For a set of axioms $\Phi$ containing $\Phi_{PA}$, the author claims the following sentence 'express' the consistency of $\Phi$: $$Con(\Phi) := \neg Prov_\Phi(0\equiv 1)$$ My question is: what is the precise relationship between consistency of $\Phi$ and the sentence $Con(\Phi)?$ My guess is Maybe […]
I have read multiple times that the tools of Abstract Algebraic Logic are most useful only when restricted to propositional logics, meaning that they are much less useful in predicate logics. Is that so? In what sense are the algebraic tools less useful? Could you give me some intuitive explanation as to why is that? […]
I know this might sound extremely basic, but I'm currently in discrete math and we are learning logic. I'm actually having a really hard time with questions that ask me to determine the truth value of certain expressions. The one in particular is the following:$$x = 2^4.$$ Apparently this is not a statement, but I'm […]
My understanding is that in the implication $$p {\implies} q,$$ $p$ is sufficient to conclude $q$, and $q$ is necessary condition for $p$. Let $p$ be "it is raining," and $q$ be "the sidewalk is wet." The sufficient condition statement "If it is raining then it is sufficient to conclude that the sidewalk is wet" […]
On any set $X$, it seems intuitive that the only functions $X^n \to X$ we can actually specify are those which follow structurally from, say, the axioms of ZFC. In particular, it seems highly intuitive that this class of functions that can actually be written down is some near-inductively defined set of functions such as […]
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