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Trigonometry and relocation entries.

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Joshua Potter 2025-01-19 21:11:02 -07:00
parent e25be7b823
commit cbeb26fbde
27 changed files with 1221 additions and 97 deletions
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58
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@ -235,7 +235,29 @@
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---
title: "2025-01-17"
---
- [ ] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)

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@ -0,0 +1,11 @@
---
title: "2025-01-19"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* More notes on relocation entries.

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@ -0,0 +1,13 @@
---
title: "2025-01-17"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Distinguish asserted and inferred RDF triples.
* Notes on the [[circle]] and [[trigonometry/index#Unit Circle|unit circle]].
* Formal definition of the [[pred-trans#Iterative|iterative]] command.

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@ -0,0 +1,11 @@
---
title: "2025-01-18"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Notes on [[relocatable#Relocation Entries|relocation entries]].

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notes/algebra/radices.md
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@ -202,6 +202,33 @@ Tags: binary::hex
<!--ID: 1707432641583-->
END%%
%%ANKI
Basic
What hexadecimal value does $2^{0 + 4(0)}$ evaluate to?
Back: `0x1`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1737232783457-->
END%%
%%ANKI
Basic
What hexadecimal value does $2^{0 + 4(2)}$ evaluate to?
Back: `0x100`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1737232783462-->
END%%
%%ANKI
Basic
Write `0x10000` in form $2^{i + 4j}$. What values of $i$ and $j$ satisfy this?
Back: $i = 0$ and $j = 4$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1737232783466-->
END%%
%%ANKI
Basic
How is the $1$ in $2^{1 + 4j}$ translated to hex?
@ -211,6 +238,33 @@ Tags: binary::hex
<!--ID: 1707432641585-->
END%%
%%ANKI
Basic
What hexadecimal value does $2^{1 + 4(2)}$ evaluate to?
Back: `0x200`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1737232783469-->
END%%
%%ANKI
Basic
What hexadecimal value does $2^{1 + 4(3)}$ evaluate to?
Back: `0x2000`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1737232783480-->
END%%
%%ANKI
Basic
Write `0x200` in form $2^{i + 4j}$. What values of $i$ and $j$ satisfy this?
Back: $i = 1$ and $j = 2$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1737232783488-->
END%%
%%ANKI
Basic
How is the $2$ (power) in $2^{2 + 4j}$ translated to hex?
@ -220,6 +274,24 @@ Tags: binary::hex
<!--ID: 1707432641586-->
END%%
%%ANKI
Basic
What hexadecimal value does $2^{2 + 4(1)}$ evaluate to?
Back: `0x40`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1737232783492-->
END%%
%%ANKI
Basic
Write `0x4000` in form $2^{i + 4j}$. What values of $i$ and $j$ satisfy this?
Back: $i = 2$ and $j = 3$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1737232783497-->
END%%
%%ANKI
Basic
How is the $3$ in $2^{3 + 4j}$ translated to hex?
@ -229,6 +301,24 @@ Tags: binary::hex
<!--ID: 1707432641587-->
END%%
%%ANKI
Basic
What hexadecimal value does $2^{3 + 4(0)}$ evaluate to?
Back: `0x8`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1737232783501-->
END%%
%%ANKI
Basic
Write `0x80` in form $2^{i + 4j}$. What values of $i$ and $j$ satisfy this?
Back: $i = 3$ and $j = 1$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1737232783505-->
END%%
%%ANKI
Basic
How is $n$ in $2^n$ factored to quickly write the decimal value's hex representation?

63
notes/algebra/sequences/triangular-numbers.md
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@ -15,7 +15,7 @@ The $n$th term of the **triangular numbers** $(T_n)_{n \geq 0}$ is the sum of wh
Basic
What is a polygonal number?
Back: A number of pebbles that can be arranged into the shape of a regular filled 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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325851-->
END%%
@ -23,7 +23,7 @@ END%%
Basic
What is a figurate number?
