80 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Integers | Obsidian::STEM | binary::integer |
|
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:
| Value | w = 8 |
w = 16 |
w = 32 |
|---|---|---|---|
UMin_w |
0x00 |
0x0000 |
0x00000000 |
UMax_w |
0xFF |
0xFFFF |
0xFFFFFFFF |
TMin_w |
0x80 |
0x8000 |
0x80000000 |
TMax_w |
0x7F |
0x7FFF |
0x7FFFFFFF |
%%ANKI Basic What is a C integral type? Back: A type representing finite ranges of integers. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016. Tags: c17
END%%
%%ANKI Basic In what two ways are C integral types usually encoded? Back: Unsigned encoding 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. Tags: c17
END%%
%%ANKI
Basic
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.
END%%
%%ANKI
Basic
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.
END%%
%%ANKI
Basic
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.
END%%
%%ANKI Basic Which of unsigned encoding or two's-complement exhibit asymmetry in their range? 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.
END%%
%%ANKI
Basic
What integral values share the same binary representation in unsigned encoding and two's-complement?
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.
END%%
%%ANKI
Basic
According to the C standard, how are unsigned integral types encoded?
Back: Using unsigned encoding.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
According to the C standard, how are signed integral types encoded?
Back: The C standard leaves this unspecified.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
According to the C standard, Is unsigned overflow well-defined?
Back: Yes.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
According to the C standard, Is signed overflow well-defined?
Back: No.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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.
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.
END%%
%%ANKI
Basic
Provide a combinatorial explanation on why UMax equals 2 \cdot TMax + 1.
Back: There is one more negative number than positive numbers represented 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.
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.
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.
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.
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.
END%%
Unsigned Encoding
Always represents nonnegative numbers. Given an integral type \vec{x} of w bits, we convert binary to its unsigned encoding with: $B2U_w(\vec{x}) = 2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$
Note we unfold the summation on the RHS by one term to make it's relationship to T2U_w clearer.
%%ANKI
Basic
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.
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.
END%%
%%ANKI
Basic
What half-open interval represents the possible w-bit unsigned decimal values?
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.
END%%
%%ANKI
Basic
What is the binary representation of the smallest 4-bit unsigned number?
Back: 0000_2
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What is the binary representation of the largest 4-bit unsigned number?
Back: 1111_2
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What is the decimal expansion of unsigned integer 1010_2?
Back: 2^3 + 2^1 = 10
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI Basic What does the "uniqueness" of unsigned encoding refer to? Back: The function used to convert integral types to their unsigned encoding is a bijection. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
How does Bryant et al. define B2U_w?
Back: B2U_w(\vec{x}) = 2^{w-1}x_{w-1} + \sum_{k=0}^{w-2} 2^kx_k
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What is B2U_w an acronym for?
Back: Binary to unsigned, width w.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What is U2B_w an acronym for?
Back: Unsigned to binary, width w.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What does w in B2U_w represent?
Back: The number of bits in the integral type being interpreted.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What is the domain of B2U_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.
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.
END%%
%%ANKI
Basic
What is the range of B2U_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.
END%%
%%ANKI
Basic
What is the range of U2B_w?
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.
END%%
%%ANKI
Basic
How is the smallest unsigned integer formatted in hexadecimal?
Back: As all 0s.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
How is the largest unsigned integer formatted in hexadecimal?
Back: As all Fs.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
How does n relate to \textasciitilde n in unsigned encoding?
Back: n + \textasciitilde n = UMax
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
%%ANKI
Basic
Using unsigned encoding, why does n + \textasciitilde n = UMax?
Back: Because the sum always yields a bit string of all 1s.
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
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.
END%%
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: $B2T_w(\vec{x}) = -2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$
%%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.
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.
END%%
%%ANKI
Basic
How do TMin and TMax relate to one another?
Back: TMin = -TMax - 1
END%%
%%ANKI
Basic
What half-open interval represents the possible w-bit two's-complement decimal values?
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.
