More integer encoding notes.

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

@@ -118,7 +118,7 @@
     "_journal/2024-02-02.md": "a3b222daee8a50bce4cbac699efc7180",
     "_journal/2024-02-01.md": "3aa232387d2dc662384976fd116888eb",
     "_journal/2024-01-31.md": "7c7fbfccabc316f9e676826bf8dfe970",
-    "logic/equiv-trans.md": "4d825c23bf54e8b1e645584d52b2b993",
+    "logic/equiv-trans.md": "08ad9c814f0a46981b218a9c7cc8a3b8",
     "_journal/2024-02-07.md": "8d81cd56a3b33883a7706d32e77b5889",
     "algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2",
     "algorithms/loop-invariant.md": "d883dfc997ee28a7a1e24b995377792b",
@@ -132,7 +132,7 @@
     "_journal/2024-02-09.md": "a798d35f0b2bd1da130f7ac766166109",
     "c/types.md": "cf3e66e5aee58a94db3fdf0783908555",
     "logic/quantification.md": "5d7579a511e9ff683edeec62bcc291b8",
-    "c/declarations.md": "7b0277d89b83fb330970deeb37e664c4",
+    "c/declarations.md": "94c85cb1a09efed2cad6b5b80e9d0be3",
     "algorithms/sorting/bubble-sort.md": "16dad1016dc6555163e42ba20f1d152d",
     "_journal/2024-02-10.md": "562b01f60ea36a3c78181e39b1c02b9f",
     "_journal/2024-01/2024-01-31.md": "7c7fbfccabc316f9e676826bf8dfe970",
@@ -159,7 +159,7 @@
     "logic/boolean-algebra.md": "f9101b2dfdedb73dc13c34c1a70a0010",
     "_journal/2024-02-13.md": "6242ed4fecabf95df6b45d892fee8eb0",
     "_journal/2024-02/2024-02-12.md": "618c0035a69b48227119379236a02f44",
-    "binary/shifts.md": "c7fb2a5722a69bb23047b603edcd0b91",
+    "binary/shifts.md": "ce6f2d80536b8a8b3f05383ce2190e72",
     "_journal/2024-02-14.md": "76d1b607470305fb3f00a47b8e9ece27",
     "_journal/2024-02/2024-02-13.md": "6242ed4fecabf95df6b45d892fee8eb0",
     "_journal/2024-02-15.md": "575ba46d692795d9606de9e635d1f4ac",
@@ -171,7 +171,7 @@
     "algorithms/binary-search.md": "08cb6dc2dfb204a665d8e8333def20ca",
     "_journal/2024-02-17.md": "7c37cb10515ed3d2f5388eaf02a67048",
     "_journal/2024-02/2024-02-16.md": "e701902e369ec53098fc2deed4ec14fd",
-    "binary/integer-encoding.md": "193566bf1b9e88257002d32bb2bc0bf2",
+    "binary/integer-encoding.md": "afe5e82b534fc5ab409b188addab36f4",
     "combinatorics/index.md": "f9de9671fdb6068ef2bb5e63051734be",
     "_journal/2024-02-18.md": "67e36dbbb2cac699d4533b5a2eaeb629",
     "_journal/2024-02/2024-02-17.md": "7c37cb10515ed3d2f5388eaf02a67048",
@@ -186,7 +186,9 @@
     "combinatorics/inclusion-exclusion.md": "4d5ba716bc90cd378c7c4c816b224c75",
     "_journal/2024-02-21.md": "b9d944ecebe625da5dd72aeea6a916a2",
     "_journal/2024-02/2024-02-20.md": "af2ef10727726200c4defe2eafc7d841",
-    "algebra/radices.md": "1171835a864103fbc4cbe21d56fcfa1f"
+    "algebra/radices.md": "0fcd901c798eaed8075ff1375e2429dd",
+    "_journal/2024-02-22.md": "e01f1d4bd2f7ac2a667cdfd500885a2a",
+    "_journal/2024-02/2024-02-21.md": "a627f267700f36abd5ceb9b4c294479e"
   },
   "fields_dict": {
     "Basic": [

notes/_journal/2024-02-21.md

@@ -1,15 +0,0 @@
----
-title: "2024-02-21"
----
-
-- [x] Anki Flashcards
-- [x] KoL
-- [ ] Sheet Music (10 min.)
-- [ ] OGS (1 Life & Death Problem)
-- [ ] Korean (Read 1 Story)
-- [ ] Interview Prep (1 Practice Problem)
-- [ ] Log Work Hours (Max 3 hours)
-
-* Read through "Stars and Bars" section in "Discrete Mathematics: An Open Introduction".
-* Added notes on radices.
-* Spent time thinking about how two's-complement works.

