From bd81364613bddb115c5a4ea44bc545a83c223da3 Mon Sep 17 00:00:00 2001
From: Youwen Wu <youwenw@gmail.com>
Date: Sat, 18 Jan 2025 00:23:11 -0800
Subject: [PATCH] auto-update(nvim): 2025-01-18 00:23:11

---
 .../by-course/math-8/course-notes/dvd.typ     | 341 ------------------
 .../by-course/math-8/course-notes/main.typ    |  49 ++-
 2 files changed, 45 insertions(+), 345 deletions(-)
 delete mode 100644 documents/by-course/math-8/course-notes/dvd.typ

diff --git a/documents/by-course/math-8/course-notes/dvd.typ b/documents/by-course/math-8/course-notes/dvd.typ
deleted file mode 100644
index 06deb27..0000000
--- a/documents/by-course/math-8/course-notes/dvd.typ
+++ /dev/null
@@ -1,341 +0,0 @@
-#import "@preview/ctheorems:1.1.3": *
-#import "@preview/showybox:2.0.3": showybox
-
-#let colors = (
-  rgb("#9E9E9E"),
-  rgb("#F44336"),
-  rgb("#E91E63"),
-  rgb("#9C27B0"),
-  rgb("#673AB7"),
-  rgb("#3F51B5"),
-  rgb("#2196F3"),
-  rgb("#03A9F4"),
-  rgb("#00BCD4"),
-  rgb("#009688"),
-  rgb("#4CAF50"),
-  rgb("#8BC34A"),
-  rgb("#CDDC39"),
-  rgb("#FFEB3B"),
-  rgb("#FFC107"),
-  rgb("#FF9800"),
-  rgb("#FF5722"),
-  rgb("#795548"),
-  rgb("#9E9E9E"),
-)
-
-#let dvdtyp(
-  title: "",
-  subtitle: "",
-  author: "",
-  abstract: none,
-  bibliography: none,
-  paper-size: "a4",
-  date: "today",
-  body,
-) = {
-  set document(title: title, author: author)
-
-  set std.bibliography(style: "springer-mathphys", title: [References])
-
-  show: thmrules
-
-  set page(
-    numbering: "1",
-    number-align: center,
-    header: locate(loc => {
-      if loc.page() == 1 {
-        return
-      }
-      box(stroke: (bottom: 0.7pt), inset: 0.4em)[#text(
-          font: "New Computer Modern",
-        )[
-          *#author* --- #datetime.today().display("[day] [month repr:long] [year]")
-          #h(1fr)
-          *#title*
-        ]]
-    }),
-    paper: paper-size,
-    // The margins depend on the paper size.
-    margin: (
-      left: (86pt / 216mm) * 100%,
-      right: (86pt / 216mm) * 100%,
-    ),
-  )
-
-
-  set heading(numbering: "1.")
-  show heading: it => {
-    set text(font: "Libertinus Serif")
-
-    block[
-      #if it.numbering != none {
-        text(rgb("#2196F3"), weight: 500)[#sym.section]
-
-        text(rgb("#2196F3"))[#counter(heading).display() ]
-      }
-      #it.body
-      #v(0.5em)
-    ]
-  }
-
-  set text(font: "New Computer Modern", lang: "en")
-
-  show math.equation: set text(weight: 400)
-
-
-  // Title row.
-  align(center)[
-    #set text(font: "Libertinus Serif")
-    #block(text(weight: 700, 26pt, title))
-
-
-    #if subtitle != none [#text(12pt, weight: 500)[#(
-          subtitle
-        )]]
-
-    #if author != none [#text(16pt)[#smallcaps(author)]]
-    #v(1.2em, weak: true)
-
-    #if date == "today" {
-      datetime.today().display("[day] [month repr:long] [year]")
-    } else {
-      date
-    }
-
-  ]
-
-  if abstract != none [
-    #v(2.2em)
-    #set text(font: "Libertinus Serif")
-    #pad(x: 14%, abstract)
-    #v(1em)
-  ]
-
-  set outline(fill: repeat[~.], indent: 1em)
-
-  show outline: set heading(numbering: none)
-  show outline: set par(first-line-indent: 0em)
-
-  show outline.entry.where(level: 1): it => {
-    text(font: "Libertinus Serif", rgb("#2196F3"))[#strong[#it]]
-  }
-  show outline.entry: it => {
-    h(1em)
-    text(font: "Libertinus Serif", rgb("#2196F3"))[#it]
-  }
-
-
-  // Main body.
-  set par(
-    justify: true,
-    spacing: 0.65em,
-    first-line-indent: 2em,
-  )
-
-  body
-
-  // Display the bibliography, if any is given.
-  if bibliography != none {
-    show std.bibliography: set text(footnote-size)
-    show std.bibliography: set block(above: 11pt)
-    show std.bibliography: pad.with(x: 0.5pt)
-    bibliography
-  }
-}
-
-#let thmtitle(t, color: rgb("#000000")) = {
-  return text(
-    font: "Libertinus Serif",
-    weight: "semibold",
-    fill: color,
-  )[#t]
-}
-#let thmname(t, color: rgb("#000000")) = {
-  return text(font: "Libertinus Serif", fill: color)[(#t)]
-}
-
-#let thmtext(t, color: rgb("#000000")) = {
-  let a = t.children
-  if (a.at(0) == [ ]) {
-    a.remove(0)
-  }
-  t = a.join()
-
-  return text(font: "New Computer Modern", fill: color)[#t]
-}
-
-#let thmbase(
-  identifier,
-  head,
-  ..blockargs,
-  supplement: auto,
-  padding: (top: 0.5em, bottom: 0.5em),
-  namefmt: x => [(#x)],
