Math

Motivation

My previous post about the definition of the exponential function has provided no connection between a well-known characterization (or an alternative definition) of the exponential function:

The term to be summed is simpler than the one in the binomial expansion of $(1 + s / n)^n$.

Solution

We want this sum to be as small as possible as $n \to \infty$.

Observe that the fraction is a product

Motivation from compound interest

Increase the number of times that the interest is compounded each year ($n$), so as to increase the final amount of money ($A$)

$t$ and $r$ are the number of years and the interest rate per annum.

As $n$ becomes large, we can approximate the amount by

The value of $A$ in the above formula is the amount compounded continuously.

Settings

Let

• $P$ be an arbitrary reference point.
• $\triangle ABC$ be a triangle.
• $\vec{a} = \overrightarrow{PA}$, $\vec{b} = \overrightarrow{PB}$, $\vec{c} = \overrightarrow{PC}$

Remark: $O$ is reserved for circumcenter.

Centroid

Verify that $(\vec{a} + \vec{b} + \vec{c})/3$ satisfy the constraints.

Orthocenter

Let

• $H$ be the point of intersection of two altitudes $AA_H$ and $BB_H$
• $\vec{h} = \overrightarrow{PH}$

Add these two equations together.

Preface

There’s a simpler derivation using

1. the geometric defintion
2. linearity from the definition
3. orthogonality of the canonical basic vectors $\vec{i}$, $\vec{j}$ and $\vec{k}$.

If I had read that, I wouldn’t have type this document in LaTeX. The following content was transcribed from a PDF file that I made three days ago.

Content

Recall: $\sum_{i=m}^{n} a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_{n},$ where $m$, $n$ are integers.

Example: $\sum_{i=3}^6 i^2 = 3^2 + 4^2 + 5^2 + 6^2 = 86$

1. my HTML slides
2. You may view the source code of this blog to see the source files.

Goal

To provide an introduction to the formal definition of functions for arts stream students.

I aim to

1. minimize the calculations
2. use daily-life examples
3. favor pictures over text.

IT skills used

Inkscape

1. C-S-a for alignment.
2. C-S-f for boundary and colors, and line patterns (i.e. arrows).
3. C-S-d for document size.
4. clipping process:
   1. prepare the object to be clipped.
   2. prepare the clip, which is on top of the previous object.
   3. select Object → Clip and then the first option from the top menu.

dvisvgm

Use the TEX → DVI → SVG route with latex and this tool with options --font-format=woff and -b 5 -R -d 2 to save file size and to ensure that the SVG can be correctly opened on Inkscape.

Background

A secondary school student posted a coordinate geometry question on a Discord homework help server. A helper replied to that question with an illustration of the Perpendicular Distance Formula

where $L$ is the line $Ax + By + C = 0$ without proof.

Goal

To provide an illustrated derivation of this formula.

Recall

A basic property of the dot product: the “algebraic definition” is equivalent to the “geometric definition”.

Goal

Show that

for all $x > 0$.

Proof by inverse functions

Think about their reflection along the line $y = x$ (i.e. their inverse function). Then we get

for all $x \in \Bbb{R}$, which is obviously true.

Proof by definite integrals

Here’s a second proof using definite integrals.

The rectangle $[1,x] \times [0,1]$ contains the region under the graph of the reciprocal $y = 1/x$ in the domain $[1,x]$.

I firsted tried with foreach, but that would create so much paths. I found them difficult to operate on later, for example, with PGF plots library fillbetween. const plot is a better solution.