END%%
%%ANKI
Cloze
[{1:0}, {2:2^w}) is to unsigned as [{1:-2^{w-1}}, {2:2^{w-1}}) is to 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.
END%%
%%ANKI
Basic
What is the binary representation of the smallest 4-bit two's-complement number?
Back: 1000_2
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What is the binary representation of the largest 4-bit two's-complement number?
Back: 0111_2
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI Cloze 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.
END%%
%%ANKI
Basic
What is the weight of the sign bit in w-bit two's-complement?
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.
END%%
%%ANKI Basic What does the "uniqueness" of two's-complement refer to? Back: The function used to convert integral types to two's-complement is a bijection. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
How does Bryant et al. define B2T_w?
Back: B2T_w(\vec{x}) = -2^{w-1}x_{w-1} + \sum_{k=0}^{w-2} 2^kx_k
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What is B2T_w an acronym for?
Back: Binary to two'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.
END%%
%%ANKI
Basic
What is T2B_w an acronym for?
Back: Two's-complement to binary, width w.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What does w in B2T_w represent?
Back: The number of bits in the integral type being interpreted.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What is the domain of B2T_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.
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.
END%%
%%ANKI
Basic
What is the range of B2T_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.
END%%
%%ANKI
Basic
What is the range of T2B_w?
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.
END%%
%%ANKI
Basic
How is the smallest two's-complement integer formatted in hexadecimal?
Back: With a leading 8 followed by 0s.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
How is the largest two's-complement integer formatted in hexadecimal?
Back: With a leading 7 followed by Fs.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
How is equality |TMin| = |TMax| modified so that both sides actually balance?
Back: |TMin| = |TMax| + 1
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI Basic Which of negative and positive numbers can two's-complement encoding express more of? Back: Negative. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
Why is two's-complement's encoding range asymmetric?
Back: Leading 1s correspond to negatives but leading 0s 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.
END%%
%%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.
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.
END%%
%%ANKI
Basic
Why is it "two's-complement" instead of "twos'-complement"?
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.
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.
END%%
%%ANKI
Basic
Let x be a w-bit two's-complement number. What is it's complement?
Back: 2^w - x
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
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.
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.
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.
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.
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.
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.
END%%
%%ANKI
Basic
Subtracting x from all 1 bits is equivalent to what bitwise operation?.
Back: ~x
Reference: Finley, Thomas. “Two’s Complement,” April 2000. https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html.
END%%
Casting
Most implementations of C cast an object of signed type to unsigned type and vice versa, most implementations simply re-interpret the object's binary representation. This casting may happen implicitly if comparing or operating on 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 do most implementations of C perform casting of `signed` and `unsigned` types?
Back: As a reinterpretation of the same byte pattern of the object being casted.
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: 1708615249879-->
END%%
%%ANKI
Basic
What is $T2U_w$ an acronym for?
Back: **T**wo's-complement 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: 1708615249883-->
END%%
%%ANKI
Basic
For what values $x$ does $T2U_w(x) = U2T_w(x) = x$?
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.
<!--ID: 1708696117167-->
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: c17
<!--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: c17
<!--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: c17
<!--ID: 1708615249903-->
END%%
%%ANKI
Cloze
For {$x < 0$}, $T2U_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: 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
Cloze
For all $x$, $T2U_w(x)=$ {$x + x_{w-1}2^w$} where $x_{w-1}$ is the most significant bit 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: 1709492205954-->
END%%
%%ANKI
Basic
How is $T2U_w$ written as a function composition?
Back: $T2U_w = B2U_w \circ T2B_w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--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%%
### Expansion
For unsigned encoding, use **zero extension** to convert numbers to larger types. For example, $1010_2$ can be expanded to 8-bit $00001010_2$.
%%ANKI
Cloze
Use {zero} extension to convert {unsigned} numbers to larger types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708697867799-->
END%%
%%ANKI
Basic
Zero extension is generally used for what type of integer encoding?
Back: Unsigned.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708697867807-->
END%%
%%ANKI
Basic
*Why* does zero extension of unsigned numbers work?