-  titlefmt: strong,
-  bodyfmt: x => x,
-  separator: [. \ ],
-  base: "heading",
-  base_level: none,
-) = {
-  if supplement == auto {
-    supplement = head
-  }
-  let boxfmt(name, number, body, title: auto, ..blockargs_individual) = {
-    if not name == none {
-      name = [ #namefmt(name)]
-    } else {
-      name = []
-    }
-    if title == auto {
-      title = head
-    }
-    if not number == none {
-      title += " " + number
-    }
-    title = titlefmt(title)
-    body = [#pad(top: 2pt, bodyfmt(body))]
-    pad(
-      ..padding,
-      showybox(
-        width: 100%,
-        radius: 0.3em,
-        breakable: true,
-        padding: (top: 0em, bottom: 0em),
-        ..blockargs.named(),
-        ..blockargs_individual.named(),
-        [
-          #title#name#titlefmt(separator)#body
-        ],
-      ),
-    )
-  }
-
-  let auxthmenv = thmenv(
-    identifier,
-    base,
-    base_level,
-    boxfmt,
-  ).with(supplement: supplement)
-
-  return auxthmenv.with(numbering: "1.1")
-}
-
-#let styled-thmbase = thmbase.with(
-  titlefmt: thmtitle,
-  namefmt: thmname,
-  bodyfmt: thmtext,
-)
-
-#let builder-thmbox(color: rgb("#000000"), ..builderargs) = styled-thmbase.with(
-  titlefmt: thmtitle.with(color: color.darken(30%)),
-  bodyfmt: thmtext.with(color: color.darken(70%)),
-  namefmt: thmname.with(color: color.darken(30%)),
-  frame: (
-    body-color: color.lighten(92%),
-    border-color: color.darken(10%),
-    thickness: 1.5pt,
-    inset: 1.2em,
-    radius: 0.3em,
-  ),
-  ..builderargs,
-)
-
-#let builder-thmline(
-  color: rgb("#000000"),
-  ..builderargs,
-) = styled-thmbase.with(
-  titlefmt: thmtitle.with(color: color.darken(30%)),
-  bodyfmt: thmtext.with(color: color.darken(70%)),
-  namefmt: thmname.with(color: color.darken(30%)),
-  frame: (
-    body-color: color.lighten(92%),
-    border-color: color.darken(10%),
-    thickness: (left: 2pt),
-    inset: 1.2em,
-    radius: 0em,
-  ),
-  ..builderargs,
-)
-
-#let problem-style = builder-thmbox(
-  color: colors.at(11),
-  shadow: (offset: (x: 2pt, y: 2pt), color: luma(70%)),
-)
-
-#let exercise = problem-style("item", "Exercise")
-#let problem = exercise
-
-#let theorem-style = builder-thmbox(
-  color: colors.at(6),
-  shadow: (offset: (x: 3pt, y: 3pt), color: luma(70%)),
-)
-
-#let example-style = builder-thmbox(
-  color: colors.at(16),
-  shadow: (offset: (x: 3pt, y: 3pt), color: luma(70%)),
-)
-
-#let theorem = theorem-style("item", "Theorem")
-#let lemma = theorem-style("item", "Lemma")
-#let corollary = theorem-style("item", "Corollary")
-
-#let definition-style = builder-thmline(color: colors.at(8))
-
-// #let definition = definition-style("definition", "Definition")
-#let proposition = definition-style("item", "Proposition")
-#let remark = definition-style("item", "Remark")
-#let observation = definition-style("item", "Observation")
-
-// #let example-style = builder-thmline(color: colors.at(16))
-
-#let example = example-style("item", "Example")
-
-#let proof(body, name: none) = {
-  v(0.5em)
-  [_Proof_]
-  if name != none {
-    [ #thmname[#name]]
-  }
-  [.]
-  body
-  h(1fr)
-
-  // Add a word-joiner so that the proof square and the last word before the
-  // 1fr spacing are kept together.
-  sym.wj
-
-  // Add a non-breaking space to ensure a minimum amount of space between the
-  // text and the proof square.
-  sym.space.nobreak
-
-  $square.stroked$
-  v(0.5em)
-}
-
-#let fact = thmplain(
-  "item",
-  "Fact",
-  titlefmt: content => [*#content.*],
-  namefmt: content => [_(#content)._],
-  separator: [],
-  inset: 0pt,
-  padding: (bottom: 0.5em, top: 0.5em),
-)
-#let abuse = thmplain(
-  "item",
-  "Abuse of Notation",
-  titlefmt: content => [*#content.*],
-  namefmt: content => [_(#content)._],
-  separator: [],
-  inset: 0pt,
-  padding: (bottom: 0.5em, top: 0.5em),
-)
-#let definition = thmplain(
-  "item",
-  "Definition",
-  titlefmt: content => [*#content.*],
-  namefmt: content => [_(#content)._],
-  separator: [],
-  inset: 0pt,
-  padding: (bottom: 0.5em, top: 0.5em),
-)
diff --git a/documents/by-course/math-8/course-notes/main.typ b/documents/by-course/math-8/course-notes/main.typ
index c2c92e2..77272e9 100644
--- a/documents/by-course/math-8/course-notes/main.typ
+++ b/documents/by-course/math-8/course-notes/main.typ
@@ -1,6 +1,6 @@
-#import "./dvd.typ": *
+#import "@youwen/zen:0.1.0": *
 
