More notes on two's-complement.

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

@@ -81,7 +81,9 @@
     "lattice-path-before-recurrence.png",
     "lattice-path-after-recurrence.png",
     "merge-sort.gif",
-    "theta-notation.png"
+    "theta-notation.png",
+    "big-o-notation.png",
+    "big-omega-notation.png"
   ],
   "File Hashes": {
     "algorithms/index.md": "cd7c7ba91fb2f961c9f2437777e8e2ac",
@@ -112,7 +114,7 @@
     "posix/regexp.md": "f5fb177c7356faf1bf768023c2563c54",
     "journal/2024-02-04.md": "e2b5678fc53d7284b71ed6820c02b954",
     "gawk/regexp.md": "d9229f1eabe1b99e965eecaa03bee86c",
-    "_templates/daily.md": "7866014e730e85683155207a02e367d8",
+    "_templates/daily.md": "5863e4524f8d012e153918d238c611a4",
     "_journal/2024-02-05.md": "f8505abd415c50fd97c81fd6153a6d4f",
     "_journal/2024-02-06.md": "1ea415f3c3f5be17f796b9a0d4df565f",
     "_journal/2024-02-04.md": "f77a3c5f3ce7969120f226738836dc92",
@@ -125,7 +127,7 @@
     "algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2",
     "algorithms/loop-invariant.md": "29f9f9090a3109890d333a78acc18b50",
     "algorithms/running-time.md": "5efc0791097d2c996f931c9046c95f65",
-    "algorithms/order-growth.md": "d5481b2611a39f255cbd0bb8a4120a70",
+    "algorithms/order-growth.md": "f4bab009ba9a6af34b9245fe595b0a2a",
     "_journal/2024-02-08.md": "19092bdfe378f31e2774f20d6afbfbac",
     "algorithms/sorting/selection-sort.md": "fcd0dc2ebaabd0a4db1baf7e7ef9f7a9",
     "algorithms/index 1.md": "6fada1f3d5d3af64687719eb465a5b97",
@@ -173,7 +175,7 @@
     "algorithms/binary-search.md": "dbbaf8d4be7aabef1ed232c1906b4c99",
     "_journal/2024-02-17.md": "7c37cb10515ed3d2f5388eaf02a67048",
     "_journal/2024-02/2024-02-16.md": "e701902e369ec53098fc2deed4ec14fd",
-    "binary/integer-encoding.md": "f4e553365bf596ddc7d24cfb12b63fab",
+    "binary/integer-encoding.md": "81b7d429c05b5bc95c0b3d4d8d3e27f0",
     "combinatorics/index.md": "9a85e8858c50c9797243d6d01e1dcbe7",
     "_journal/2024-02-18.md": "67e36dbbb2cac699d4533b5a2eaeb629",
     "_journal/2024-02/2024-02-17.md": "7c37cb10515ed3d2f5388eaf02a67048",
@@ -204,7 +206,7 @@
     "_journal/2024-02/2024-02-24.md": "ee92fd63ebbda5f8ffa0abd06700ead1",
     "_journal/2024-02-26.md": "ccbee662c1b2c42027bce1ed6f9a5ac4",
     "_journal/2024-02/2024-02-25.md": "426be827fe8483e4ab432304a2aa6df3",
-    "_journal/2024-02-27.md": "275c99a69ac426820df1803a462154ba",
+    "_journal/2024-02-27.md": "851180f1f7c1aaacd4c0b0ec2639bdf2",
     "_journal/2024-02/2024-02-26.md": "417b84be5d96f9d8adcdedca3b68b141"
   },
   "fields_dict": {

notes/_journal/2024-02-27.md

@@ -8,4 +8,7 @@ title: "2024-02-27"
 - [x] Go (1 Life & Death Problem)
 - [ ] Korean (Read 1 Story)
 - [ ] Interview Prep (1 Practice Problem)
-- [ ] Log Work Hours (Max 3 hours)
+- [x] Log Work Hours (Max 3 hours)
+
+* Added many flashcards on $\Theta$-, $O$-, and $\Omega$-notation.
+* Yet more notes on two's-complement.