Back: The weights of additional bits are zeroed out.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708697867810-->
END%%
%%ANKI
Basic
*Why* does zero extension of two's-complement numbers generally not work?
Back: A negative value would have its new sign bit be positive.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708697867814-->
END%%
%%ANKI
Basic
How is $\langle x_3, x_2, x_1, x_0 \rangle$ zero extended to 8 bits?
Back: As $\langle 0, 0, 0, 0, x_3, x_2, x_1, x_0 \rangle$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708697867818-->
END%%
For two's-complement, use **sign extension** to convert numbers to larger types. This means the additional leftmost bits are set to match the sign bit of the original number. For example, $1010_2$ can be expanded to 8-bit $11111010_2$.
%%ANKI
Cloze
Use {sign} extension to convert {two's-complement} numbers to larger types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708697867821-->
END%%
%%ANKI
Basic
Sign extension is generally used for what type of integer 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: 1708697867825-->
END%%
%%ANKI
Basic
*Why* does sign extension of two's-complement numbers work?
Back: The new sign bit weight is equal to the swing in the previous sign bit weight.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708697867829-->
END%%
%%ANKI
Basic
*Why* does sign extension of unsigned numbers generally not work?
Back: If new bits have value `1`, we're adding powers of $2$ to the interpreted value.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708697867833-->
END%%
%%ANKI
Basic
How is $\langle x_3, x_2, x_1, x_0 \rangle$ sign extended to 8 bits?
Back: As $\langle x_3, x_3, x_3, x_3, x_3, x_2, x_1, x_0 \rangle$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708697867839-->
END%%
### Truncation
Let $$\begin{align*}
x & = \langle x_{w-1}, \ldots, x_1, x_0 \rangle \\
x' & = \langle x_{k-1}, \ldots, x_1, x_0 \rangle
\end{align*}$$
Then in unsigned encoding, truncating $x$ to $k$ bits is equal to $x \bmod 2^k$. This is because $x_i \bmod 2^k = 0$ for all $i \geq k$ meaning $$B2U_k(x') = B2U_w(x) \bmod 2^k$$
%%ANKI
Basic
What bit string results from truncating $\langle x_{w-1}, \ldots, x_1, x_0 \rangle$ to $k$ bits?
Back: $\langle x_{k-1}, \ldots, x_1, x_0 \rangle$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708700130849-->
END%%
%%ANKI
Basic
What is the decimal value of truncating unsigned $x$ to $k$ bits?
Back: $x \bmod 2^k$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708700130856-->
END%%
%%ANKI
Basic
*Why* does truncating unsigned $x$ to $k$ bits yield $x \bmod 2^k$?
Back: $\bmod 2^k$ is a convenient way of "zero-ing" out bits $x_{w-1}, \ldots, x_k$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708700130859-->
END%%
%%ANKI
Basic
How is the following equality balanced for $k \leq w$? $$B2U_w(\langle x_{w-1}, \ldots, x_1, x_0 \rangle) = B2U_k(\langle x_{k-1}, \ldots, x_1, x_0 \rangle)$$
Back: $$B2U_w(\langle x_{w-1}, \ldots, x_1, x_0 \rangle) \bmod 2^k = B2U_k(\langle x_{k-1}, \ldots, x_1, x_0 \rangle)$$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708700225123-->
END%%
In two's-complement encoding, truncating $x$ to $k$ bits is equal to $U2T_k(T2U_w(x) \bmod 2^k)$. Like with unsigned truncation, $B2U_k(x') = B2U_w(x) \bmod 2^k$. Therefore $$U2T_k(B2U_k(x')) = U2T_k(B2U_w(x) \bmod 2^k)$$
%%ANKI
Basic
What is the $k$-truncation of $w$-bit two's-complement $x$?
Back: $U2T_k(x \bmod 2^k)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708701087974-->
END%%
%%ANKI
Cloze
Two's-complement $k$-truncation of $w$-bit $x$ is {$U2T_k$}$(${$x \bmod 2^k$}$)$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708701087985-->
END%%
%%ANKI
Basic
What is the purpose of $U2T_k$ in two's-complement truncation expression $U2T_k(x \bmod 2^k)$?