-#show: dvdtyp.with(
+#show: zen.with(
   title: "Math 8 Course Notes",
   author: "Youwen Wu",
   date: "Winter 2025",
@@ -308,13 +308,31 @@ $ P => Q and Q => P $
 
 #theorem("Fundamental Theorem of Arithmetic")[
   $forall x in ZZ, x > 1$, $x$ can be written as a product of prime factors.
+  Additionally, these prime factors are unique, i.e. there is only one set of
+  prime factors that uniquely factorizes $x$. Hence it is sometimes called the
+  _unique factorization theorem_ or the _prime factorization theorem_.
 ]
 
 #example[
   Assume $p$ is prime. Then $p | b$ iff $p | b^2$.
 
   #proof[
+    First let us show $p | b => p | b^2$. We know $b$ can be written as the
+    product of the prime $p$ and some other integer $n$.
+    $ b = p n $
+    Then
+    $ b^2 = p^2 n^2 $
+    which implies $p$ is a factor of and divides $b^2$. So,
+    $ p | b => p | b^2 $
 
+    Now let us show the converse, i.e. $p | b^2 => p | b$.
+
+    By the unique factorization theorem, $b^2$ can be written as a unique
+    product of primes, one of which is $p$. But we also know $b^2 = b dot b$
+    and so at least one $b$ must have a prime factor $p$. But $b$ has unique
+    prime factors (again by the same theorem) so $b$ always has a prime factor
+    $p$. Hence,
+    $ p | b^2 => p | b $
   ]
 ]
 
@@ -391,9 +409,14 @@ non-perfect square is irrational.
   Show that $sqrt(15)$ is irrational.
 ]
 
-#exercise("Euclid's Theorem")[
+#problem("Euclid's Theorem")[
   Show that there are an infinite amount of prime numbers.
-]
+]<euclid>
+
+#problem[
+  Show that in general, given any integer $n$ that is not a perfect square,
+  i.e. $ exists.not a in ZZ, n = a^2 $ $sqrt(n)$ is irrational.
+]<perfectsquare>
 
 == Proofs involving quantifiers
 
@@ -443,3 +466,21 @@ $ exists x in U, P(x) $
   Prove that between any two rational numbers $x$ and $y$ there is another
   rational number $z$.
 ]
+
+== Solutions
+
+Solutions to selected problems and exercises.
+
+#linebreak()
+
+*@euclid.* We begin by considering primes $p_1, p_2, ..., p_n$. Let $P = p_1 dot p_2 dot ... dot p_n$. Then let $q = P + 1$.
+
+Then if $q$ is prime, we have an additional prime not in the original list.
+
+Otherwise, $q$ is not prime and we have a unique prime factorization of $q$.
+Without loss of generality, take one such prime to be $p_k$. $p_k$ cannot be in
+the original list $p_1, p_2, ..., p_n$.
+
+If $p_k$ were in the original list, then since $P$ is divisible by $p_k$, and $P
++ 1$ is also divisible by $p_k$, 1 must be divisible by $p_k$ which is
+impossible. So $p_k$ is a new prime.