Back: Polygonal numbers and their generalizations 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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325856-->
END%%
@ -31,7 +31,7 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325859-->
END%%
@ -39,7 +39,7 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325862-->
END%%
@ -47,7 +47,7 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325865-->
END%%
@ -55,7 +55,7 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325874-->
END%%
@ -64,7 +64,7 @@ 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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325878-->
END%%
@ -73,7 +73,7 @@ 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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325883-->
END%%
@ -81,15 +81,24 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325887-->
END%%
%%ANKI
Basic
How do the triangular numbers correspond to *equilateral* triangles?
Back:
![[triangular-gnomon.png]]
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1737233067030-->
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325891-->
END%%
@ -97,7 +106,7 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325904-->
END%%
@ -111,7 +120,7 @@ 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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325909-->
END%%
@ -119,7 +128,7 @@ END%%
Basic
How is the $n$th triangular number written as a summation?
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325914-->
END%%
@ -127,7 +136,7 @@ END%%
Basic
What polygonal sequence is the summation analog 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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325918-->
END%%
@ -135,7 +144,7 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325922-->
END%%
@ -143,14 +152,14 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325931-->
END%%
@ -158,7 +167,7 @@ END%%
Basic
What closed formula is traditionally used to compute the $n$th triangular number?
Back: $\large{\frac{n(n + 1)}{2}}$
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325936-->
END%%
@ -166,7 +175,7 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709422558652-->
END%%
@ -174,7 +183,7 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709422558656-->
END%%
@ -196,7 +205,7 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325949-->
END%%
@ -204,7 +213,7 @@ END%%
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325956-->
END%%
@ -213,7 +222,7 @@ Basic
Where in Pascal's triangle are the natural numbers embedded?
Back: Along the second leftward diagonal:
![[pascals-triangle.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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325963-->
END%%
@ -222,7 +231,7 @@ Basic
Where in Pascal's triangle are the triangular numbers embedded?
Back: Along the third leftward diagonal:
![[pascals-triangle.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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325970-->
END%%
@ -236,7 +245,7 @@ for (int i = 1; i <= n; ++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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325976-->
END%%
@ -250,7 +259,7 @@ Back: $2 \cdot T_n$ is the number of units in an $n \times (n + 1)$ rectangle, e
* * - - -
* - - - -
```
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).
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
<!--ID: 1709419325981-->
END%%
@ -265,4 +274,4 @@ END%%
## Bibliography
* 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).
* “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).

1
notes/algorithms/dfs.md
View File

@ -534,6 +534,7 @@ END%%
%%ANKI
Basic
Why is edge $\langle b, s \rangle$ classified as a back edge?
![[dfs-edge-classification.png]]
Back: Because $s$ is an ancestor of $b$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

71
notes/combinators/index.md
View File

@ -335,6 +335,77 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
<!--ID: 1735413657674-->
END%%
%%ANKI
Basic
Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}\mathbf{S}))\mathbf{K})$ a combinator?
Back: N/A. It is.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1737212699921-->
END%%
%%ANKI
Basic
Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}x))((\mathbf{S}\mathbf{K})\mathbf{K}))$ a combinator?
Back: It contains atom $x$ which isn't a basic combinator.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1737212699927-->
END%%
%%ANKI
Basic
Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}0))((\mathbf{S}\mathbf{K})\mathbf{K}))$ a combinator?
Back: It contains atom $0$ which isn't a basic combinator.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1737212699932-->
END%%
%%ANKI
Basic
Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}\mathbf{S}))\mathbf{K})$ a closed term?
Back: N/A. It is.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1737212699937-->
END%%
%%ANKI
Basic
Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}x))((\mathbf{S}\mathbf{K})\mathbf{K}))$ a closed term?
Back: It contains variable $x$.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1737212699942-->
END%%
%%ANKI
Basic
Assume basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. *Why* isn't $((\mathbf{S}(\mathbf{K}0))((\mathbf{S}\mathbf{K})\mathbf{K}))$ a closed term?
Back: N/A. It is.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1737212699947-->
END%%
%%ANKI
Cloze
By convention, parentheses in combinatory logic are {left}-associative.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1737245400699-->
END%%
%%ANKI
Basic
How is $CL$-term $UVWX$ written with parentheses reintroduced?
Back: $(((UV)W)X)$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1737245400709-->
END%%
%%ANKI
Basic
In combinatory logic, is $UVW \equiv ((UV)W)$?