Back: To reinterpret the sign bit correctly.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708702794304-->
END%%
%%ANKI
Basic
Why isn't $T2U_w$ in two's-complement truncation $U2T_k(T2U_w(x) \bmod 2^k)$ strictly necessary?
Back: $x \bmod 2^k$ will always yield an integer in range $[0, 2^k)$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708702794313-->
END%%
%%ANKI
Basic
What additional steps does calculating two's-complement truncation have?
Back: Casting to and from 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: 1708701087982-->
END%%
## Arithmetic
### Addition
Addition of two unsigned or two two's-complement numbers operate in much the same way as grade-school arithmetic. Digits are added one-by-one and overflows "carried" to the next summation. Overflows are truncated; the final carry bit is discarded in the underlying bit adder.
%%ANKI
Basic
*Why* is addition overflow of $w$-bit integral types equivalent to $w$-bit truncation?
Back: The underlying bit adder discards any final carry bit.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678721-->
END%%
%%ANKI
Basic
Why should you generally prefer `x < y` over `x - y < 0`?
Back: The former avoids possible overflows.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678725-->
END%%
%%ANKI
Basic
How is `x - y < 0` rewritten more safely?
Back: `x < y`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678728-->
END%%
%%ANKI
Basic
What hardware-level advantage does two's-complement introduce over other signed encodings?
Back: The same circuits can be used for unsigned and two's-complement arithmetic.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678732-->
END%%
%%ANKI
Basic
What representational-level advantage does two's-complement introduce over other signed encodings?
Back: `0` is encoded in only one way.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678736-->
END%%
Unsigned addition of $w$-bit integral types, denoted $+_w^u$, behaves like so:
$$x +_w^u y = \begin{cases}
x + y - 2^w & \text{if } x + y \geq 2^w \\
x + y & \text{otherwise}
\end{cases}$$
This is more simply expressed as $x +_w^u y = (x + y) \bmod 2^w$.
%%ANKI
Basic
What kind of overflow does unsigned addition potentially exhibit?
Back: Positive overflow.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678739-->
END%%
%%ANKI
Basic
Why is unsigned addition overflow *not* UB?
Back: Because the C standard enforces unsigned encoding of `unsigned` data types.
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: 1708799678742-->
END%%
%%ANKI
Basic
What does $+_w^u$ denote?
Back: Unsigned addition of $w$-bit integral types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678745-->
END%%
%%ANKI
Basic
Unsigned addition overflow is equivalent to what bit-level manipulation tactic?
Back: Truncation.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678748-->
END%%
%%ANKI
Basic
What is the result of $x +_w^u y$?
Back: $(x + y) \bmod 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: 1708799678751-->
END%%
%%ANKI
Basic
*Why* does $x +_w^u y = (x + y) \bmod 2^w$?
Back: Because discarding any carry bit is equivalent to truncating the sum to $w$ bits.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678755-->
END%%
%%ANKI
Cloze
Without using modular arithmetic, $x +_w^u y =$ {$x + y$} if {$x + y < 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: 1708799678758-->
END%%
%%ANKI
Cloze
Without using modular arithmetic, $x +_w^u y =$ {$x + y - 2^w$} if {$x + y \geq 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: 1708799678761-->
END%%
%%ANKI
Basic
How do you detect whether unsigned addition $s \coloneqq x +_w^u y$ overflowed?
Back: Overflow occurs if and only if $s < x$ (or $s < y$).
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678765-->
END%%
%%ANKI
Basic
How would you complete the body of this function?
```c
/* Determine whether arguments can be added without overflow */
int uadd_ok(unsigned x, unsigned y);
```
Back:
```c
return (x + y) >= x;
```
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678769-->
END%%
%%ANKI
Basic
Does unsigned overflow detection depend on the left or right operand of $s \coloneqq x +_w^u y$?
Back: Either.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678772-->
END%%
%%ANKI
Basic
Why can we choose to compare $s$ to either $x$ or $y$ when detecting overflow of $s \coloneqq x +_w^u y$?
