Compare commits
6 Commits
attributio
...
2f6a44627f
Author | SHA1 | Date | |
---|---|---|---|
2f6a44627f | |||
313c4fb4c6
|
|||
c4ea1aa5c5
|
|||
5b84604d79
|
|||
03a2e920f8
|
|||
0ae3cdfb22
|
3
.vscode/settings.json
vendored
3
.vscode/settings.json
vendored
@ -1,5 +1,4 @@
|
||||
{
|
||||
"latex-workshop.latex.recipe.default": "latexmk (xelatex)",
|
||||
"tinymist.formatterPrintWidth": 80,
|
||||
"tinymist.typstExtraArgs": ["--package-path=./lib/typst"]
|
||||
"tinymist.formatterPrintWidth": 80
|
||||
}
|
||||
|
@ -1,3 +0,0 @@
|
||||
[authors."mark"]
|
||||
email = "mark@betalupi.com"
|
||||
webpage = "betalupi.com"
|
@ -1,6 +0,0 @@
|
||||
[package]
|
||||
name = "handout"
|
||||
version = "0.1.0"
|
||||
entrypoint = "handout.typ"
|
||||
authors = []
|
||||
license = "GPL"
|
@ -1,5 +1,3 @@
|
||||
/// Typst handout library, used for all documents in this repository.
|
||||
|
||||
|
||||
/// If false, hide instructor info.
|
||||
///
|
||||
@ -240,8 +238,8 @@
|
||||
|
||||
set page(
|
||||
margin: 20mm,
|
||||
width: 8in,
|
||||
height: 11.5in,
|
||||
width: 8.5in,
|
||||
height: 11in,
|
||||
footer: align(
|
||||
center,
|
||||
context counter(page).display(),
|
@ -1,4 +1,4 @@
|
||||
#import "@local/handout:0.1.0": *
|
||||
#import "./handout.typ": *
|
||||
#import "@preview/cetz:0.3.1"
|
||||
|
||||
|
||||
|
@ -1,4 +1,4 @@
|
||||
#import "@local/handout:0.1.0": *
|
||||
#import "./handout.typ": *
|
||||
|
||||
#show: doc => handout(
|
||||
doc,
|
||||
|
@ -1,4 +1,4 @@
|
||||
#import "@local/handout:0.1.0": *
|
||||
#import "../handout.typ": *
|
||||
#import "../macros.typ": *
|
||||
|
||||
= Tropical Arithmetic
|
||||
@ -62,7 +62,8 @@ Let's expand $#sym.RR$ to include a tropical additive identity.
|
||||
#problem()
|
||||
Do tropical additive inverses exist? \
|
||||
#note([
|
||||
Is there an inverse $y$ for every $x$ so that $x #tp y = #sym.infinity$?
|
||||
Is there an inverse $y$ for every $x$ so that $x #tp y = #sym.infinity$? \
|
||||
Remember that $#sym.infinity$ is the additive identity.
|
||||
])
|
||||
|
||||
#solution([
|
||||
@ -277,7 +278,7 @@ Fill the following tropical addition and multiplication tables
|
||||
|
||||
#problem()
|
||||
Expand and simplify $f(x) = (x #tp 2)(x #tp 3)$, then evaluate $f(1)$ and $f(4)$ \
|
||||
Adjacent parenthesis imply tropical multiplication
|
||||
#hint([Adjacent parenthesis imply tropical multiplication])
|
||||
|
||||
#solution([
|
||||
$
|
||||
|
@ -1,4 +1,4 @@
|
||||
#import "@local/handout:0.1.0": *
|
||||
#import "../handout.typ": *
|
||||
#import "../macros.typ": *
|
||||
#import "@preview/cetz:0.3.1"
|
||||
|
||||
@ -20,7 +20,7 @@ for some nonnegative integer $n$ and coefficients $c_0, c_1, ..., c_n$. \
|
||||
The _degree_ of a polynomial is the largest $n$ for which $c_n$ is nonzero.
|
||||
|
||||
#theorem()
|
||||
The _fundamental theorem of algebra_ implies that any non-constant polynomial with real coefficients
|
||||
The _fundamental theorem of algebra_ states that any non-constant polynomial with real coefficients
|
||||
can be written as a product of polynomials of degree 1 or 2 with real coefficients.