Back: Yes.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1737245400737-->
END%%
## Basic Combinators
The combinatory logic is a notation that eliminate the need for quantified variables. We start with basis $\mathbf{S}$, $\mathbf{K}$, and $\mathbf{I}$. These **basic combinators** are defined as:

7
notes/computability/index.md
View File

@ -247,11 +247,10 @@ Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third ed
END%%
%%ANKI
Basic
A language is a set satisfying what?
Back: It contains strings over some alphabet.
Cloze
A {language} is a set containing {strings} over some {alphabet}.
Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
<!--ID: 1734903366709-->
<!--ID: 1737232847428-->
END%%
%%ANKI

40
notes/data-models/rdf/rdfs.md
View File

@ -8,7 +8,7 @@ tags:
## Overview
The simplest extension to RDF that allows a modeler to manage inference is **RDF Schema** (RDFS).
The simplest extension to RDF that allows a modeler to manage inference is **RDF Schema** (RDFS). Triples that are inserted directly into an underlying RDF store are called **asserted triples**. Triples that are derived from inference rules are called **inferred triples**.
%%ANKI
Basic
@ -26,6 +26,44 @@ Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web
<!--ID: 1736629473653-->
END%%
%%ANKI
Cloze
An {asserted} triple is contrary to an {inferred} triple.
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: 1737167693491-->
END%%
%%ANKI
Basic
What does it mean for a triple to be asserted?
Back: The triple exists directly in the underlying RDF store.
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: 1737167693496-->
END%%
%%ANKI
Basic
What does it mean for a triple to be inferred?
Back: The triple is derived from some set of inference rules.
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: 1737167693501-->
END%%
%%ANKI
Cloze
{Asserted} triples are used to derive {inferred} triples.
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: 1737167693506-->
END%%
%%ANKI
Basic
When might a triple be both asserted and inferred?
Back: When the inference engine infers an already existing triple.
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: 1737167693510-->
END%%
## Classes
All things described by RDF are called **resources**. Resources may be divided into groups called **classes**. Classes are themselves resources. The `rdf:type` property may be used to state that a resource is an instance of a class. Associated with a class is a set, called the **class extension** of the class, which is the set of the instances of the class. A class may be a member of its own class extension.

16
notes/geometry/cartesian.md
View File

@ -213,22 +213,6 @@ Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” acces
<!--ID: 1734620943926-->
END%%
%%ANKI
Basic
Which of horizontal and/or vertical transformations "act inversely"?
Back: Horizontal transformations.
Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
<!--ID: 1734620943931-->
END%%
%%ANKI
Basic
Which of horizontal and/or vertical transformations "act normally"?
Back: Vertical transformations.
Reference: James Jones, “Shifting, Reflecting, and Stretching Graphs,” accessed December 6, 2024, [https://people.richland.edu/james/lecture/m116/functions/translations.html](https://people.richland.edu/james/lecture/m116/functions/translations.html).
<!--ID: 1734620943937-->
END%%
%%ANKI
Basic
How is the graph of $y = f(x)$ transformed in the graph of $y = \frac{1}{2}f(\frac{x}{3})$?

200
notes/geometry/circle.md Normal file
View File

@ -0,0 +1,200 @@
---
title: Circle
TARGET DECK: Obsidian::STEM
FILE TAGS: geometry::circle
tags:
- circle
- geometry
---
## Overview
A **circle** is a shape consisting of all points in a plane at a given distance from a given point, i.e. the **center**. In this way, a circle is characterized by its center and **radius**, the distance from the center to any point of the circle. The perimeter of a circle is known as its **circumference**.
On the [[cartesian|Cartesian coordinate system]], the equation of a circle with radius $r$ and center $\langle h, k \rangle$ is $$(x - h)^2 + (y - k)^2 = r^2.$$
Such a circle has circumference $2\pi r$ and [[area]] $\pi r^2$.
%%ANKI
Basic
A circle is characterized by what two properties?
Back: Its center and its radius.
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693438-->
END%%
%%ANKI
Basic
In plain English, describe what a circle is.
Back: A set of points at a given distance from some given point.