Back: Because unsigned addition is commutative.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678776-->
END%%
%%ANKI
Basic
Given integer $0 < x < 2^w$, what is $x$'s unsigned additive inverse?
Back: $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: 1708808252010-->
END%%
%%ANKI
Basic
Which unsigned integer is its own additive inverse?
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: 1708808252017-->
END%%
%%ANKI
Basic
What bitwise operations yield the additive inverse of an unsigned number $x$?
Back: `~x + 1`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709042784783-->
END%%
%%ANKI
Basic
Given unsigned integer `x`, what is the value of `x + ~x`?
Back: $UMax$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709042784788-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the largest possible value of $x +_w^u y$?
Back: $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: 1709492205961-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the smallest possible value of $x +_w^u y$?
Back: $w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205964-->
END%%
%%ANKI
Basic
Is $+_w^u$ commutative?
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: 1710680824725-->
END%%
%%ANKI
Basic
Is $+_w^u$ associative?
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: 1710680824728-->
END%%
Two's-complement addition, denoted $+_w^t$ operates similarly:
$$x +_w^u y = \begin{cases}
x + y - 2^w & \text{if } x + y \geq 2^{w-1} \\
x + y + 2^w & \text{if } x + y < -2^{w-1} \\
x + y & \text{otherwise}
\end{cases}$$
Unlike with unsigned addition, there is no simpler modulus operation that can be applied.
%%ANKI
Basic
What kind of overflows does two's-complement addition potentially exhibit?
Back: Positive and negative overflow.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376220-->
END%%
%%ANKI
Basic
Why is signed addition overflow UB?
Back: Because the C standard does not mandate any particular signed integer encoding.
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: 1708964376225-->
END%%
%%ANKI
Basic
What does $+_w^t$ denote?
Back: Two's-complement addition of $w$-bit integral types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376228-->
END%%
%%ANKI
Basic
*Why* doesn't two's-complement addition perform modular arithmetic?
Back: Because negative values are representable.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376231-->
END%%
%%ANKI
Cloze
$x +_w^t y =$ {$x + y - 2^w$} if {$x + y \geq 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: 1708964376235-->
END%%
%%ANKI
Cloze
$x +_w^t y =$ {$x + y + 2^w$} if {$x + y < -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: 1708964376238-->
END%%
%%ANKI
Cloze
$x +_w^t y =$ {$x + y$} if {$-2^{w-1} \leq x + y < 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: 1708964376242-->
END%%
%%ANKI
Basic
How do we detect $x +_w^t y$ positive overflowed?
Back: This happens iff $x > 0$, $y > 0$, and $x +_w^t y \leq 0$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376246-->
END%%
%%ANKI
Basic
How do we detect $x +_w^t y$ negative overflowed?
Back: This happens iff $x < 0$, $y < 0$, and $x +_w^t y \geq 0$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376250-->
END%%
%%ANKI
Basic
How can we write $x +_w^t y$ in terms of unsigned addition?
Back: $x +_w^t y = U2T_w(T2U_w(x) +_w^u T2U_w(y))$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376254-->
END%%
%%ANKI
Basic
How is the following expressed more simply (i.e. using more standard algebra)? $$x +_w^t y = U2T_w(T2U_w(x) +_w^u T2U_w(y))$$
Back: $x +_w^t y = U2T_w((x + y) \bmod 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: 1709492205967-->
END%%
%%ANKI
*Why* are we able to characterize $+_w^t$ in terms of $+_w^u$?
Back: Because two's-complement addition has the same bit-level representation as unsigned addition.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
How would you complete the body of this function?
```c
/* Determine whether arguments can be added without overflow */
int tadd_ok(int x, int y);
```
Back:
```c
int pos_over = x > 0 && y > 0 && (x + y) <= 0;
int neg_over = x < 0 && y < 0 && (x + y) >= 0;
return !pos_over && !neg_over;
```
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376259-->
END%%
%%ANKI
Basic
Given integer $-2^{w-1} < x < 2^{w-1}$, what is $x$'s two's-complement additive inverse?
