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Add more notes on two's-complement.

c-declarations
Joshua Potter 2024-02-21 13:00:19 -07:00
parent 2678c94882
commit d93f0d3cb6
3 changed files with 170 additions and 4 deletions
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notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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notes/_journal/2024-02-21.md
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- [ ] OGS (1 Life & Death Problem) - [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story) - [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem) - [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours) - [ ] 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.

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notes/binary/integer-encoding.md
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## Overview ## Overview
Integers are typically encoded using either **unsigned encoding** or **two's complement**. The following table highlights how the min and max of these encodings behave: Integers are typically encoded using either **unsigned encoding** or **two's-complement**. The following table highlights how the min and max of these encodings behave:
Value | $w = 8$ | $w = 16$ | $w = 32$ Value | $w = 8$ | $w = 16$ | $w = 32$
-------- | ------- | -------- | ------------ -------- | ------- | -------- | ------------
@ -108,6 +108,38 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708455064696--> <!--ID: 1708455064696-->
END%% END%%
%%ANKI
Basic
What does $TMin_w$ evaluate to?
Back: $-2^{w-1}$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708545383252-->
END%%
%%ANKI
Basic
What does $TMax_w$ evaluate to?
Back: $2^{w-1} - 1$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708545383255-->
END%%
%%ANKI
Basic
What does $UMin_w$ evaluate to?
Back: $0$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708545383256-->
END%%
%%ANKI
Basic
What does $UMax_w$ evaluate to?
Back: $2^w - 1$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708545383258-->
END%%
### Unsigned Encoding ### 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: $$B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$$
@ -240,6 +272,30 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708453398403--> <!--ID: 1708453398403-->
END%% END%%
%%ANKI
Basic
How does `n` relate to `~n` in unsigned encoding?
Back: `~n = UMax - 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: 1708545383259-->
END%%
%%ANKI
Basic
With 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-->
END%%
%%ANKI
Basic
Regardless of word size, what bitwise operations yield $UMax$?
Back: `~0`
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: 1708545383261-->
END%%
### Two's-Complement ### Two's-Complement
Represents negative numbers along with nonnegative ones. Given an integral type $\vec{x}$ of $w$ bits, we convert binary to its twos'-complement encoding with: $$B2T_w(\vec{x}) = -2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$$ Represents negative numbers along with nonnegative ones. Given an integral type $\vec{x}$ of $w$ bits, we convert binary to its twos'-complement encoding with: $$B2T_w(\vec{x}) = -2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$$
@ -451,6 +507,112 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708453398469--> <!--ID: 1708453398469-->
END%% 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 an integer is the other 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?
Back: `-1` and `0`
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: 1708545383262-->
END%%
%%ANKI
Basic
In two's-complement, how many negative values can be encoded compared to nonnegative values?
Back: The same amount.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708545383264-->
END%%
%%ANKI
Cloze
In two's-complement, the {sign bit} partitions the encoding range into two sets.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708545383265-->
END%%
%%ANKI
Basic
How does `n` relate to `~n` in two's-complement?
Back: `~n = -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: 1708545383267-->
END%%
%%ANKI
Basic
What value does not have a two's-complement?
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: 1708545383268-->
END%%
%%ANKI
Basic
*Why* doesn't $TMin$ have a two's-complement?
Back: Because there is one less representable positive number than negative number.
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: 1708545383270-->
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-->
END%%
%%ANKI
Basic
What identity show $3$ and $5$ are complements with respect to $2^3$?
Back: $3 + 5 = 8$
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: 1708545383280-->
END%%
## References ## References
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
* “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).