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693441-->
END%%
%%ANKI
Basic
The perimeter of a circle is known as what?
Back: Its circumference.
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693445-->
END%%
%%ANKI
Basic
What is the Cartesian equation of a circle with radius $r$ and center $\langle h, k \rangle$?
Back: $(x - h)^2 + (y - k)^2 = r^2$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693448-->
END%%
%%ANKI
Basic
Given indeterminates $x$ and $y$, the following equation describes what shape? $$(x - h)^2 + (y - k)^2 = r^2$$
Back: A circle with radius $r$ and center $\langle h, k \rangle$.
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693451-->
END%%
%%ANKI
Basic
Consider a circle with radius $r$. What does its circumference evaluate to?
Back: $2 \pi r$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693455-->
END%%
%%ANKI
Basic
Consider a circle with radius $r$. What is its area?
Back: $\pi r^2$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693458-->
END%%
%%ANKI
Basic
Consider a circle with diameter $d$. What does its circumference evaluate to?
Back: $\pi d$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693462-->
END%%
%%ANKI
Basic
Consider a circle with diameter $d$. What its area?
Back: $\pi \left(\frac{d}{2}\right)^2$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693465-->
END%%
%%ANKI
Basic
Consider a circle with radius $r$. What does the following evaluate to? $$2 \pi r$$
Back: Its circumference.
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693468-->
END%%
%%ANKI
Basic
Consider a circle with radius $r$. What does the following evaluate to? $$\pi r^2$$
Back: Its area.
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693472-->
END%%
%%ANKI
Basic
What is the Cartesian equation of the following shape?
![[unit-circle.png]]
Back: $x^2 + y^2 = 1$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693476-->
END%%
%%ANKI
Basic
What is the Cartesian equation of the following shape?
![[circle-right.png]]
Back: $(x - 2)^2 + y^2 = 4$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693481-->
END%%
%%ANKI
Basic
What is the Cartesian equation of the following shape?
![[circle-left-down.png]]
Back: $(x + 1)^2 + (y + 1)^2 = 4$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167693485-->
END%%
%%ANKI
Basic
What is the Cartesian equation of the following shape?
![[circle-left-up.png]]
Back: $(x + 1)^2 + (y - 1)^2 = 4$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737167904156-->
END%%
%%ANKI
Basic
Rewrite equation $x^2 + y^2 = 1$ shifted left by $a > 0$.
Back: $(x + a)^2 + y^2 = 1$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737168379559-->
END%%
%%ANKI
Basic
Rewrite equation $x^2 + y^2 = 1$ shifted up by $b > 0$.
Back: $x^2 + (y - b)^2 = 1$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737168379562-->
END%%
%%ANKI
Basic
Rewrite equation $x^2 + y^2 = 1$ shifted right by $a > 0$.
Back: $(x - a)^2 + y^2 = 1$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737168379565-->
END%%
%%ANKI
Basic
Rewrite equation $x^2 + y^2 = 1$ shifted down by $b > 0$.
Back: $x^2 + (y + b)^2 = 1$
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737168379568-->
END%%
%%ANKI
Basic
In what direction(s) is the unit circle shifted in the following? $$(x - 3)^2 + (y - 3)^2 = 1$$
Back: Right by $3$ and up by $3$.
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737168379571-->
END%%
%%ANKI
Basic
In what direction(s) is the unit circle shifted in the following? $$(x + 3)^2 + (y - 3)^2 = 1$$
Back: Left by $3$ and up by $3$.
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737168379574-->
END%%
%%ANKI
Basic
In what direction(s) is the unit circle shifted in the following? $$(x - 3)^2 + (y + 3)^2 = 1$$
Back: Right by $3$ and down by $3$.
Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).
<!--ID: 1737168379578-->
END%%
## Bibliography
* “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102).

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2
notes/lambda-calculus/index.md
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@ -214,7 +214,7 @@ END%%
%%ANKI
Basic
How is expression $\lambda x. \lambda y. MN$ written with parentheses reintroduced?
How is $\lambda$-term $\lambda x. \lambda y. MN$ written with parentheses reintroduced?
