Complex numbers and the iterative command.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -241,7 +241,9 @@
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+    "algebra/complex.md": "37befb93643418cd8905c49e2f53627b"
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notes/_journal/2025-01-02.md

@@ -0,0 +1,12 @@
+---
+title: "2025-01-02"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Add a number of basic facts about [[complex|complex numbers]].
+* Introductory notes on the [[pred-trans#Iterative|iterative command]].

notes/algebra/complex.md

@@ -0,0 +1,246 @@
+---
+title: Complex Numbers
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algebra::complex
+tags:
+  - algebra
+  - complex
+---
+
+## Overview
+
+The set $\mathbb{C}$ of **complex numbers** is defined by $$\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\},$$
+where $i$ is the **imaginary number** defined as $i^2 = -1$.
+
+%%ANKI
+Basic
+How is set the complex numbers denoted?
+Back: As $\mathbb{C}$.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487309-->
+END%%
+
+%%ANKI
+Basic
+How is set $\mathbb{C}$ defined in set-builder notation?
+Back: $\mathbb{C} = \{a + bi \mid a, b \in \mathbb{R}\}$
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487353-->
+END%%
+
+%%ANKI
+Basic
+Which of $\mathbb{R}$ or $\mathbb{C}$ is a subset of the other?
+Back: $\mathbb{R} \subseteq \mathbb{C}$
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487367-->
+END%%
+
+%%ANKI
+Basic
+What is $i$ called?
+Back: The imaginary number.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487383-->
+END%%
+
+%%ANKI
+Basic
+How is the imaginary number typically denoted?
+Back: As $i$.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487411-->
+END%%
+
+%%ANKI
+Basic
+$i$ was invented to provide a solution to what equation?
+Back: $x^2 = -1$
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487426-->
+END%%
+
+%%ANKI
+Basic
+What is the solution of $x^2 = -1$?
+Back: $i$
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487437-->
+END%%
+
+%%ANKI
+Cloze
+Real number {$r$} is identified with complex number {$r + 0i$}.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487452-->
+END%%
+
+%%ANKI
+Cloze
+What real number is identified with $-\pi + 0i$?
+Back: $-\pi$
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487459-->
+END%%
+
+%%ANKI
+Cloze
+What real number is identified with $\pi + 2i$?
+Back: N/A.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487463-->
+END%%
+
+%%ANKI
+Basic
+What is the horizontal axis of the complex plane typically called?
+Back: The real axis.
+Reference: “Complex Plane,” in _Wikipedia_, December 14, 2024, [https://en.wikipedia.org/w/index.php?title=Complex_plane](https://en.wikipedia.org/w/index.php?title=Complex_plane&oldid=1263031649).
+<!--ID: 1735870487466-->
+END%%
+
+%%ANKI
+Basic
+What is the vertical axis of the complex plane typically called?
+Back: The imaginary axis.
+Reference: “Complex Plane,” in _Wikipedia_, December 14, 2024, [https://en.wikipedia.org/w/index.php?title=Complex_plane](https://en.wikipedia.org/w/index.php?title=Complex_plane&oldid=1263031649).
+<!--ID: 1735870487469-->
+END%%
+
+%%ANKI
+Cloze
+The complex plane is formed from the {$x$}-axis and {$yi$}-axis.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487472-->
+END%%
+
+%%ANKI
+Basic
+Which number is plotted on the complex plane below?
+![[complex-plane-point.png]]
+Back: $2 + i$
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487475-->
+END%%
+
+%%ANKI
+Cloze
+Real numbers are plotted on a {line} whereas complex numbers are plotted on a {plane}.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487478-->
+END%%
+
+## Operations
+
+Addition and multiplication of complex numbers are done in the natural way. Given complex numbers $a + bi$ and $c + di$, we have that $$\begin{align*} (a + bi) + (c + di) & = (a + c) + (b + d)i \\ (a + bi) \cdot (c + di) & = (ac -bd) + (ad + bc)i \end{align*}$$
+
+The **absolute value** of $a + bi$, denoted $\lvert a + bi \rvert$, is defined as $$\lvert a + bi \rvert = \sqrt{a^2 + b^2}.$$
+
+%%ANKI
+Basic
+Let $a + bi$ and $c + di$ be complex numbers. What is their sum?
+Back: $(a + c) + (b + d)i$
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487481-->
+END%%
+
+%%ANKI
+Basic
+Let $a + bi$ and $c + di$ be complex numbers. What is their product?
+Back: $(ac - bd) + (ad + bc)i$
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487484-->
+END%%
+
+%%ANKI
+Basic
+Is addition of complex numbers commutative?
+Back: Yes.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487487-->
+END%%
+
+%%ANKI
+Basic
+Is addition of complex numbers associative?
+Back: Yes.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487491-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for addition of complex numbers to be commutative?
+Back: For $z_1, z_2 \in \mathbb{C}$, it follows that $z_1 + z_2 = z_2 + z_1$.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487494-->
+END%%
+
+%%ANKI
+Basic
+Is multiplication of complex numbers commutative?
+Back: Yes.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487498-->
+END%%
+
+%%ANKI
+Basic
+Is multiplication of complex numbers associative?
+Back: Yes.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487502-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for multiplication of complex numbers to be associative?
+Back: For $z_1, z_2, z_3 \in \mathbb{C}$, it follows that $z_1(z_2z_3) = (z_1z_2)z_3$.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870487506-->
+END%%
+
+%%ANKI
+Basic
+How is the absolute value of complex number $z \in \mathbb{C}$ denoted?
+Back: As $\lvert z \rvert$.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870829668-->
+END%%
+
+%%ANKI
+Basic
+Let $z \in \mathbb{C}$. How is $\lvert z \rvert$ defined?
+Back: Assuming $z = a + bi$, as $\lvert z \rvert = \sqrt{a^2 + b^2}$.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870829671-->
+END%%
+
+%%ANKI
+Basic
+Geometrically speaking, what does the absolute value of $z \in \mathbb{C}$ correspond to?
+Back: $z$'s distance from the complex plane's origin.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870829675-->
+END%%
+
+%%ANKI
+Basic
+Let $a + bi$ be a complex number. How is $\sqrt{a^2 + b^2}$ more compactly written?
+Back: As $\lvert a + bi \rvert$.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870829679-->
+END%%
+
+%%ANKI
+Basic
+What geometric theorem motivates the definition of complex numbers' absolute values?
+Back: The Pythagorean theorem.
+Reference: John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).
+<!--ID: 1735870829684-->
+END%%
+
+## Bibliography
+
+* “Complex Plane,” in _Wikipedia_, December 14, 2024, [https://en.wikipedia.org/w/index.php?title=Complex_plane](https://en.wikipedia.org/w/index.php?title=Complex_plane&oldid=1263031649).
+* John B. Fraleigh, _A First Course in Abstract Algebra_, Seventh edition, Pearson new international edition (Harlow: Pearson, 2014).

