auto-update(nvim): 2025-01-18 00:23:11
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2 changed files with 45 additions and 345 deletions
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#import "@preview/ctheorems:1.1.3": *
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#import "@preview/showybox:2.0.3": showybox
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#let colors = (
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rgb("#9E9E9E"),
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rgb("#F44336"),
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rgb("#E91E63"),
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rgb("#9C27B0"),
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rgb("#673AB7"),
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rgb("#3F51B5"),
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rgb("#2196F3"),
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rgb("#03A9F4"),
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rgb("#00BCD4"),
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rgb("#009688"),
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rgb("#4CAF50"),
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rgb("#8BC34A"),
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rgb("#CDDC39"),
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rgb("#FFEB3B"),
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rgb("#FFC107"),
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rgb("#FF9800"),
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rgb("#FF5722"),
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rgb("#795548"),
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rgb("#9E9E9E"),
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)
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#let dvdtyp(
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title: "",
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subtitle: "",
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author: "",
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abstract: none,
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bibliography: none,
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paper-size: "a4",
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date: "today",
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body,
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) = {
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set document(title: title, author: author)
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set std.bibliography(style: "springer-mathphys", title: [References])
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show: thmrules
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set page(
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numbering: "1",
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number-align: center,
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header: locate(loc => {
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if loc.page() == 1 {
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return
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}
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box(stroke: (bottom: 0.7pt), inset: 0.4em)[#text(
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font: "New Computer Modern",
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)[
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*#author* --- #datetime.today().display("[day] [month repr:long] [year]")
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#h(1fr)
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*#title*
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]]
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}),
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paper: paper-size,
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// The margins depend on the paper size.
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margin: (
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left: (86pt / 216mm) * 100%,
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right: (86pt / 216mm) * 100%,
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),
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)
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set heading(numbering: "1.")
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show heading: it => {
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set text(font: "Libertinus Serif")
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block[
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#if it.numbering != none {
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text(rgb("#2196F3"), weight: 500)[#sym.section]
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text(rgb("#2196F3"))[#counter(heading).display() ]
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}
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#it.body
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#v(0.5em)
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]
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}
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set text(font: "New Computer Modern", lang: "en")
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show math.equation: set text(weight: 400)
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// Title row.
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align(center)[
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#set text(font: "Libertinus Serif")
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#block(text(weight: 700, 26pt, title))
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#if subtitle != none [#text(12pt, weight: 500)[#(
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subtitle
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)]]
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#if author != none [#text(16pt)[#smallcaps(author)]]
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#v(1.2em, weak: true)
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#if date == "today" {
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datetime.today().display("[day] [month repr:long] [year]")
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} else {
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date
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}
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]
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if abstract != none [
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#v(2.2em)
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#set text(font: "Libertinus Serif")
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#pad(x: 14%, abstract)
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#v(1em)
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]
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set outline(fill: repeat[~.], indent: 1em)
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show outline: set heading(numbering: none)
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show outline: set par(first-line-indent: 0em)
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show outline.entry.where(level: 1): it => {
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text(font: "Libertinus Serif", rgb("#2196F3"))[#strong[#it]]
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}
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show outline.entry: it => {
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h(1em)
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text(font: "Libertinus Serif", rgb("#2196F3"))[#it]
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}
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// Main body.
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set par(
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justify: true,
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spacing: 0.65em,
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first-line-indent: 2em,
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)
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body
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// Display the bibliography, if any is given.