Back: $(\lambda x. (\lambda y. (MN)))$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1716498992530-->

53
notes/linkers/index.md
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@ -10,10 +10,8 @@ tags:
To build an executable, a linker must perform two main tasks:
1. **Symbol resolution**. The linker must associate each symbol reference with exactly one symbol definition.
2. **Relocation**. The linker must relocate code and data sections by associating a memory location with each symbol definition, and then modifying all of the references to those symbols so that they point to this memory location.
The linker blindly performs relocations using detailed instructions generated by the assembler called **relocation entries**.
1. **Symbol resolution**. The linker phase in which each symbol reference is associated with exactly one symbol definition.
2. **Relocation**. The linker phase in which code and data sections across input modules are combined. Each section, along with the symbols defined in them, are assigned unique run-time memory addresses. Additionally, references to symbols are updated so they point to the correct run-time addresses.
%%ANKI
Basic
@ -31,6 +29,14 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1733671136078-->
END%%
%%ANKI
Basic
After which linker phase is the size of the code and data sections of each input object module known?
Back: Symbol resolution.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1737257167718-->
END%%
%%ANKI
Basic
What is the goal of symbol resolution?
@ -39,6 +45,13 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1733671136081-->
END%%
%%ANKI
Cloze
The {symbol resolution} phase associates each {symbol reference} with exactly one {symbol definition}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1733671136092-->
END%%
%%ANKI
Basic
What is the goal of relocation?
@ -55,13 +68,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1733671136088-->
END%%
%%ANKI
Cloze
The {symbol resolution} phase associates each {symbol reference} with exactly one {symbol definition}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1733671136092-->
END%%
%%ANKI
Cloze
The {relocation} phase assigns a {memory location} to each symbol and {updates references} accordingly.
@ -79,24 +85,25 @@ END%%
%%ANKI
Basic
What is emitted by the assembler to help the linker relocate sections?
Back: Relocation entries.
Which linker phase combines code and data sections of the same type into a new aggregate section?
Back: Relocation.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1733671136107-->
<!--ID: 1737257167723-->
END%%
%%ANKI
Basic
Which linker phase is responsible for building up the `.symtab` section?
Back: Symbol resolution.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1737257167727-->
END%%
%%ANKI
Cloze
The assembler outputs {relocation entries} to guide the linker during {relocation}.
{Relocatable} object files are merged together to make an {executable} object file.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1733671136112-->
END%%
%%ANKI
Cloze
The {1:assembler} outputs relocation entries to guide the {1:linker} during relocation.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1733671136117-->
<!--ID: 1737257167732-->
END%%
## Object Files

329
notes/linkers/relocatable.md
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@ -637,6 +637,335 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1736632025924-->
END%%
## Relocation Entries
Whenever the assembler encounters a reference to an object whose ultimate location is unknown, it generates a **relocation entry** that tells the linker how to modify the reference when it merges the object file into an executable. Each entry looks something like:
```c
struct Elf64_Rela {
long offset; // Offset of the reference to relocate
long type : 32, // Relocation type
symbol : 32; // Symbol table index
long addend; // Additional constant used to bias the value
};
```
%%ANKI
Basic
What is emitted by the assembler to help the linker relocate sections?
Back: Relocation entries.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1733671136107-->
END%%
%%ANKI
Cloze
The assembler outputs {relocation entries} to guide the linker during {relocation}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1733671136112-->
END%%
%%ANKI
Cloze
The {1:assembler} outputs relocation entries to guide the {1:linker} during relocation.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1733671136117-->
END%%
%%ANKI
Basic
Which component of a compiler driver produces relocation entries?
Back: The assembler.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1737257167650-->
END%%
%%ANKI
Basic
How many relocation entries are produced for any given object module?
Back: One for every reference whose ultimate address is unknown.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1737257167677-->
END%%
%%ANKI
Basic
*Why* aren't relocation entries relevant for executable object files?
Back: All memory addresses should be resolved once the executable is produced.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1737257167683-->
END%%
%%ANKI
Cloze
Relocation entries for {data} are placed in {`.rel.data`}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1737257167690-->
END%%
%%ANKI
Cloze
Relocation entries for {code} are placed in {`.rel.text`}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1737257167696-->
END%%
%%ANKI
Basic
At what point during linking are relocation entries no longer necessary?