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usetikzlibrary{patterns}
\usepgfplotslibrary{fillbetween}
\begin{document}
\newcommand\myN{8}
\pgfplotsset{
    axis lines=center,
    legend style={at={(1,1)},anchor=north east,fill=none},
    title style={at={(0.5,1.05)}},
    every axis x label/.style={
        at={(ticklabel* cs:1)},
        anchor=west,
    },
    every axis y label/.style={
        at={(ticklabel* cs:1)},
        anchor=south,
    },
}
\begin{tikzpicture}
\begin{axis}[
    title={sum of hatched region converges to Euler's constant},
    xlabel={$x$},
    ylabel={$y$},
    xmin=0,
    xmax={\the\numexpr\myN+2},
    ymin=0,
    ymax=1.3,
    xtick=\empty,
    ytick=\empty,
    extra x ticks={1, \myN},
    extra x tick labels={$1$, $n$},
    extra y ticks={1},
]
\addplot[name path=A,domain=1:\myN,samples=501,smooth] {1/x} \closedcycle;
\addlegendentry{$y = \frac{1}{x}$};
\addplot+[domain=1:\myN+1,samples=\myN+1,jump mark left,blue,mark=*,mark options={draw=blue,fill=blue}] {1/x};
\addlegendentry{$y = \frac{1}{\lfloor x \rfloor}$};
\addplot+[name path=B,domain=1:\myN+1,samples=\myN+1,jump mark left,mark=none, draw=none] {1/x} \closedcycle;
\addplot[pattern=north east lines] fill between [of=A and B];
\end{axis}
\end{tikzpicture}
\end{document}

Background

In my previous post about unit circle and trigonometric functions, I included a graph with three trigonometry functions. I’m quite satisfied with my TikZ picture.

Problem

Unluckily, a secondary school student found that my diagram was too complicated.

Solution

\documentclass[tikz,border=2pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usetikzlibrary{calc}
\newcommand\mytheta{110} %angle theta
\begin{document}
\begin{tikzpicture}[scale=2]
\coordinate[label=left:$O$] (O) at (0,0);
\coordinate[label=above left:${A = (\cos\theta, \sin\theta)}$] (A) at (\mytheta:1);
\coordinate[label=below left:${B = (\cos(-\theta), \sin(-\theta))}$] (B) at (-\mytheta:1);
\coordinate[label=right:$E$] (E) at (1,0);
\draw (O) circle (1);
\draw (A) -- (O)  node [midway, left] {$1$}
    -- (E);
\draw (B) -- (O);
\draw[-stealth] ($(O)!0.3!(E)$) arc (0:\mytheta:0.3) node[midway, above] {$\theta$};
\draw[-stealth] ($(O)!0.25!(E)$) arc (0:{-\mytheta}:0.25) node[midway, below] {$-\theta$};
\end{tikzpicture}
\end{document}

I thought the code would be easy, but it turned out that I spent one hour and half to draw this figure.

\documentclass[border=2pt,preview]{standalone}
\usepackage{tikz}
\usepackage{caption}
\newcommand{\sideA}{2}
\newcommand{\sideB}{3}
\newcommand{\sideC}{6}
\colorlet{myred}{red!40!yellow}
\colorlet{mygreen}{green!40}
\colorlet{myblue}{blue!20}
\begin{document}
\begin{figure}
\centering
\begin{tikzpicture}[thick]
\coordinate (A) at (0,0);
\coordinate (B) at (\sideC,0);
\draw[<->,>=stealth,myred] (A) -- ++ (-\sideA, 0) node [pos=0.5,below] {$\color{myred}{a}$};
\draw[<->,>=stealth,mygreen] (B) -- ++ (\sideB, 0) node [pos=0.5,below] {$\color{mygreen}{b}$};
\draw[<->,>=stealth,myblue] (A) --  (B) node [pos = 0.5, below]{$\color{myblue}{c}$};
\draw[dashed,myred] (A) circle (\sideA);
\draw[dashed,mygreen] (B) circle (\sideB);
\end{tikzpicture}
\captionsetup{labelformat=empty}
\caption{triangle inequality ${\color{myred}{a}} + {\color{mygreen}{b}} >
  {\color{myblue}{c}}$ with $\color{myred}{a = \sideA}$,
  $\color{mygreen}{b = \sideB}$, $\color{myblue}{c = \sideC}$}
\end{figure}
\end{document}

The colors are customized for the Discord bot TeXit.