notes/_journal/2024-02-22.md

@@ -0,0 +1,19 @@
+---
+title: "2024-02-22"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] OGS (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [x] Interview Prep (1 Practice Problem)
+- [x] Log Work Hours (Max 3 hours)
+
+* Briefly discussed first chapter of "Designing Data-Intensive Applications" with Mike.
+* Continued reading more on integer encodings.
+	* Focused on conversions between unsigned encoding and two's-complement, though still need to create flashcards for these.
+* Did Leetcode problems
+	* [3Sum Closest](https://leetcode.com/problems/3sum-closest/description/)
+	* [Letter Combinations of a Phone Number](https://leetcode.com/problems/letter-combinations-of-a-phone-number/description/)
+	* [Remove Nth Node From End of List](https://leetcode.com/problems/remove-nth-node-from-end-of-list/description/)

notes/_journal/2024-02/2024-02-21.md

@@ -0,0 +1,20 @@
+---
+title: "2024-02-21"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] OGS (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [x] Interview Prep (1 Practice Problem)
+- [ ] Log Work Hours (Max 3 hours)
+
+* Read through "Stars and Bars" section in "Discrete Mathematics: An Open Introduction".
+* Added notes on radices.
+* Spent time thinking about how two's-complement works.
+* Read the first chapter on "Designing Data-Intensive Applications".
+* Leetcode Problems
+	* [Integer to Roman](https://leetcode.com/problems/integer-to-roman/description/)
+	* [Roman to Integer](https://leetcode.com/problems/roman-to-integer/description/)
+	* [Container With Most Water](https://leetcode.com/problems/container-with-most-water/description/)

notes/algebra/radices.md

@@ -267,6 +267,24 @@ Tags: binary::hex
 <!--ID: 1707432641596-->
 END%%
 
+%%ANKI
+Basic
+Which hexadecimal digits have a leading `1` bit?
+Back: `8` through `F`
+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: 1708631918825-->
+END%%
+
+%%ANKI
+Basic
+Which hexadecimal digits have a leading `0` bit?
+Back: `0` through `7`
+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: 1708631918829-->
+END%%
+
 ## References
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

notes/binary/integer-encoding.md

@@ -54,7 +54,7 @@ END%%
 
 %%ANKI
 Basic
-An integral value of 0 has what encoding?
+An integral value of $0_{10}$ likely has what encoding?
 Back: Either unsigned or two's-complement.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708177246105-->
@@ -62,7 +62,7 @@ END%%
 
 %%ANKI
 Basic
-An integral value of 100 has what encoding?
+An integral value of $100_{10}$ likely has what encoding?
 Back: Either unsigned or two's-complement.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708177246109-->
@@ -70,7 +70,7 @@ END%%
 
 %%ANKI
 Basic
-An integral value of -100 has what encoding?
+An integral value of $-100_{10}$ likely has what encoding?
 Back: Two's-complement.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708177246114-->
@@ -87,7 +87,7 @@ END%%
 %%ANKI
 Basic
 What integral values share the same binary representation in unsigned encoding and two's-complement?
-Back: Nonnegative values $\leq |TMax|$.
+Back: Nonnegative values $\leq TMax$.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708454709515-->
 END%%
@@ -108,9 +108,83 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1708455064696-->
 END%%
 
+%%ANKI
+Basic
+According to the C standard, Is `unsigned` underflow/overflow safe?
+Back: Yes
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708551236389-->
+END%%
+
+%%ANKI
+Basic
+According to the C standard, Is `signed` underflow/overflow safe?
+Back: No
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708551236392-->
+END%%
+
+%%ANKI
+Basic
+Why is `signed` underflow/overflow considered UB?
+Back: Because there is no requirement on how `signed` integers are encoded.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708551236395-->
+END%%
+
+%%ANKI
+Basic
+How does $UMax$ relate to $TMax$?
+Back: $UMax = 2 \cdot TMax + 1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398445-->
+END%%
+
+%%ANKI
+Basic
+*Why* is it $UMax = 2 \cdot TMax + 1$?
+Back: All bit patterns denoting negative numbers in two's-complement are positive in unsigned encoding.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708613447880-->
+END%%
+
+%%ANKI
+Basic
+What are the binary encodings of $UMax_4$ and $TMax_4$?
+Back: $1111_2$ and $0111_2$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398449-->
+END%%
+
+%%ANKI
+Basic
+Reinterpret $TMax$ in unsigned encoding. What arithmetic operations yield $UMax$?
+Back: Multiply by two and add one.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398454-->
+END%%
+
+%%ANKI
+Basic
+Reinterpret $TMax$ in unsigned encoding. What bitwise operations yield $UMax$?
+Back: One-bit left shift and add one.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398459-->
+END%%
+
+%%ANKI
+Basic
+Reinterpret $UMax$ in two's-complement. What decimal value do you have?
+Back: $-1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708453398469-->
+END%%
+
 ### Unsigned Encoding
 