Back: When the fully merged executable object file is produced.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1737257167702-->
END%%
%%ANKI
Basic
Relocation entries are included in what kind of object module?
Back: Relocatable object modules.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1737257167707-->
END%%
%%ANKI
Basic
A relocation entry corresponds to what kind of C construct?
Back: A `struct`.
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: 1737257167712-->
END%%
%%ANKI
Basic
Consider the following ELF relocation entry. What is the purpose of `offset`?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: It is the offset of the reference to relocate.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385837-->
END%%
%%ANKI
Basic
Consider the following ELF relocation entry. What is the `offset` measured relative to?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: The start of the section the reference is located in.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385840-->
END%%
%%ANKI
Basic
Consider the following ELF relocation entry. What is the purpose of `symbol`?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: It identifies the symbol the modified reference should point to.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385843-->
END%%
%%ANKI
Basic
Consider the following ELF relocation entry. `symbol` is an index into what table?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: `.symtab`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385846-->
END%%
%%ANKI
Basic
Consider the following ELF relocation entry. What is the purpose of `addend`?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: To bias the value of the modified reference.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385849-->
END%%
%%ANKI
Basic
Consider the following ELF relocation entry. What is the purpose of `type`?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: To indicate to the linker what algorithm should be used to compute the relocated address.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385853-->
END%%
%%ANKI
Basic
Consider the following ELF relocation entry. What are the two most basic values of `type`?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: `R_X86_64_32` and `R_X86_64_PC32`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385856-->
END%%
%%ANKI
Cloze
{1:`R_X86_64_32`} is to {2:absolute} whereas {2:`R_X86_64_PC32`} is to {1:relative}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: linker::elf x86-64
<!--ID: 1737320385860-->
END%%
%%ANKI
Basic
What is the significance of `R` in type `R_X86_64_32`?
Back: It is the prefix used for relocation entry `type` values.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385864-->
END%%
%%ANKI
Basic
What is the significance of `32` in type `R_X86_64_32`?
Back: The reference is relocated using a 32-bit absolute address.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385867-->
END%%
%%ANKI
Basic
What is the significance of `PC` in type `R_X86_64_PC32`?
Back: It is short for **p**rogram **c**ounter.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385871-->
END%%
%%ANKI
Basic
What is the significance of `PC32` in type `R_X86_64_PC32`?
Back: The reference is relocated using a 32-bit PC-relative address.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385875-->
END%%
%%ANKI
Basic
`R_X86_64_32` is a possible value for what field in the following ELF relocation entry?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: `type`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385879-->
END%%
%%ANKI
Basic
`R_X86_64_PC32` is a possible value for what field in the following ELF relocation entry?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: `type`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385883-->
END%%
%%ANKI
Basic
Consider the following ELF relocation entry. What should `type` be if relocating PC-relatively?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: `R_X86_64_PC32`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385887-->
END%%
%%ANKI
Basic
Consider the following ELF relocation entry. What should `type` be if relocating absolutely?
```c
struct Elf64_Rela {
long offset;
long type : 32;
long symbol : 32;
long addend;
};
```
Back: `R_X86_64_32`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c17 linker::elf x86-64
<!--ID: 1737320385894-->
END%%
## Bibliography
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

127
notes/programming/pred-trans.md
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@ -1035,6 +1035,9 @@ END%%
The general form of the **iterative command** is: $$\begin{align*} \textbf{do } & B_1 \rightarrow S_1 \\ \textbf{ | } & B_2 \rightarrow S_2 \\ & \quad\cdots \\ \textbf{ | } & B_n \rightarrow S_n \\ \textbf{od } & \end{align*}$$
We denote the iterative command as $\text{DO}$ and define $\text{DO}$ in terms of $wp$ as: $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$
where $H_k$ is given [[algebra/sequences/index|recursive definition]]: $$\begin{align*} H_0(R) & = \neg (B_1 \lor \cdots \lor B_n) \land R \\ H_{k+1}(R) & = H_0(R) \lor wp(\text{IF}, H_k(R)) \end{align*}$$
%%ANKI
Basic
The conventional `while` statement corresponds to what command?