notes/data-models/rdf.md

@@ -201,12 +201,12 @@ END%%
 
 ## Blank Nodes
 
-A **blank node** (bnode) is a node in an RDF graph representing a resource for which a [[uri|URI]] is not specified. That is, the represented resource is anonymous. Such a node can only be used as a subject or object in an RDF triple.
+A **blank node** (bnode) is a node in an RDF graph representing a resource for which a [[uri|IRI]] is not specified. That is, the represented resource is anonymous. Such a node can only be used as a subject or object in an RDF triple.
 
 %%ANKI
 Basic
 What is a blank node?
-Back: A node in an RDF graph representing a resource with an unspecified URI.
+Back: A node in an RDF graph representing a resource with an unspecified IRI.
 Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
 <!--ID: 1735162429073-->
 END%%
@@ -214,7 +214,6 @@ END%%
 %%ANKI
 Cloze
 A {bnode} is shorthand for a {blank node}.
-Back: A node in an RDF graph representing a resource with an unspecified URI.
 Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
 <!--ID: 1735162429077-->
 END%%

notes/encoding/floating-point.md

@@ -850,6 +850,15 @@ Tags: c17
 <!--ID: 1710605798327-->
 END%%
 
+%%ANKI
+Basic
+Let `float x = 1.0`. What does `x`'s exponent *value* equal?
+Back: $0$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1735827481751-->
+END%%
+
 %%ANKI
 Basic
 Let `double x = 1.0`. What is the bit representation of `x`'s exponent *field*?

notes/linkers/relocatable.md

@@ -225,21 +225,23 @@ Basic
 What C variables are marked `COMMON` instead of put in `.bss`?
 Back: Global uninitialized variables.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
 <!--ID: 1735343812827-->
 END%%
 
 %%ANKI
 Basic
 What C variables are put in `.bss` instead of marked `COMMON`?
-Back: Static variables or global variables initialized to zero.
+Back: Static variables and global variables initialized to zero.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
 <!--ID: 1735343812828-->
 END%%
 
 %%ANKI
 Basic
 Assuming `-fcommon`, what kind of C variables does the `.bss` section contain?
-Back: Static variables or global and static variables initialized to zero.
+Back: Static variables and global variables initialized to zero.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: c17
 <!--ID: 1735343812829-->

notes/programming/pred-trans.md

@@ -878,6 +878,14 @@ The general form of the **alternative command** is: $$\begin{align*} \textbf{if
 