|
||||
|
||||
#v(2mm)
|
||||
@ -30,8 +30,8 @@ can be written as $(x^2 + 2x+5)(x-2)(x+4)(x+4)$
|
||||
|
||||
#v(2mm)
|
||||
A similar theorem exists for polynomials with complex coefficients. \
|
||||
These coefficients may be found using the roots of this polynomial. \
|
||||
As it turns out, there are formulas that determine the roots of quadratic, cubic, and quartic #note([(degree 2, 3, and 4)]) polynomials. There are no formulas for the roots of polynomials with larger degrees---in this case, we usually rely on approximate roots found by computers.
|
||||
These coefficients may be found using the _roots_ of this polynomial. \
|
||||
As you already know, there are formulas that determine the roots of quadratic, cubic, and quartic #note([(degree 2, 3, and 4)]) polynomials. There are no formulas for the roots of polynomials with larger degrees---in this case, we usually rely on appropriate roots found by computers.
|
||||
|
||||
#v(2mm)
|
||||
In this section, we will analyze tropical polynomials:
|
||||
@ -100,7 +100,7 @@ In other words, find $r$ and $s$ so that
|
||||
),
|
||||
)
|
||||
|
||||
we will call $r$ and $s$ the _roots_ of $f$.
|
||||
#note([Naturally, we will call $r$ and $s$ the _roots_ of $f$.])
|
||||
|
||||
#solution([
|
||||
Because $(x #tp r)(x #tp s) = x^2 #tp (r #tp s)x #tp s r$, we must have $r #tp s = 1$ and $r #tm s = 4$. \
|
||||
@ -116,7 +116,8 @@ we will call $r$ and $s$ the _roots_ of $f$.
|
||||
#v(1fr)
|
||||
|
||||
#problem()
|
||||
How can we use the graph to determine these roots?
|
||||
Can you see the roots of this polynomial in the graph? \
|
||||
#hint([Yes, you can. What "features" do the roots correspond to?])
|
||||
|
||||
#solution([The roots are the corners of the graph.])
|
||||
|
||||
@ -316,7 +317,7 @@ into linear factors.
|
||||
#v(2mm)
|
||||
|
||||
Whenever we say "the roots of $f$", we really mean "the roots of $accent(f, macron)$." \
|
||||
$f$ and $accent(f, macron)$ might be the same polynomial.
|
||||
Also, $f$ and $accent(f, macron)$ might be the same polynomial.
|
||||
|
||||
#problem()
|
||||
If $f(x) = a x^2 #tp b x #tp c$, then $accent(f, macron)(x) = a x^2 #tp B x #tp c$ for some $B$. \
|
||||
|
@ -1,4 +1,4 @@
|
||||
#import "@local/handout:0.1.0": *
|
||||
#import "../handout.typ": *
|
||||
#import "../macros.typ": *
|
||||
#import "@preview/cetz:0.3.1"
|
||||
|
||||
|
Binary file not shown.
Binary file not shown.
@ -4,8 +4,3 @@ title = "Odd Dice"
|
||||
[publish]
|
||||
handout = true
|
||||
solutions = true
|
||||
|
||||
[[attribution]]
|
||||
who = "mark"
|
||||
when = 2024-02-13
|
||||
what = "Initial version of handout"
|
||||
|
@ -143,12 +143,10 @@ def build_typst(source_dir: Path, out_subdir: Path) -> IndexEntry | None:
|
||||
[
|
||||
TYPST_PATH,
|
||||
"compile",
|
||||
"--package-path",
|
||||
f"{ROOT}/lib/typst",
|
||||
"--ignore-system-fonts",
|
||||
"main.typ",
|
||||
"--input",
|
||||
"show_solutions=false",
|
||||
"main.typ",
|
||||
f"{out}/{handout_file}",
|
||||
],
|
||||
cwd=source_dir,
|
||||
@ -166,8 +164,6 @@ def build_typst(source_dir: Path, out_subdir: Path) -> IndexEntry | None:
|
||||
[
|
||||
TYPST_PATH,
|
||||
"compile",
|
||||
"--package-path",
|
||||
f"{ROOT}/lib/typst",
|
||||
"--ignore-system-fonts",
|
||||
"main.typ",
|
||||
f"{out}/{solutions_file}",
|
||||
|
Reference in New Issue
Block a user