-Always represents nonnegative numbers. Given an integral type $\vec{x}$ of $w$ bits, we convert binary to its unsigned encoding with: $$B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$$
+Always represents nonnegative numbers. Given an integral type $\vec{x}$ of $w$ bits, we convert binary to its unsigned encoding with: $$B2U_w(\vec{x}) = 2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$$
+
+Note we unfold the summation on the RHS by one term to make it's relationship to $T2U_w$ clearer.
 
 %%ANKI
 Basic
@@ -171,7 +245,7 @@ END%%
 %%ANKI
 Basic
 How does Bryant et al. define $B2U_w$?
-Back: $B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$
+Back: $B2U_w(\vec{x}) = 2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708179147785-->
 END%%
@@ -179,11 +253,19 @@ END%%
 %%ANKI
 Basic
 What is $B2U_w$ an acronym for?
-Back: **B**inary "to" **u**nsigned.
+Back: **B**inary to **u**nsigned, width $w$.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708179147791-->
 END%%
 
+%%ANKI
+Basic
+What is $U2B_w$ an acronym for?
+Back: **U**nsigned to **b**inary, width $w$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708613447885-->
+END%%
+
 %%ANKI
 Basic
 What does $w$ in $B2U_w$ represent?
@@ -195,11 +277,19 @@ END%%
 %%ANKI
 Basic
 What is the domain of $B2U_w$?
-Back: Bit vectors of size $w$.
+Back: Bit strings of size $w$.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708179147798-->
 END%%
 
+%%ANKI
+Basic
+What is the domain of $U2B_w$?
+Back: $[0, 2^w)$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708613447888-->
+END%%
+
 %%ANKI
 Basic
 What is the range of $B2U_w$?
@@ -210,10 +300,10 @@ END%%
 
 %%ANKI
 Basic
-What is the hexadecimal representation of $UMin_4$?
+What is the range of $U2B_w$?
-Back: `0x0000`
+Back: Bit strings of length $w$.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708453398386-->
+<!--ID: 1708613447891-->
 END%%
 
 %%ANKI
@@ -224,14 +314,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1708453398392-->
 END%%
 
-%%ANKI
-Basic
-What is the hexadecimal representation of $UMax_4$?
-Back: `0xFFFF`
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708453398397-->
-END%%
-
 %%ANKI
 Basic
 How is the largest unsigned integer formatted in hexadecimal?
@@ -250,7 +332,7 @@ END%%
 
 %%ANKI
 Basic
-With unsigned encoding, *why* does `n + ~n = UMax`?
+Using unsigned encoding, *why* does `n + ~n = UMax`?
 Back: Because this always yields a bit string of all `1`s.
 Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
 <!--ID: 1708545574154-->
@@ -284,6 +366,13 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1708545383255-->
 END%%
 
+%%ANKI
+Basic
+How do $TMin$ and $TMax$ relate to one another?
+Back: $TMin = -TMax - 1$
+<!--ID: 1708609869518-->
+END%%
+
 %%ANKI
 Basic
 What half-open interval represents the possible $w$-bit two's-complement decimal values?
@@ -326,7 +415,6 @@ END%%
 Basic
 What is the weight of the sign bit in $w$-bit two's-complement?
 Back: $-2^{w-1}$
-The {sign bit} refers to the {most significant bit} in two's-complement.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708179649887-->
 END%%
@@ -350,11 +438,19 @@ END%%
 %%ANKI
 Basic
 What is $B2T_w$ an acronym for?
-Back: **B**inary "to" **t**wo's-complement.
+Back: **B**inary to **t**wo's-complement, width $w$.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708179649907-->
 END%%
 
