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.
The question: Let f be a mapping that takes as input a Boolean proposition (no quantifiers) and outputs the same proposition but with all ∧ symbols replaced by ∨. For example: $$ f(x_1 ∧ (x_2 → ¬x_5) = (x_1 ∨ (x_2 → ¬x_5) $$ Prove: Let α be a proposition containing only Boolean connectives ∧,∨. […]
As in the question, is it the case that a predicate calculus with these properties is decidable? n-ary predicates quantifiers finite domain of discourse impredicativity disallowed no function terms If so, is it possible to encode the semantics of this logic in Simply Typed Lambda Calculus ($\lambda\to$)?
If we can prove that $¬Y \implies ¬X,$ then is it always possible to prove that $X \implies Y$ without first proving that $¬Y \implies ¬X$ ? Motivation: when studying analysis, there are some problems/theorems where we must prove $X\implies Y$ and then I try to prove it directly, spend hours stuck without being able […]
On a circle, write all the numbers in ascending order from $1$ to $1000$, clockwise. Starting from $1$, colour in all the numbers in a clockwise direction $k$-th number ($1$, $k + 1$, $2k + 1$, ...). The procedure continues until a number is reached which is already is already coloured. At the end of […]
Question. I have trouble understanding how Patrick Suppes in his "Introduction to logic" derives the important sentence (A2) from another two, (A1) and (CrI). Introduction. He uses criterion I: (Cr1): "Given a set of premises, the rules of logical derivation must permit to infer ONLY those conclusions which logically follow from the premises." He also […]
Has anyone systematically studied how powerful hypercomputation can overcome "incompleteness proof theory"? For example: A deductive system for second-order logic in standard semantics with soundness and completeness, How powerful hypercomputation can correctly decide whether a given sequence of symbols is a proof or not? A deductive system for Infinitary logic with soundness and Barwise completeness, […]
$f\in B_n$ is called linear if $f(x_1,\cdots, x_n)=a_0+a_1x_1+\cdots+a_nx_n$ for suitable coefficients $a_0,\cdots, a_n\in \{0,1\}.$ Here $+$ denotes exclusive disjunction (addition modulo 2) and the not written multiplication is conjunction (i.e. $a_ix_i=x_i$ for $a_i=1$ and $a_ix_i=0$ for $a_i=0$) Show that the above representation of a linear function $f$ is unique. Notation meaning: Here $B_n$ stands for […]
I'm trying to solve the following question: Let $A=\left \{ a_1,..,a_n \right \}$ be a finite set of arbitrary elements and $B_1,...,B_m$ subsets of A. Let $k\leq m$ be a natural number. Write a proposition $\gamma$ in propositional logic, such that $\gamma$ is a tautology if and only if there exist $k$ $B_i$ sets, such […]
On page 40 in Kunens "Set Theory An Introduction to Independence Proofs", we are given the following theorem: Gödel. If $\phi(x)$ is any formula with one free variable, $x$, then there is a sentence $\psi$ such that $$\sf{ZF} \vdash \psi \leftrightarrow \phi(\lceil \psi \rceil)$$ (*): (Should really be \textopencorner and \textcorner here, but this is […]
Negate the statement "He will sink unless he swims". How do we write this statement as an if-then statement ? Once it is written as if-then, then it could be using the formula $\neg (P \implies Q) \equiv P \wedge (\neg Q)$.
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