notes/algorithms/order-growth.md

@@ -155,6 +155,20 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
 <!--ID: 1708974221797-->
 END%%
 
+%%ANKI
+What does it mean for function $f(n)$ to be asymptotically nonnegative?
+Back: $f(n) \geq 0$ whenever $n$ is sufficiently large.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+END%%
+
+%%ANKI
+Basic
+What does it mean for function $f(n)$ to be asymptotically positive?
+Back: $f(n) > 0$ whenever $n$ is sufficiently large.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221871-->
+END%%
+
 ## $\Theta$-notation
 
 ![[theta-notation.png]]
@@ -163,7 +177,7 @@ $\Theta$-notation refers to a strict lower- and upper-bound. It is defined as se
 
 %%ANKI
 Basic
-What kind of mathematical object is $\Theta(n)$?
+What kind of mathematical object is $\Theta(g(n))$?
 Back: A set.
 Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
 <!--ID: 1708974221801-->
@@ -250,14 +264,14 @@ END%%
 
 %%ANKI
 Cloze
-Given $f(n) = \Theta(g(n))$, we say {1:$g(n)$} is an asymptotically {2:tight} bound for {1:$f(n)$}.
+Given $f(n) = \Theta(g(n))$, we say {1:$g(n)$} is an asymptotic {2:tight} bound for {1:$f(n)$}.
 Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
 <!--ID: 1708974221851-->
 END%%
 
 %%ANKI
 Basic
-Which notation corresponds to asymptotically tight bounds?
+Which notation corresponds to asymptotic tight bounds?
 Back: $\Theta$-notation.
 Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
 <!--ID: 1708974221857-->
@@ -271,20 +285,6 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
 <!--ID: 1708974221864-->
 END%%
 
-%%ANKI
-What does it mean for function $f(n)$ to be asymptotically nonnegative?
-Back: $f(n) \geq 0$ whenever $n$ is sufficiently large.
-Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
-END%%
-
-%%ANKI
-Basic
-What does it mean for function $f(n)$ to be asymptotically positive?
-Back: $f(n) > 0$ whenever $n$ is sufficiently large.
-Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
-<!--ID: 1708974221871-->
-END%%
-
 %%ANKI
 Basic
 What condition must $g(n)$ satisfy such that otherwise $\Theta(g(n))$ is empty?
@@ -336,11 +336,333 @@ END%%
 