@ -1123,6 +1126,130 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
<!--ID: 1735873599850-->
END%%
%%ANKI
Cloze
{1:$\text{IF}$} is to {2:$abort$} whereas {2:$\text{DO}$} is to {1:$skip$}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609860-->
END%%
%%ANKI
Basic
Given associated recursive definition $H_k$, what is the formal definition of $\text{DO}$?
Back: For some predicate $R$, $wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609865-->
END%%
%%ANKI
Basic
In the following definition, what does $H_k(R)$ represent? $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$
Back: The set of states in which execution of $\text{DO}$ terminates in $k$ or fewer iterations.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609868-->
END%%
%%ANKI
Basic
In the following definition, how is $H_0$ defined? $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$
Back: Given guards $B_1, \ldots, B_n$, as $H_0 = \neg (B_1 \lor \cdots \lor B_n) \land R$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609870-->
END%%
%%ANKI
Basic
In the following definition, what set of states does $H_0(R)$ correspond to? $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$
Back: Those in which $\text{DO}$ finishes execution in $0$ iterations with $R$ true.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737188069810-->
END%%
%%ANKI
Basic
In the following definition, how is $H_{k+1}$ defined? $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$
Back: As $H_{k+1}(R) = H_0(R) \lor wp(\text{IF}, H_k(R))$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609873-->
END%%
%%ANKI
Basic
In the following definition, what set of states does $H_k(R)$ correspond to? $$wp(\text{DO}, R) = \exists k \geq 0, H_k(R)$$
Back: Those in which $\text{DO}$ finishes execution in $k$ or fewer iterations with $R$ true.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737188069815-->
END%%
%%ANKI
Basic
Let $H_k$ denote the associated recursive definition of $wp(\text{DO}, R)$. *Why* does $H_k \Rightarrow H_{k+1}$?
Back: $H_{k+1}$ is the set of states in which $\text{DO}$ finishes execution in $k$ *or fewer* iterations.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737188069819-->
END%%
%%ANKI
Basic
Let $H_k$ denote the associated recursive definition of $wp(\text{DO}, R)$. *Why* does $H_{k + 1} \Rightarrow H_k$?
Back: N/A. It doesn't.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737188069822-->
END%%
%%ANKI
Basic
How is the associated recursive definition of $wp(\text{DO}, R)$ described in plain English?
Back: As the set of states in which execution of $\text{DO}$ terminates in a finite number of iterations with $R$ true.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609876-->
END%%
%%ANKI
Cloze
Iterative command {$\textbf{do od}$} is equivalent to command {$skip$}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609879-->
END%%
%%ANKI
Basic
*Why* does command $\textbf{do od}$ skip?
Back: The $\text{DO}$ command iterates until no guard is satisfied.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609882-->
END%%
%%ANKI
Basic
What does $wp(\textbf{do } T \rightarrow skip \textbf{ od}, R)$ evaluate to?
Back: $F$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609885-->
END%%
%%ANKI
Basic
*Why* does $wp(\textbf{do } T \rightarrow skip \textbf{ od}, R)$ evaluate to $F$?
Back: $\textbf{do } T \rightarrow skip \text{ od}$ never terminates.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609888-->
END%%
%%ANKI
Cloze
The {$\text{DO}$} command corresponds to zero or more {$\text{IF}$} commands.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609892-->
END%%
%%ANKI
Cloze
Let $R$ be a predicate. Then $wp(\text{DO}, R) = \exists k,$ {$k \geq 0$} $\land$ $H_k(R)$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1737187609895-->
END%%
## Bibliography
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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%%ANKI
Basic
Let $A$ and $B$ be disjoint sets. $f \colon A \rightarrow B$ is an operation on what set?
Back: N/A.
Back: N/A. $f$ is not an operation.
Reference: “Operation (Mathematics).” In _Wikipedia_, October 10, 2024. [https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)](https://en.wikipedia.org/w/index.php?title=Operation_(mathematics)&oldid=1250395938).
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END%%

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---
title: Trigonometry
TARGET DECK: Obsidian::STEM
FILE TAGS: trigonometry
tags:
- trigonometry
---
## Overview
Trigonometry was originally derived from a Greek word meaning "triangle measuring". It has generalized to studying periodicity.