+%%ANKI
+Basic
+What is $T2B_w$ an acronym for?
+Back: **T**wo's-complement to **b**inary, width $w$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708613447895-->
+END%%
+
 %%ANKI
 Basic
 What does $w$ in $B2T_w$ represent?
@@ -366,11 +462,19 @@ END%%
 %%ANKI
 Basic
 What is the domain of $B2T_w$?
-Back: Bit vectors of size $w$.
+Back: Bit strings of size $w$.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708179649921-->
 END%%
 
+%%ANKI
+Basic
+What is the domain of $T2B_w$?
+Back: $[-2^{w-1}, 2^{w-1})$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708613447899-->
+END%%
+
 %%ANKI
 Basic
 What is the range of $B2T_w$?
@@ -381,10 +485,10 @@ END%%
 
 %%ANKI
 Basic
-What is the hexadecimal representation of $TMin_4$?
+What is the range of $T2B_w$?
-Back: `0x8000`
+Back: Bit strings of length $w$.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708453398408-->
+<!--ID: 1708613447903-->
 END%%
 
 %%ANKI
@@ -395,14 +499,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1708453398413-->
 END%%
 
-%%ANKI
-Basic
-What is the hexadecimal representation of $TMax_4$?
-Back: `0x7FFF`
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708453398418-->
-END%%
-
 %%ANKI
 Basic
 How is the largest two's-complement integer formatted in hexadecimal?
@@ -429,54 +525,12 @@ END%%
 
 %%ANKI
 Basic
-Why is two's-complement encoding's range asymmetric?
+Why is two's-complement's encoding range asymmetric?
-Back: Leading `1`s correspond to negatives but leading `0`s corerspond to nonnegative numbers (which include $0$.)
+Back: Leading `1`s correspond to negatives but leading `0`s corerspond to nonnegative numbers (which include $0$).
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1708453398440-->
 END%%
 
-%%ANKI
-Basic
-How does $UMax$ relate to $TMax$?
-Back: $|UMax| = 2|TMax| + 1$
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708453398445-->
-END%%
-
-%%ANKI
-Basic
-What are the binary encodings of $UMax_4$ and $TMax_4$?
-Back: $1111_2$ and $0111_2$
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708453398449-->
-END%%
-
-%%ANKI
-Basic
-Reinterpret $TMax$ in unsigned encoding. What arithmetic operations yield $UMax$?
-Back: Multiply by two and add one.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708453398454-->
-END%%
-
-%%ANKI
-Basic
-Reinterpret $TMax$ in unsigned encoding. What bitwise operations yield $UMax$?
-Back: One-bit left shift and add one.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708453398459-->
-END%%
-
-%%ANKI
-Basic
-Reinterpret $UMax$ in two's-complement. What decimal value do you have?
-Back: $-1$
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708453398469-->
-END%%
-
-The "two's-complement" of an integer often refers to the binary representation of the integer's additive inverse. For example, $0110_2 = 6$ has two's-complement $1010_2 = -6$. In contrast, the "complement" of a nonnegative integer is the other nonnegative integer such that together add up to $2^{w-1}$. For example, $0110_2 = 6$ has complement $0010_2 = 2$ (with respect to $2^3 = 8$).
-
 %%ANKI
 Basic
 What are the median values of two's-complement's encoding range?
@@ -492,90 +546,122 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1708545383265-->
 END%%
 
+## Casting
+
+Most implementations of C cast an object of one type to another by simply re-interpreting their binary representation. This casting may happen implicitly if comparing or operating on e.g. `signed` and `unsigned` objects in the same expression. $T2U$ and $U2T$ reflect this method of casting:
+
+$$T2U_w(x) = \begin{cases}
+x + 2^w & x < 0 \\
+x & x \geq 0
+\end{cases}$$
+
+$$U2T_w(x) = \begin{cases}
+x & x \leq TMax_w \\
+x - 2^w & x > TMax_w
+\end{cases}$$
+
 %%ANKI
 Basic
-How does `n` relate to `~n` in two's-complement?
+How do most implementations of C perform casting?
-Back: `~n = -n - 1`
+Back: As a reinterpretation of the same byte pattern of the object being casted.
-Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708545383267-->
+Tags: c
+<!--ID: 1708615249879-->
 END%%
 
 %%ANKI
 Basic
-What value does not have a two's-complement?
+What is $T2U_w$ an acronym for?
-Back: $TMin$
+Back: **T**wo's-complement to **u**nsigned, width $w$.
-Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708545383268-->
+<!--ID: 1708615249883-->
 END%%
 