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if bibliography != none {
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show std.bibliography: set text(footnote-size)
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show std.bibliography: set block(above: 11pt)
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show std.bibliography: pad.with(x: 0.5pt)
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bibliography
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}
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}
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#let thmtitle(t, color: rgb("#000000")) = {
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return text(
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font: "Libertinus Serif",
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weight: "semibold",
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fill: color,
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)[#t]
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}
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#let thmname(t, color: rgb("#000000")) = {
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return text(font: "Libertinus Serif", fill: color)[(#t)]
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}
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#let thmtext(t, color: rgb("#000000")) = {
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let a = t.children
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if (a.at(0) == [ ]) {
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a.remove(0)
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}
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t = a.join()
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return text(font: "New Computer Modern", fill: color)[#t]
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}
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#let thmbase(
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identifier,
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head,
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..blockargs,
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supplement: auto,
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padding: (top: 0.5em, bottom: 0.5em),
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namefmt: x => [(#x)],
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titlefmt: strong,
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bodyfmt: x => x,
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separator: [. \ ],
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base: "heading",
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base_level: none,
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) = {
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if supplement == auto {
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supplement = head
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}
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let boxfmt(name, number, body, title: auto, ..blockargs_individual) = {
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if not name == none {
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name = [ #namefmt(name)]
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} else {
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name = []
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}
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if title == auto {
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title = head
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}
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if not number == none {
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title += " " + number
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}
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title = titlefmt(title)
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body = [#pad(top: 2pt, bodyfmt(body))]
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pad(
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..padding,
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showybox(
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width: 100%,
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radius: 0.3em,
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breakable: true,
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padding: (top: 0em, bottom: 0em),
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..blockargs.named(),
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..blockargs_individual.named(),
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[
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#title#name#titlefmt(separator)#body
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],
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),
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)
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}
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let auxthmenv = thmenv(
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identifier,
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base,
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base_level,
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boxfmt,
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).with(supplement: supplement)
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return auxthmenv.with(numbering: "1.1")
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}
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#let styled-thmbase = thmbase.with(
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titlefmt: thmtitle,
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namefmt: thmname,
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bodyfmt: thmtext,
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)
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#let builder-thmbox(color: rgb("#000000"), ..builderargs) = styled-thmbase.with(
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titlefmt: thmtitle.with(color: color.darken(30%)),
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bodyfmt: thmtext.with(color: color.darken(70%)),
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namefmt: thmname.with(color: color.darken(30%)),
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frame: (
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body-color: color.lighten(92%),
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border-color: color.darken(10%),
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thickness: 1.5pt,
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inset: 1.2em,
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radius: 0.3em,
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),
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..builderargs,
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)
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#let builder-thmline(
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color: rgb("#000000"),
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..builderargs,
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) = styled-thmbase.with(
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titlefmt: thmtitle.with(color: color.darken(30%)),
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bodyfmt: thmtext.with(color: color.darken(70%)),
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namefmt: thmname.with(color: color.darken(30%)),
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frame: (
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body-color: color.lighten(92%),
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border-color: color.darken(10%),
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thickness: (left: 2pt),
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inset: 1.2em,
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radius: 0em,
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),
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..builderargs,
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)
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#let problem-style = builder-thmbox(
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color: colors.at(11),
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shadow: (offset: (x: 2pt, y: 2pt), color: luma(70%)),
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)
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#let exercise = problem-style("item", "Exercise")
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#let problem = exercise
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#let theorem-style = builder-thmbox(
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color: colors.at(6),
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shadow: (offset: (x: 3pt, y: 3pt), color: luma(70%)),
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)
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#let example-style = builder-thmbox(
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color: colors.at(16),
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shadow: (offset: (x: 3pt, y: 3pt), color: luma(70%)),
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)
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#let theorem = theorem-style("item", "Theorem")
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#let lemma = theorem-style("item", "Lemma")
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#let corollary = theorem-style("item", "Corollary")
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#let definition-style = builder-thmline(color: colors.at(8))
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// #let definition = definition-style("definition", "Definition")
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#let proposition = definition-style("item", "Proposition")
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#let remark = definition-style("item", "Remark")
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#let observation = definition-style("item", "Observation")
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// #let example-style = builder-thmline(color: colors.at(16))
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#let example = example-style("item", "Example")
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#let proof(body, name: none) = {
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v(0.5em)
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[_Proof_]
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if name != none {
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[ #thmname[#name]]
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}
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[.]
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body
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h(1fr)
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// Add a word-joiner so that the proof square and the last word before the
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// 1fr spacing are kept together.