Back: $-x$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709040965774-->
END%%
%%ANKI
Basic
What is the additive inverse of $TMin$?
Back: $TMin$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709040965804-->
END%%
%%ANKI
Basic
What is the additive inverse of $TMax$?
Back: $-TMax$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709040965810-->
END%%
%%ANKI
Basic
Which two's-complement integers are their own additive inverse?
Back: $TMin$ and $0$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709040965815-->
END%%
%%ANKI
Basic
What bitwise operations yield the additive inverse of two's-complement number $x$?
Back: `~x + 1`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709042784791-->
END%%
%%ANKI
Basic
Given two's-complement integer `x`, what is the value of `x + ~x`?
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: 1709042784794-->
END%%
%%ANKI
Basic
What "splitting" approach to $x$'s two's-complement negation does Bryant et al. describe?
Back: Find the rightmost $1$ in $x$'s bit string representation and complement the bits to its left.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709042784797-->
END%%
%%ANKI
Basic
Where do we "split" $x$'s binary representation to perform two's-complement negation?
Back: At the rightmost $1$ in $x$'s binary representation.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709042784800-->
END%%
%%ANKI
Basic
Using *just* `~`, what is the two's-complement negation of $\langle x_{w-1}, \ldots, x_{k+1}, 1, 0, \ldots, 0\rangle$?
Back: $\langle \textasciitilde x_{w-1}, \ldots, \textasciitilde x_{k+1}, 1, 0, \ldots, 0 \rangle$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709042784803-->
END%%
%%ANKI
Basic
*Why* does complementing and adding one yield integer $x$'s additive inverse?
Back: `x + ~x` yields a bit string of all `1`s. Adding `1` to this overflows.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709042784806-->
END%%
%%ANKI
Basic
What decimal value does two's-complement `~x` evaluate to?
Back: `-x - 1`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709044103781-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the largest possible value of $x +_w^t y$?
Back: $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: 1709492205970-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the smallest possible value of $x +_w^t y$?
Back: $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: 1709492205974-->
END%%
%%ANKI
Basic
Is $+_w^t$ commutative?
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: 1710680824730-->
END%%
%%ANKI
Basic
Is $+_w^t$ associative?
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: 1710680824733-->
END%%
### Shifting
Left shift operations (`<<`) drop the `k` most significant bits and fills the right end of the result with `k` zeros. Right shift operations (`>>`) are classified in two ways:
* **Logical**
* Drops the `k` least significant bits and fills the left end of the result with `k` zeros.
* This mode is always used when calling `>>` on unsigned data.
* Sometimes denoted as `>>>` to disambiguate from arithmetic right shifts.
* **Arithmetic**
* Drops the `k` least significant bits and fills the left end of the result with `k` copies of the most significant bit.
* This mode is usually used when calling `>>` on signed data.
%%ANKI
Basic
How is decimal value $2^n$ written in binary?
Back: As `1` followed by $n$ zeros.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707432641574-->
END%%
%%ANKI
Basic
What kinds of left shift operations are there?
Back: Just logical.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589773-->
END%%
%%ANKI
Basic
How many significant bits are dropped on a left shift by `k`?
Back: `k`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708784904518-->
END%%
%%ANKI
Basic
How many `0`s exist in the result of a left shift by `k`?
Back: At least `k`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708784904521-->
END%%
%%ANKI
Basic
What kinds of right shift operations are there?
Back: Logical and arithmetic.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589784-->
END%%
%%ANKI
Basic
What is a logical right shift operation?
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%%
%%ANKI
Basic
What is an arithmetic right shift operation?
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%%
%%ANKI
Basic
What kind of right shift operation is *usually* applied to signed numbers?
Back: Arithmetic.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589801-->
END%%
%%ANKI
Basic
What kind of right shift operation is applied to unsigned numbers?
Back: Logical.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589804-->
END%%
%%ANKI
Basic
What portability issue do shift operations introduce?
Back: There is no standard on whether right shifts of signed numbers are logical or arithmetic.