 %%ANKI
 Basic
-*Why* doesn't $TMin$ have a two's-complement?
+For what decimal values $x$ does $T2U_w$ and $U2T_w$ serve as the identity function?
-Back: Because there is one less representable positive number than negative number.
+Back: $0 \leq x \leq TMax_w$
-Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708545383270-->
+<!--ID: 1708615249887-->
+END%%
+
+%%ANKI
+Basic
+What values $x$ are unaffected when casting from `signed` to `unsigned`?
+Back: $0 \leq x \leq TMax_w$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c
+<!--ID: 1708615249891-->
+END%%
+
+%%ANKI
+Basic
+What values $x$ are unaffected when casting from `unsigned` to `signed`?
+Back: $0 \leq x \leq TMax_w$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c
+<!--ID: 1708615249897-->
+END%%
+
+%%ANKI
+Basic
+How are casts implicitly performed in operations containing `signed` and `unsigned` objects?
+Back: `signed` objects are cast to `unsigned` objects.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c
+<!--ID: 1708615249903-->
 END%%
 
 %%ANKI
 Cloze
-$TMin$ is called the "{weird number}" because {it is it's own two's-complement}.
+For {$x < 0$}, $T2U_w(x) =$ {$x + 2^w$}.
-Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
-END%%
-
-%%ANKI
-Basic
-What is the result of applying the two's-complement operation on $TMin$?
-Back: $TMin$
-Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
-<!--ID: 1708545383271-->
-END%%
-
-%%ANKI
-Basic
-What does the two's-complement of $n$ refer to?
-Back: The binary representation of $-n$.
-Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
-<!--ID: 1708545383273-->
-END%%
-
-%%ANKI
-Basic
-Using two's-complement, what is the *complement* of $0101_2$?
-Back: $0011_2$
-Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
-<!--ID: 1708545383274-->
-END%%
-
-%%ANKI
-Basic
-Using two's-complement, the *complement* of $0101_2$ is found with respect to what value?
-Back: $2^3 = 8$
-Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
-<!--ID: 1708545383276-->
-END%%
-
-%%ANKI
-Basic
-What bitwise operations yield the two's-complement of $n$?
-Back: `~n + 1`
-Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
-<!--ID: 1708545383277-->
-END%%
-
-%%ANKI
-Basic
-Why is two's-complement named the way it is?
-Back: For $x \geq 0$, $-x$'s $w$-bit representation is found using $2^w - x$.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708545383279-->
+<!--ID: 1708615249908-->
+END%%
+
+%%ANKI
+Cloze
+For {$x \geq 0$}, $T2U_w(x) =$ {$x$}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708615249914-->
 END%%
 
 %%ANKI
 Basic
-What identity show $3$ and $5$ are complements with respect to $2^3$?
+How is $T2U_w$ written as a function composition?
-Back: $3 + 5 = 8$
+Back: $T2U_w = B2U_w \circ T2B_w$
-Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708545383280-->
+<!--ID: 1708615249920-->
+END%%
+
+%%ANKI
+Basic
+What is $U2T_w$ an acronym for?
+Back: **U**nsigned to **t**wo's-complement, width $w$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708615249925-->
+END%%
+
+%%ANKI
+Basic
+How is $U2T_w$ written as a function composition?
+Back: $U2T_w = B2T_w \circ U2B_w$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708615249930-->
+END%%
+
+%%ANKI
+Cloze
+For {$x > TMax_w$}, $U2T_w(x) =$ {$x - 2^w$}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708615249935-->
+END%%
+
+%%ANKI
+Cloze
+For {$x \leq TMax_w$}, $U2T_w(x) =$ {$x$}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708615249939-->
 END%%
 
 ## References

notes/binary/shifts.md

@@ -45,7 +45,7 @@ END%%
 %%ANKI
 Basic
 What is a logical right shift operation?
-Back: One that fills the left end of the result with `k` zeros.
+Back: One that fills the left end of the result with zeros.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1707854589786-->
 END%%
@@ -53,7 +53,7 @@ END%%
 %%ANKI
 Basic
 What is an arithmetic right shift operation?
-Back: One that fills the left end of the result with `k` copies of the most significant bit.
+Back: One that fills the left end of the result with copies of the most significant bit.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 <!--ID: 1707854589789-->
 END%%

notes/c/declarations.md

@@ -28,6 +28,14 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1707835869728-->
 END%%
 
