diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index b34d610..07c87b9 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -88,7 +88,10 @@ "ordering-repetition.jpg", "ordering-y-repetition-n.jpg", "ordering-n-repetition-y.jpg", - "ordering-n-repetition-n.jpg" + "ordering-n-repetition-n.jpg", + "triangular-gnomon.png", + "pascals-triangle.webp", + "square-gnomon.png" ], "File Hashes": { "algorithms/index.md": "cd7c7ba91fb2f961c9f2437777e8e2ac", @@ -219,13 +222,17 @@ "filesystems/cas.md": "34906013a2a60fe5ee0e31809b4838aa", "git/objects.md": "b95228a78744d3f9fe173e575aa0445a", "git/index.md": "83d2d95fc549d9e8436946c7bd058d15", - "encoding/integer.md": "5dd3a961bf259e53da207209051b8335", + "encoding/integer.md": "3107074a5070670f2f8fab17abc05fbb", "_journal/2024-02-29.md": "f610f3caed659c1de3eed5f226cab508", "_journal/2024-02/2024-02-28.md": "7489377c014a2ff3c535d581961b5b82", "_journal/2024-03-01.md": "a532486279190b0c12954966cbf8c3fe", "_journal/2024-02/2024-02-29.md": "0e502a2c8baf90c2f12859b03f10b5a1", "algebra/sequences.md": "97c217823aacf8910a1a37bde694ecfe", - "algebra/sequences/index.md": "e5a7cdfbcb61709ce2963c4b5e53a8f2" + "algebra/sequences/index.md": "e5a7cdfbcb61709ce2963c4b5e53a8f2", + "_journal/2024-03-02.md": "08c3cae1df0079293b47e1e9556f1ce1", + "_journal/2024-03/2024-03-01.md": "70da812300f284df72718dd32fc39322", + "algebra/sequences/triangular-numbers.md": "18925b0ecae151c3e6d38bc018c632c4", + "algebra/sequences/square-numbers.md": "886fb22fb8dbfffdd2cd233558ea3424" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-03-02.md b/notes/_journal/2024-03-02.md new file mode 100644 index 0000000..d895dfa --- /dev/null +++ b/notes/_journal/2024-03-02.md @@ -0,0 +1,13 @@ +--- +title: "2024-03-02" +--- + +- [x] Anki Flashcards +- [x] KoL +- [ ] Sheet Music (10 min.) +- [ ] Go (1 Life & Death Problem) +- [ ] Korean (Read 1 Story) +- [ ] Interview Prep (1 Practice Problem) +- [ ] Log Work Hours (Max 3 hours) + +* Notes on triangular and square numbers. \ No newline at end of file diff --git a/notes/_journal/2024-03-01.md b/notes/_journal/2024-03/2024-03-01.md similarity index 83% rename from notes/_journal/2024-03-01.md rename to notes/_journal/2024-03/2024-03-01.md index b564999..8c06514 100644 --- a/notes/_journal/2024-03-01.md +++ b/notes/_journal/2024-03/2024-03-01.md @@ -8,7 +8,7 @@ title: "2024-03-01" - [ ] Go (1 Life & Death Problem) - [ ] Korean (Read 1 Story) - [ ] Interview Prep (1 Practice Problem) -- [ ] Log Work Hours (Max 3 hours) +- [x] Log Work Hours (Max 3 hours) * Reviewed database reading/videos from yesterday with Kevin. * Finished planning soft skills course with Gus. diff --git a/notes/algebra/sequences/images/square-gnomon.png b/notes/algebra/sequences/images/square-gnomon.png new file mode 100644 index 0000000..7f3c86a Binary files /dev/null and b/notes/algebra/sequences/images/square-gnomon.png differ diff --git a/notes/algebra/sequences/images/triangular-gnomon.png b/notes/algebra/sequences/images/triangular-gnomon.png new file mode 100644 index 0000000..5ee5863 Binary files /dev/null and b/notes/algebra/sequences/images/triangular-gnomon.png differ diff --git a/notes/algebra/sequences/square-numbers.md b/notes/algebra/sequences/square-numbers.md new file mode 100644 index 0000000..a44550f --- /dev/null +++ b/notes/algebra/sequences/square-numbers.md @@ -0,0 +1,132 @@ +--- +title: Square Numbers +TARGET DECK: Obsidian::STEM +FILE TAGS: algebra::sequence +tags: + - algebra + - sequence +--- + +## Overview + +The $n$th term of the **square numbers** $(s_n)_{n \geq 0}$ is $n^2$. The first few terms are $$0, 1, 4, 9, 16, 25, 36, 49, 64, \ldots$$ + +%%ANKI +Basic +What shape do gnomons associated with square numbers take on? +Back: L-shapes. +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +%%ANKI +Basic +How are gnomons of the square numbers visualized? +Back: +![[square-gnomon.png]] +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +%%ANKI +Basic +What general term refers to the different colored segments in the following? +![[square-gnomon.png]] +Back: Gnomons. +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +%%ANKI +Basic +What are the first five square numbers $(s_n)_{n \geq 0}$? +Back: $0, 1, 4, 9, 16$ +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +%%ANKI +Basic +How is square number $16$ graphically depicted? +Back: +``` +* * * * +* * * * +* * * * +* * * * +``` +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +%%ANKI +Basic +What closed formula is used to find the $n$th square number? +Back: $n^2$ +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +%%ANKI +Basic +What is the recurrence relation in the recursive definition of square numbers $(s_n)_{n \geq 0}$? +Back: $s_n = s_{n-1} + (2n - 1)$ +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +%%ANKI +What is the initial condition(s) in the recursive definition of square numbers $(s_n)_{n \geq 0}$? +Back: $s_0 = 0$ +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). +END%% + +%%ANKI +Basic +How is the $n$th square number $s_n$ represented with sigma notation? +Back: $s_n = \sum_{k=1}^n (2k - 1)$ +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +%%ANKI +Basic +Which polygonal numbers are the "next" generalization of triangular numbers? +Back: The square numbers. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Cloze +The sum of {1:natural numbers} is to {2:triangular numbers} whereas the sum of {2:odd natural numbers} is to {1:square numbers}. +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +%%ANKI +Basic +What polygonal number is $k$ equal to after the following `for` loops? +```c +int k = 0; +for (int i = 1; i <= n; ++i) { + k += 2 * i - 1; +} +``` +Back: The $n$th square number. +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +%%ANKI +Basic +*Why* are square numbers a sum of odd numbers? +Back: The gnomon of a square number is twice the width of the previous square, plus the corner. +Reference: “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). + +END%% + +## References + +* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). +* “Square Number,” in _Wikipedia_, May 10, 2023, [https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731](https://en.wikipedia.org/w/index.php?title=Square_number&oldid=1154182731). \ No newline at end of file diff --git a/notes/algebra/sequences/triangular-numbers.md b/notes/algebra/sequences/triangular-numbers.md new file mode 100644 index 0000000..420bd41 --- /dev/null +++ b/notes/algebra/sequences/triangular-numbers.md @@ -0,0 +1,260 @@ +--- +title: Triangular Numbers +TARGET DECK: Obsidian::STEM +FILE TAGS: algebra::sequence +tags: + - algebra + - sequence +--- + +## Overview + +The $n$th term of the **triangular numbers** $(T_n)_{n \geq 0}$ is the sum of whole numbers $\sum_{k=0}^n k$. The first few terms are $$0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \ldots$$ + +%%ANKI +Basic +What is a polygonal number? +Back: A number of pebbles that can be arranged into the shape of a regular polygon. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What is a figurate number? +Back: Polygonal numbers or generalizations of polygonal numbers to other dimensions. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What are considered the simplest polygonal numbers? +Back: The triangular numbers. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +How do polygonal numbers relate to figurate numbers? +Back: Polygonal numbers are a subset of the figurate numbers. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What is a gnomon? +Back: The "piece" added to a figurate number to transform it to the next larger one. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What shape do gnomons associated with triangular numbers take on? +Back: Lines. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +How are gnomons of the triangular numbers visualized? +Back: +![[triangular-gnomon.png]] +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What general term refers to the highlighted portion of pebbles in the following? +![[triangular-gnomon.png]] +Back: Gnomons. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +The triangular numbers correspond to what kind of triangles? +Back: Equilateral triangles. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What is the first triangular *and* square number? +Back: $36$ +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What are the first five triangular numbers $(T_n)_{n \geq 0}$? +Back: $0, 1, 3, 6, 10$ +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +How is triangular number $10$ graphically depicted? +Back: +``` + * + * * + * * * +* * * * +``` +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +Algebraically speaking, *what* is the $n$th triangular number? +Back: $\sum_{k=1}^n k$. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What polygonal sequence is the summation analogue of factorial? +Back: The triangular numbers. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What notation does Knuth introduce to denote the $n$th triangular number? +Back: $n?$ +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What name does Knuth give the LHS of $n? = \sum_{k=1}^n k$? +Back: The termial. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Cloze +The {1:term}ial is to {2:$n?$} as the {2:factor}ial is to {1:$n!$}. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What closed formula is traditionally used to compute the $n$th triangular number? +Back: $\frac{n(n + 1)}{2}$ +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What is the recurrence relation in the recursive definition of triangular numbers $(T_n)_{n \geq 0}$? +Back: $T_n = T_{n-1} + n$ +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What is the initial condition(s) in the recursive definition of triangular numbers $(T_n)_{n \geq 0}$? +Back: $T_0 = 0$ +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +How do you expand sum $\sum_{k=1}^n k$ to derive closed formula $\frac{n(n + 1)}{2}$? +Back: +$$\begin{matrix} +1 & + & 2 & + & \cdots & + & n \\ +n & + & (n - 1) & + & \cdots & + & 1 \\ +\hline +(n + 1) & + & (n + 1) & + & \cdots & + & (n + 1) +\end{matrix}$$ +Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). + +END%% + +%%ANKI +Basic +What combinatorial closed formula is used to compute the $n$th triangular number? +Back: $\binom{n + 1}{2}$ +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What is the combinatorial explanation as to why the $n$th triangular number is $\binom{n + 1}{2}$? +Back: $\sum_{k=1}^n k$ is the number of ways distinct pairs can be made from $n + 1$ objects. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +Where in Pascal's triangle are the natural numbers embedded? +Back: Along the second leftward diagonal: +![[pascals-triangle.webp]] +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +Where in Pascal's triangle are the triangular numbers embedded? +Back: Along the third leftward diagonal: +![[pascals-triangle.webp]] +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +What polygonal number is $k$ equal to after the following `for` loops? +```c +int k = 0; +for (int i = 1; i <= n; ++i) { + k += i; +} +``` +Back: The $n$th triangular number. +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +%%ANKI +Basic +Why is $n(n + 1)$ geometrically significant w.r.t. the $n$th triangular number? +Back: $2 \cdot T_n$ is the number of units in an $n \times (n + 1)$ rectangle, e.g. +``` +* * * * - +* * * - - +* * - - - +* - - - - +``` +Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). + +END%% + +## References + +* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). +* “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122). \ No newline at end of file diff --git a/notes/combinatorics/images/pascals-triangle.webp b/notes/combinatorics/images/pascals-triangle.webp new file mode 100644 index 0000000..758e98d Binary files /dev/null and b/notes/combinatorics/images/pascals-triangle.webp differ diff --git a/notes/encoding/integer.md b/notes/encoding/integer.md index 6a7a968..c4029e3 100644 --- a/notes/encoding/integer.md +++ b/notes/encoding/integer.md @@ -949,7 +949,7 @@ END%% %%ANKI Basic Why should you generally prefer `x < y` over `x - y < 0`? -Back: The former avoids possible underflows. +Back: The former avoids possible overflows. Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. END%%