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I imagine that in just a few years, 50% of the people on earth will have a virtual AI helper. Like a fairy-godmother, elf, or coach that resides in your computer who does whatever it can to help you. Or it might be a stand alone speaker like Amazon-Alexa or Amazon-Echo.
Math is immutable truth, and it s everywhere in physics, chemistry, statistics, and engineering, subjects that I love. Once a theorem is proven, it does not change. The laws of physics seem to change a bit every 100 years. The recommendations of medicine change all the time. Math theorems are permanent and very well-defined. The language [ ]
I haven t played poker for many years, but I thought I might get back into it, so I decided to write out a few theorems for myself that might be useful. Basic Calling Theorem Theorem: Suppose in a poker game that you have the option of folding or paying $B$ dollars to call a [ ]
In part 1, I presented a proof that sin(2°)/2 sin(3°)/3 using only trigonometry. In part 2, I presented a proof using the Taylor series expansion of sine. In part 3 below, I present three additional proofs: a proof using the average value of a function and the mean value theorem, a proof by Reddit user Dala_The_Pimp [ ]
In part 1, I proved that sin(2°)/2 sin(3°)/3 using trigonometry. Here is a proof using power series. In Some Simple Lemmas For Bounding Power Series (Part 1) Example 1, we applied the lemmas to prove that $$x-x^3/6 + x^5/120 \sin(x) x-x^3/6$$when $0 x \sqrt{3}.$ (That s actually true for all $x 0$, but I did not prove it.) Let $\epsilon [ ]
Often functions are defined or approximated by infinitely long sums which can be thought of as polynomials with an infinite degree. For example, $$\cos(x) = 1- x^2/2! + x^4/4! +x^6/6! + .$$ (where $n! = n \cdot (n-1) \cdot (n-2) \cdots \cdot1$ and $4!=4\cdot 3\cdot 2\cdot 1$.) These sums are usually referred to as infinite [ ]
A few days ago, I wrote several proofs that $\sin(2°)/2 \sin(3°)/3$. One of those proofs involves power series and two simple, useful lemmas. Leibniz bound on alternating series (1682) Theorem: Suppose $a_1, a_2, \ldots$ is a sequence of real numbers such that: 1) $ a_i \geq a_{i+1} $ for positive all integers $i$, 2) $a_1 0$, [ ]
So, on Reddit, user Straight-Ad-7750 posted the question, Which is bigger without using a calculator: $\frac{\sin(2°)}{2}$ or $\frac{\sin(3°)}{3}$? (Note: $2°= 2\cdot \frac{\pi}{180}$ and $3°= 3\cdot \frac{\pi}{180}$.) There are many ways to prove that one is smaller than the other. These proofs are great because each proof uses one or more useful lemma or commonly known identity. Below we [ ]
This post can also be found in a nicer format at GitHub (LaTex), Github blog format, and Reddit. There are three great deck builder card games: Dominion, Slay the Spire, and Magic the Gathering. There are four different characters in Slay the Spire. My favorite is the Silent character who often uses poison to defeat [ ]