%%ANKI
Basic
Trigonometry was originally the study of what geometric shape?
Back: Triangles.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693405-->
END%%
## Unit Circle
On the [[cartesian|Cartesian coordinate system]], the **unit circle** is the [[circle]] with center at the origin and radius $1$.
%%ANKI
Basic
On the Cartesian coordinate system, what is the unit circle?
Back: The circle with center at the origin and radius $1$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693410-->
END%%
%%ANKI
Basic
On the Cartesian coordinate system, where is the center of the unit circle located?
Back: At $\langle 0, 0 \rangle$, i.e. the origin.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693413-->
END%%
%%ANKI
Basic
What is the radius of the unit circle?
Back: $1$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693416-->
END%%
%%ANKI
Basic
What is the diameter of the unit circle?
Back: $2$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693419-->
END%%
%%ANKI
Basic
What is the circumference of the unit circle?
Back: $2\pi$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693428-->
END%%
%%ANKI
Basic
What is the area of the unit circle?
Back: $\pi$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737167693435-->
END%%
%%ANKI
Basic
Which real numbers does the point $\langle 0, 0 \rangle$ on the unit circle map to?
Back: N/A. This point is not on the circle itself.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795237-->
END%%
%%ANKI
Basic
Which real numbers does the point $\langle 1, 0 \rangle$ on the unit circle map to?
Back: $2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795241-->
END%%
%%ANKI
Basic
Which point on the unit circle does number $2\pi$ map to?
Back: $\langle 1, 0 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795265-->
END%%
%%ANKI
Basic
Which point on the unit circle does number $\frac{3\pi}{2}$ map to?
Back: $\langle 0, -1 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795244-->
END%%
%%ANKI
Basic
Which real numbers does the point $\langle 0, -1 \rangle$ on the unit circle map to?
Back:$\frac{3\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795269-->
END%%
%%ANKI
Basic
Which real numbers does the point $\langle 0, 1 \rangle$ on the unit circle map to?
Back: $\frac{\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795248-->
END%%
%%ANKI
Basic
Which point on the unit circle does number $\frac{\pi}{2}$ map to?
Back: $\langle 0, 1 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795261-->
END%%
%%ANKI
Basic
Which point on the unit circle does number $\pi$ map to?
Back: $\langle -1, 0 \rangle$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795252-->
END%%
%%ANKI
Basic
Which real numbers does the point $\langle -1, 0 \rangle$ on the unit circle map to?
Back: $\pi + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737168795256-->
END%%
%%ANKI
Basic
Which real numbers correspond to the highlighted point on the unit circle?
![[unit-circle-1-0.png]]
Back: $2 \pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169243685-->
END%%
%%ANKI
Basic
Which real numbers correspond to the highlighted point on the unit circle?
![[unit-circle-0-1.png]]
Back: $\frac{\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169243690-->
END%%
%%ANKI
Basic
Which real numbers correspond to the highlighted point on the unit circle?
![[unit-circle-n1-0.png]]
Back: $\pi + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169243692-->
END%%
%%ANKI
Basic
Which real numbers correspond to the highlighted point on the unit circle?
![[unit-circle-0-n1.png]]
Back: $\frac{3\pi}{2} + 2\pi k$ for all $k \in \mathbb{Z}$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169243695-->
END%%
%%ANKI
Basic
*Why* does point $\langle 1, 0 \rangle$ on the unit circle coincide with real number $2\pi$?
Back: Because the circumference of the unit circle is $2\pi$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169683142-->
END%%
%%ANKI
Basic
*Why* does point $\langle -1, 0 \rangle$ on the unit circle coincide with real number $\pi$?
Back: Because half the circumference of the unit circle is $\pi$.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737169683151-->
END%%
%%ANKI
Basic
What is the "periodicity" of the unit circle?
Back: $2 \pi$
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737230158153-->
END%%
%%ANKI
Basic
What property of the unit circle does its periodicity correspond to?
Back: Its circumference.
Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.
<!--ID: 1737230158163-->
END%%
## Bibliography
* Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.