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: 1707854589808-->
END%%
%%ANKI
Cloze
{1:Arithmetic} right shifts are to {1:signed} numbers whereas {2:logical} right shifts are to {2:unsigned} numbers.
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: 1707854589813-->
END%%
In C, it is undefined behavior to shift by more than the width $w$ of an integral type or by a negative value.
%%ANKI
Basic
Assuming two's-complement, what is the result of shifting an `int32_t` value by `32`?
Back: It is undefined behavior.
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: 1708785613342-->
END%%
%%ANKI
Basic
Assuming two's-complement, what is the result of shifting `int32_t x = 1` left by `31`?
Back: $-2^{31}$
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: 1708785613370-->
END%%
%%ANKI
Basic
What is the result of shifting an `int32_t` value by `-1`?
Back: It is undefined behavior.
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: 1708785613376-->
END%%
%%ANKI
Basic
What is the result of shifting an `uint32_t` value by `32`?
Back: It is undefined behavior.
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: 1708785613383-->
END%%
%%ANKI
Basic
What is the result of shifting an `uint32_t` value by `31`?
Back: $2^{31}$
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: 1708785613389-->
END%%
%%ANKI
Basic
What is the result of shifting an `uint32_t` value by `-1`?
Back: It is undefined behavior.
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: 1708785613393-->
END%%
%%ANKI
Basic
How is $2^n$ written using bitwise shift operators?
Back: `1 << n`
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: 1708784904524-->
END%%
%%ANKI
Basic
What decimal value does `1 << n` translate to?
Back: $2^n$
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: 1708784904526-->
END%%
%%ANKI
Basic
How is $x \bmod 2^k$ equivalently written as a bit mask?
Back: `x & ((1 << k) - 1)`
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: 1707873410780-->
END%%
### Multiplication
Unsigned multiplication, denoted with the $*_w^u$ operator, is defined as follows: $$x *_w^u y = (x \cdot y) \bmod 2^w$$
%%ANKI
Basic
Given decimal integers $m$ and $n$, how many digits exist in $m \cdot n$?
Back: At most the number of digits in $m$ plus the number of digits in $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: 1709563221438-->
END%%
%%ANKI
Basic
Given binary integers $m$ and $n$ of width $w$, how many bits exist in $m \cdot n$?
Back: At most $2w$.
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: 1709563221442-->
END%%
%%ANKI
Basic
What does $*_w^u$ denote?
Back: Unsigned multiplication of $w$-bit integral types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205977-->
END%%
%%ANKI
Basic
How do you multiply $10_2 \cdot 10_2$ to a $4$-bit unsigned result by hand?
Back:
```
10
x 10
-----
00
+ 10
-----
0100
```
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: 1709563221444-->
END%%
%%ANKI
Basic
What is the result of $x *_w^u y$?
Back: $(x \cdot y) \bmod 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: 1709492205981-->
END%%
%%ANKI
Basic
*Why* does $x *_w^u y = (x \cdot y) \bmod 2^w$ (at least in C)?
Back: Because unsigned multiplication is *defined* to be the result truncated to $w$ bits.
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: 1709492205984-->
END%%
%%ANKI
Basic
How do $+_w^u$ and $*_w^u$ behave similarly?
Back: Letting $\square$ denote either $+$ or $*$, both satisfy $x \;\square_w^u\; y = (x \;\square\; y) \bmod 2^w$.
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: 1709492205988-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the largest possible value of $x *_w^u y$?
Back: $2w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205991-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the smallest possible value of $x *_w^u y$?
Back: $w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205995-->
END%%
%%ANKI
Basic
Given unsigned `x`, what arithmetic operation is equivalent to `x << k`?
Back: $x *_w^u 2^k$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428810-->
END%%
%%ANKI
Basic
What bitwise operation is equivalent to $x *_w^u 2^k$?
Back: `x << k`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428815-->
END%%
%%ANKI
Basic
How is `unsigned x` equivalently modified using bitwise operations?