 %%ANKI
 Basic
-What values does the $y$-axis implicitly range over in the following?
+For $n < n_0$, what values does the $y$-axis take on?
+![[theta-notation.png]]
+Back: Indeterminate.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221909-->
+END%%
+
+%%ANKI
+Basic
+For $n \geq n_0$, what values does the $y$-axis take on?
 ![[theta-notation.png]]
 Back: Nonnegative values.
 Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
-<!--ID: 1708974221909-->
+<!--ID: 1709053894064-->
+END%%
+
+## $O$-notation
+
+![[big-o-notation.png]]
+
+$O$-notation refers to a strict upper-bound. It is defined as set $$O(g(n)) = \{ f(n) \mid \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq f(n) \leq cg(n) \}$$
+
+%%ANKI
+Basic
+What kind of mathematical object is $O(g(n))$?
+Back: A set.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894066-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers found in $O(g(n))$, what values do $c$ and $n_0$ take on?
+Back: Positive constants.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894068-->
+END%%
+
+%%ANKI
+Basic
+What names are usually given to the existentially quantified identifers in $O(g(n))$'s definition?
+Back: $c$ and $n_0$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894070-->
+END%%
+
+%%ANKI
+Basic
+What name is usually given to the universally quantified identifer in $O(g(n))$'s definition?
+Back: $n$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894072-->
+END%%
+
+%%ANKI
+Cloze
+Using typical identifiers, $f(n) = O(g(n))$ satisfies {$0$} $\leq$ {$f(n)$} $\leq$ {$cg(n)$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894074-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the lower bound of $cg(n)$ in $O(g(n))$?
+Back: $f(n)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894076-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the upper bound of $cg(n)$ in $O(g(n))$?
+Back: N/A
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894078-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the upper bound of $f(n)$ in $O(g(n))$?
+Back: $0$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894080-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the upper bound of $f(n)$ in $O(g(n))$?
+Back: $cg(n)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+END%%
+
+%%ANKI
+Cloze
+Given $f(n) = O(g(n))$, we say {1:$g(n)$} is an asymptotic {2:upper} bound for {1:$f(n)$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894084-->
+END%%
+
+%%ANKI
+Basic
+Which notation corresponds to asymptotic upper bounds?
+Back: $O$-notation.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894088-->
+END%%
+
+%%ANKI
+Basic
+Every member of $O(g(n))$ is expected to be asymptotically what?
+Back: Nonnegative.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894091-->
+END%%
+
+%%ANKI
+Basic
+What condition must $g(n)$ satisfy such that otherwise $O(g(n))$ is empty?
+Back: $g(n)$ must be asymptotically nonnegative.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894093-->
+END%%
+
+%%ANKI
+Basic
+How is $O(g(n))$ defined?
+Back: $\{ \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq f(n) \leq cg(n) \}$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894096-->
+END%%
+
+%%ANKI
+Basic
+Which asymptotic notation is this image demonstrating?
+![[big-o-notation.png]]
+Back: $O$-notation
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894098-->
+END%%
+
+%%ANKI
+Basic
+When is it guaranteed $y$-values are nonnegative in the following?
+![[big-o-notation.png]]
+Back: When $n \geq n_0$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894100-->
+END%%
+
+%%ANKI
+Basic
+In set-theoretic notation, what does it mean for $\Theta$-notation to be stronger than $O$-notation?
+Back: $\Theta(g(n)) \subseteq O(g(n))$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894101-->
+END%%
+
+%%ANKI
+Basic
+What notation corresponds to worst-case running times?
+Back: $O$-notation
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894103-->
+END%%
+
+%%ANKI
+Basic
+Why does Cormen et al. say "insertion sort's running time is $O(n^2)$" is an abuse of notation?
+Back: Because technically its running time depends on the particular input of size $n$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709053894105-->
+END%%
+
+## $\Omega$-notation
+
+![[big-omega-notation.png]]
+
+$\Omega$-notation refers to a strict lower-bound. It is defined as set $$\Omega(g(n)) = \{ f(n) \mid \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) \leq f(n) \}$$
+
+%%ANKI
+Basic
+What kind of mathematical object is $\Omega(g(n))$?
+Back: A set.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157375-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers found in $\Omega(g(n))$, what values do $c$ and $n_0$ take on?
+Back: Positive constants.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157377-->
+END%%
+
+%%ANKI
+Basic
+What names are usually given to the existentially quantified identifers in $\Omega(g(n))$'s definition?
+Back: $c$ and $n_0$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157379-->
+END%%
+
+%%ANKI
+Basic
+What name is usually given to the universally quantified identifer in $\Omega(g(n))$'s definition?
+Back: $n$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157381-->
+END%%
+
+%%ANKI
+Cloze
+Using typical identifiers, $f(n) = \Omega(g(n))$ satisfies {$0$} $\leq$ {$cg(n)$} $\leq$ {$f(n)$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157383-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the lower bound of $cg(n)$ in $\Omega(g(n))$?
+Back: $0$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157384-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the upper bound of $cg(n)$ in $\Omega(g(n))$?
+Back: $f(n)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157386-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the lower bound of $f(n)$ in $\Omega(g(n))$?
+Back: $cg(n)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157388-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the upper bound of $f(n)$ in $\Omega(g(n))$?
+Back: N/A
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157390-->
+END%%
+
+%%ANKI
+Cloze
+Given $f(n) = \Omega(g(n))$, we say {1:$g(n)$} is an asymptotic {2:lower} bound for {1:$f(n)$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157392-->
+END%%
+
+%%ANKI
+Basic
+Which notation corresponds to asymptotic lower bounds?
+Back: $\Omega$-notation.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157393-->
+END%%
+
+%%ANKI
+Basic
+Every member of $\Omega(g(n))$ is expected to be asymptotically what?
+Back: Nonnegative.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157394-->
+END%%
+
+%%ANKI
+Basic
+What condition must $g(n)$ satisfy such that otherwise $\Omega(g(n))$ is empty?
+Back: $g(n)$ must be asymptotically nonnegative.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157396-->
+END%%
+
+%%ANKI
+Basic
+How is $\Omega(g(n))$ defined?
+Back: $\{ \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) \leq f(n) \}$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157397-->
+END%%
+
+%%ANKI
+Basic
+Which asymptotic notation is this image demonstrating?
+![[big-omega-notation.png]]
+Back: $\Omega$-notation
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157399-->
+END%%
+
+%%ANKI
+Basic
+In set-theoretic notation, what does it mean for $\Theta$-notation to be stronger than $\Omega$-notation?
+Back: $\Theta(g(n)) \subseteq \Omega(g(n))$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157401-->
+END%%
+
+%%ANKI
+Basic
+What notation corresponds to best-case running times?
+Back: $\Omega$-notation
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157402-->
+END%%
+
+%%ANKI
+Cloze
+{1:$O$}-notation is to asymptotic {2:upper}-bounds whereas {2:$\Omega$}-notation is to asymptotic {1:lower}-bounds.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157404-->
+END%%
+
+%%ANKI
+Basic
+What theorem relates $\Theta(g(n))$, $O(g(n))$, and $\Omega(g(n))$?
+Back: $f(n) = \Theta(g(n))$ if and only if $f(n) \in O(g(n))$ and $f(n) \in \Omega(g(n))$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1709055157406-->
 END%%
 
