Notes on triangular and square numbers.

2024-03-02 16:37:08 -07:00 · 2024-03-02 16:37:08 -07:00 · aff3dbf43d
parent d27412d5c7
commit aff3dbf43d
9 changed files with 417 additions and 5 deletions
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@ -88,7 +88,10 @@
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-    "ordering-n-repetition-n.jpg"
+    "ordering-n-repetition-n.jpg",
+    "triangular-gnomon.png",
+    "pascals-triangle.webp",
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--- a/notes/_journal/2024-03-02.md
+++ 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.
--- a/notes/_journal/2024-03/2024-03-01.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.
--- a/notes/algebra/sequences/images/square-gnomon.png
+++ b/notes/algebra/sequences/images/square-gnomon.png
--- a/notes/algebra/sequences/images/triangular-gnomon.png
+++ b/notes/algebra/sequences/images/triangular-gnomon.png
--- a/notes/algebra/sequences/square-numbers.md
+++ 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).
+<!--ID: 1709422558613-->
+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).
+<!--ID: 1709422558619-->
+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).
+<!--ID: 1709422558622-->
+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).
+<!--ID: 1709422558625-->
+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).
+<!--ID: 1709422558628-->
+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).
+<!--ID: 1709422558631-->
+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).
+<!--ID: 1709422558634-->
+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).
+<!--ID: 1709422558638-->
+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).
+<!--ID: 1709419325898-->
+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).
+<!--ID: 1709422558641-->
+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).
+<!--ID: 1709422558645-->
+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).
+<!--ID: 1709422558648-->
+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).
--- a/notes/algebra/sequences/triangular-numbers.md
+++ 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).
+<!--ID: 1709419325851-->
+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).
+<!--ID: 1709419325856-->
+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).
+<!--ID: 1709419325859-->
+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).
+<!--ID: 1709419325862-->
+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).
+<!--ID: 1709419325865-->
+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).
+<!--ID: 1709419325874-->
+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).
+<!--ID: 1709419325878-->
+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).
+<!--ID: 1709419325883-->
+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).
+<!--ID: 1709419325887-->
+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).
+<!--ID: 1709419325891-->
+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).
+<!--ID: 1709419325904-->
+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).
+<!--ID: 1709419325909-->
+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).
+<!--ID: 1709419325914-->
+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).
+<!--ID: 1709419325918-->
+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).
+<!--ID: 1709419325922-->
+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).
+<!--ID: 1709419325927-->
+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).
+<!--ID: 1709419325931-->
+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).
+<!--ID: 1709419325936-->
+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).
+<!--ID: 1709422558652-->
+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).
+<!--ID: 1709422558656-->
+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).
+<!--ID: 1709419325942-->
+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).
+<!--ID: 1709419325949-->
+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).
+<!--ID: 1709419325956-->
+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).
+<!--ID: 1709419325963-->
+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).
+<!--ID: 1709419325970-->
+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).
+<!--ID: 1709419325976-->
+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).
+<!--ID: 1709419325981-->
+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).
--- a/notes/combinatorics/images/pascals-triangle.webp
+++ b/notes/combinatorics/images/pascals-triangle.webp
--- 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.
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