+%%ANKI
+Basic
+Which header file contains `INT32_MAX`?
+Back: `<stdint.h>`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708615249870-->
+END%%
+
 %%ANKI
 Basic
 What does the "width" of an integer type refer to?
@@ -195,6 +203,147 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1707493017244-->
 END%%
 
+## Integer Literals
+
+Negative integer literals are typed in a counterintuitive way. When the compiler sees a number of form `-X`, the type of `X` is first determined *before* then being negated. Promotion rules are as follows:
+
+C90 (Decimal)   | C90 (Other)     | C99 (Decimal) | C99 (Other)
+--------------- | --------------- | ------------- | ---------
+`int`           | `int`           | `int`         | `int`
+`long`          | `unsigned`      | `long`        | `unsigned`
+`unsigned`      | `long`          | `long long`   | `long`
+`unsigned long` | `unsigned long` | `-`           | `unsigned long`
+`-`             | `-`             | `-`           | `long long`
+`-`             | `-`             | `-`           | `unsigned long long`
+
+%%ANKI
+Basic
+How does the compiler process integer literal `-X`?
+Back: By first determining the type of `X` and then negating the value.
+Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820805-->
+END%%
+
+%%ANKI
+Basic
+What simplification did C99 introduce to decimal integer literals?
+Back: The integer constant is guaranteed a `signed` type.
+Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820816-->
+END%%
+
+%%ANKI
+Basic
+Since what standard was it guaranteed decimal integer literals were `signed`?
+Back: C99
+Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820820-->
+END%%
+
+%%ANKI
+Basic
+In ISO C90, what integer literals are guaranteed `signed`?
+Back: None.
+Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820823-->
+END%%
+
+%%ANKI
+Basic
+In ISO C99, what integer literals are guaranteed `signed`?
+Back: Decimal integer constants.
+Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820826-->
+END%%
+
+%%ANKI
+Basic
+Why avoid negative octal integer literals?
+Back: Depending on value, the resulting type may be `unsigned`.
+Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820829-->
+END%%
+
+%%ANKI
+Basic
+Why avoid negative hexadecimal integer literals?
+Back: Depending on value, the resulting type may be `unsigned`.
+Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820833-->
+END%%
+
+%%ANKI
+Basic
+Which header file contains `INT_MAX`?
+Back: `<limits.h>`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708615249864-->
+END%%
+
+%%ANKI
+Cloze
+{`INT_MAX`} is to `signed` whereas {`UINT_MAX`} is to `unsigned`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820837-->
+END%%
+
+%%ANKI
+Basic
+How does `<limits.h>` define `INT_MIN`?
+Back: As `(-INT_MAX - 1)`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820840-->
+END%%
+
+%%ANKI
+Basic
+*Why* is `INT_MIN` defined as `(-INT_MAX - 1)` instead of directly as e.g. `-2147483648`?
+Back: Because `2147483648` (without `-`) would be sized as a non-`int` before being negated.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820843-->
+END%%
+
+%%ANKI
+Cloze
+`INT_MAX` is to {`<limits.h>`} whereas `INT32_MAX` is to {`<stdint.h>`}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708615249873-->
+END%%
+
+%%ANKI
+Basic
+What suffix can be used to denote an `unsigned` integer literal?
+Back: Case-insensitive `U`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708615249876-->
+END%%
+
+%%ANKI
+Basic
+What suffix can be used to denote a `long` integer literal?
+Back: Case-insensitive `L`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820847-->
+END%%
+
+%%ANKI
+Basic
+What suffix can be used to denote a `long long` integer literal?
+Back: Case-insensitive `LL`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820850-->
+END%%
+
+%%ANKI
+Basic
+What suffix can be used to denote an `unsigned long long` integer literal?
+Back: Case-insensitive `ULL`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708631820856-->
+END%%
+
+## Pointers
+
 Pointers have the same size as the machine's word size since it should be able to refer to any virtual address.
 
 %%ANKI

notes/logic/equiv-trans.md

@@ -788,7 +788,7 @@ END%%
 %%ANKI
 Basic
 How is $(s; x{:}v)$ written instead using set-theoretical notation?
-Back: $(s - \{(x, s(x))\}) \cup \{(x, v)\}$
+Back: $(s - \{\langle x, s(x) \rangle\}) \cup \{\langle x, v \rangle\}$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1707937867053-->
 END%%