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sym.wj
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// Add a non-breaking space to ensure a minimum amount of space between the
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// text and the proof square.
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sym.space.nobreak
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$square.stroked$
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v(0.5em)
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}
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#let fact = thmplain(
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"item",
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"Fact",
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titlefmt: content => [*#content.*],
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namefmt: content => [_(#content)._],
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separator: [],
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inset: 0pt,
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padding: (bottom: 0.5em, top: 0.5em),
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)
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#let abuse = thmplain(
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"item",
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"Abuse of Notation",
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titlefmt: content => [*#content.*],
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namefmt: content => [_(#content)._],
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separator: [],
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inset: 0pt,
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padding: (bottom: 0.5em, top: 0.5em),
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)
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#let definition = thmplain(
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"item",
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"Definition",
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titlefmt: content => [*#content.*],
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namefmt: content => [_(#content)._],
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separator: [],
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inset: 0pt,
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padding: (bottom: 0.5em, top: 0.5em),
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)
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@ -1,6 +1,6 @@
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#import "./dvd.typ": *
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#import "@youwen/zen:0.1.0": *
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#show: dvdtyp.with(
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#show: zen.with(
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title: "Math 8 Course Notes",
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author: "Youwen Wu",
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date: "Winter 2025",
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@ -308,13 +308,31 @@ $ P => Q and Q => P $
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#theorem("Fundamental Theorem of Arithmetic")[
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$forall x in ZZ, x > 1$, $x$ can be written as a product of prime factors.
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Additionally, these prime factors are unique, i.e. there is only one set of
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prime factors that uniquely factorizes $x$. Hence it is sometimes called the
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_unique factorization theorem_ or the _prime factorization theorem_.
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]
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#example[
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Assume $p$ is prime. Then $p | b$ iff $p | b^2$.
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#proof[
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First let us show $p | b => p | b^2$. We know $b$ can be written as the
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product of the prime $p$ and some other integer $n$.
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$ b = p n $
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Then
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$ b^2 = p^2 n^2 $
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which implies $p$ is a factor of and divides $b^2$. So,
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$ p | b => p | b^2 $
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Now let us show the converse, i.e. $p | b^2 => p | b$.
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By the unique factorization theorem, $b^2$ can be written as a unique
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product of primes, one of which is $p$. But we also know $b^2 = b dot b$
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and so at least one $b$ must have a prime factor $p$. But $b$ has unique
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prime factors (again by the same theorem) so $b$ always has a prime factor
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$p$. Hence,
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$ p | b^2 => p | b $
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]
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]
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@ -391,9 +409,14 @@ non-perfect square is irrational.
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Show that $sqrt(15)$ is irrational.
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]
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#exercise("Euclid's Theorem")[
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#problem("Euclid's Theorem")[
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Show that there are an infinite amount of prime numbers.
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]
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]<euclid>
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#problem[
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Show that in general, given any integer $n$ that is not a perfect square,
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i.e. $ exists.not a in ZZ, n = a^2 $ $sqrt(n)$ is irrational.
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]<perfectsquare>
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== Proofs involving quantifiers
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@ -443,3 +466,21 @@ $ exists x in U, P(x) $
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Prove that between any two rational numbers $x$ and $y$ there is another
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rational number $z$.
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]
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== Solutions
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Solutions to selected problems and exercises.
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#linebreak()
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*@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 $q = P + 1$.
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Then if $q$ is prime, we have an additional prime not in the original list.
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Otherwise, $q$ is not prime and we have a unique prime factorization of $q$.
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Without loss of generality, take one such prime to be $p_k$. $p_k$ cannot be in
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the original list $p_1, p_2, ..., p_n$.
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If $p_k$ were in the original list, then since $P$ is divisible by $p_k$, and $P
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+ 1$ is also divisible by $p_k$, 1 must be divisible by $p_k$ which is
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impossible. So $p_k$ is a new prime.
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Reference in a new issue