 ## References

notes/binary/integer-encoding.md

@@ -530,6 +530,116 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1708545383265-->
 END%%
 
+%%ANKI
+Basic
+Why is two's-complement named the way it is?
+Back: Because there is only one $2$ in $2^w - x$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709060837130-->
+END%%
+
+%%ANKI
+Basic
+Given two's-complement $x \geq 0$, what is the significance of $2^w - x$?
+Back: The result is the binary representation of $-x$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709060849456-->
+END%%
+
+%%ANKI
+Basic
+Let $x$ be a $w$-bit two's-complement number. What is it's complement?
+Back: The number $y$ such that $x + y = 2^w$.
+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).
+<!--ID: 1709060837141-->
+END%%
+
+%%ANKI
+Basic
+What is the precise definition of the two's-complement of a $w$-bit number?
+Back: The complement of $x$ with respect to $2^w$.
+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).
+<!--ID: 1709060837145-->
+END%%
+
+%%ANKI
+Basic
+With respect to two's-complement encoding, what is the "weird number"?
+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: 1709060837149-->
+END%%
+
+%%ANKI
+Basic
+Why is $TMin$ called the "weird number"?
+Back: It is the only number that is it's own complement.
+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: 1709060837151-->
+END%%
+
+%%ANKI
+Basic
+How is $2^w - x$ written schematically, fixed to $w = 8$ bits?
+Back:
+```
+  00000000
+-        x
+----------
+       ...
+```
+Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
+<!--ID: 1709060837154-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten to emphasize why "two's-complement" is named the way it is?
+```
+  00000000
+- 01010101
+----------
+       ...
+```
+Back:
+```
+  100000000
+-  01010101
+-----------
+        ...
+```
+Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
+<!--ID: 1709060837156-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten to emphasize two's-complement's idea of "invert and add one"?
+```
+  100000000
+-  01010101
+-----------
+        ...
+```
+Back:
+```
+         1
++ 11111111
+- 01010101
+----------
+       ...
+```
+Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
+<!--ID: 1709060837160-->
+END%%
+
+%%ANKI
+Cloze
+Subtracting $x$ from {a number of all `1` bits} is equivalent to {`~x`}.
+Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
+<!--ID: 1709061032778-->
+END%%
+
 ## Casting
 
 Most implementations of C cast an object of one type to another by simply re-interpreting the object's 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:
@@ -1397,5 +1507,6 @@ END%%
 ## References
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+* Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
 * Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
 * “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).