 Each $B_i \rightarrow S_i$ is called a **guarded command**. To execute the alternative command, find one true guard and execute the corresponding command. Notice this is nondeterministic. We denote the alternative command as $\text{IF}$ and define $\text{IF}$ in terms of $wp$ as: $$\begin{align*} wp(\text{IF}, R) = \;& (\forall i, 1 \leq i \leq n \Rightarrow domain(B_i)) \;\land \\ & (\exists i, 1 \leq i \leq n \land B_i) \;\land \\ & (\forall i, 1 \leq i \leq n \Rightarrow (B_i \Rightarrow wp(S_i, R))) \end{align*}$$
 
+%%ANKI
+Basic
+The conventional `if` statement corresponds to what command?
+Back: The alternative command.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377633-->
+END%%
+
 %%ANKI
 Basic
 How is the alternative command compactly denoted?
@@ -960,6 +968,14 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1722256906214-->
 END%%
 
+%%ANKI
+Basic
+Suppose two guards of an alternative command is true. Which is chosen?
+Back: Either is permitted.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377661-->
+END%%
+
 %%ANKI
 Basic
 When *might* the following alternative command abort? $$\begin{align*} \textbf{if } & x > 0 \rightarrow z \coloneqq x \\ \textbf{ | } & x < 0 \rightarrow z \coloneqq -x \\ \textbf{fi } & \end{align*}$$
@@ -1015,6 +1031,98 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1722259243640-->
 END%%
 
+### Iterative
+
+The general form of the **iterative command** is: $$\begin{align*} \textbf{do } & B_1 \rightarrow S_1 \\ \textbf{ | } & B_2 \rightarrow S_2 \\ & \quad\cdots \\ \textbf{ | } & B_n \rightarrow S_n \\ \textbf{od } & \end{align*}$$
+
+%%ANKI
+Basic
+The conventional `while` statement corresponds to what command?
+Back: The iterative command.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377664-->
+END%%
+
+%%ANKI
+Cloze
+{1:$\text{IF}$} is to the {2:alternative} command whereas {2:$\text{DO}$} is to the {1:iterative} command.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377667-->
+END%%
+
+%%ANKI
+Basic
+How is the iterative command compactly denoted?
+Back: As $\text{DO}$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377671-->
+END%%
+
+%%ANKI
+Basic
+What kind of command is $\text{DO}$ a representation of?
+Back: An iterative command.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377674-->
+END%%
+
+%%ANKI
+Basic
+What is the general form of the iterative command?
+Back: $$\begin{align*} \textbf{do } & B_1 \rightarrow S_1 \\ \textbf{ | } & B_2 \rightarrow S_2 \\ & \quad\cdots \\ \textbf{ | } & B_n \rightarrow S_n \\ \textbf{od } & \end{align*}$$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377678-->
+END%%
+
+%%ANKI
+Basic
+How are iterative commands executed?
+Back: By repeatedly choosing any true guard and executing the corresponding command.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377683-->
+END%%
+
+%%ANKI
+Basic
+What does it mean to "perform an iteration" of an iterative command?
+Back: Choosing a true guard and executing its command.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377687-->
+END%%
+
+%%ANKI
+Basic
+In what way is the iterative command's execution different from traditional loop statements?
+Back: It is nondeterministic.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377691-->
+END%%
+
+%%ANKI
+Basic
+Suppose two guards of an iterative command is true. Which is chosen?
+Back: Either is permitted.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377695-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten to have just one iterative guard? $$\begin{align*} \textbf{do } & B_1 \rightarrow S_1 \\ \textbf{ | } & B_2 \rightarrow S_2 \\ & \quad\cdots \\ \textbf{ | } & B_n \rightarrow S_n \\ \textbf{od } & \end{align*}$$
+Back: Given $BB = B_1 \lor \cdots \lor B_n$, as $$\begin{align*} \textbf{do } & BB \rightarrow \textbf{if } B_1 \rightarrow S_1 \\ & \quad\quad\quad \textbf{ | } B_2 \rightarrow S_2 \\ & \quad\quad\quad \quad\cdots \\ & \quad\quad\quad \textbf{ | } B_n \rightarrow S_n \\ & \quad\quad\quad \textbf{fi } \\ \textbf{od } & \end{align*}$$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873377699-->
+END%%
+
+%%ANKI
+Basic
+Which command is demonstrated in the following diagram?
+![[iterative-command.png]]
+Back: The iterative command.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1735873599850-->
+END%%
+
 ## Bibliography
 
 * Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.