```c
x = x * pow(2, k);
```
Back:
```c
x = (x << k);
```
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: 1709570428818-->
END%%
%%ANKI
Basic
How is `unsigned x` equivalently modified using arithmetic operations?
```c
x = (x << k);
```
Back:
```c
x = x * pow(2, k);
```
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: 1709831032382-->
END%%
%%ANKI
Basic
Is $*_w^u$ commutative?
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: 1710680824735-->
END%%
%%ANKI
Basic
Is $*_w^u$ associative?
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: 1710680824737-->
END%%
%%ANKI
Basic
Does $*^u_w$ distribute over $+^u_w$?
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: 1710680824740-->
END%%
Similarly, two's-complement multiplication is defined as follows: $$x *_w^t y = U2T_w((x \cdot y) \bmod 2^w)$$
%%ANKI
Basic
What does $*_w^t$ denote?
Back: Two's-complement multiplication of $w$-bit integral types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492205998-->
END%%
%%ANKI
Basic
What is the result of $x *_w^t y$?
Back: $U2T_w((x \cdot y) \bmod 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: 1709492206002-->
END%%
%%ANKI
Basic
How do you multiply $10_2 \cdot 01_2$ to a $4$-bit two's-complement result by hand?
Back:
```
1110
x 0001
-------
1110
+ 0000
-------
1110
```
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: 1709563221447-->
END%%
%%ANKI
Basic
What pre-processing step is done when multiplying to a $w$-bit two's-complement result by hand?
Back: Sign extend the factors to width $2w$.
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: 1709563221449-->
END%%
%%ANKI
Basic
When performing two's-complement multiplication by hand, why prefer multiplying by a positive value?
Back: Sign extension of a positive value yields `0`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: 1709563221452-->
END%%
%%ANKI
Basic
How do $+_w^t$ and $*_w^t$ behave similarly?
Back: Letting $\square$ denote either $+$ or $*$, both satisfy $x \;\square_w^t\; y = U2T_w((x \;\square\; y) \bmod 2^w)$.
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: 1709492206006-->
END%%
%%ANKI
Basic
How can we write $x *_w^t y$ in terms of unsigned multiplication?
Back: $x *_w^t y = U2T_w(T2U_w(x) *_w^u T2U_w(y))$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206012-->
END%%
%%ANKI
Basic
How is the following expressed more simply (i.e. using more standard algebra)? $$x *_w^t y = U2T_w(T2U_w(x) *_w^u T2U_w(y))$$
Back: $x *_w^t y = U2T_w((x \cdot y) \bmod 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: 1709492206017-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the largest possible value of $x *_w^t y$?
Back: $2w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206024-->
END%%
%%ANKI
Basic
Ignoring overflow, what is the width of the smallest possible value of $x *_w^t y$?
Back: $2w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709492206031-->
END%%
%%ANKI
Basic
Given two's-complement `x`, what arithmetic operation is equivalent to `x << k`?
Back: $x *_w^t 2^k$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428822-->
END%%
%%ANKI
Basic
What bitwise operation is equivalent to $x *_w^t 2^k$?
Back: `x << k`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428825-->
END%%
%%ANKI
Basic
In two's-complement, how is `x` equivalently modified using bitwise operators?
```c
x = x * pow(2, k);
```
Back:
```c
x = (x << k);
```
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: 1709570428828-->
END%%
%%ANKI
Basic
In two's-complement, how is `x` equivalently modified using arithmetic operations?
```c
x = (x << k);
```
Back:
```c
x = x * pow(2, k);
```
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: 1709831032386-->
END%%
%%ANKI
Basic
Is $*_w^t$ commutative?
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: 1710680824742-->
END%%
%%ANKI
Basic
Is $*_w^t$ associative?
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: 1710680824744-->
END%%
%%ANKI
Basic
Does $*^t_w$ distribute over $+^t_w$?
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: 1710680824746-->
END%%
%%ANKI
Basic
How can we rewrite $x \cdot 1101_2$ as an expression of *only* `<<` and `+`?
Back: `(x << 3) + (x << 2) + (x << 0)`
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: 1709570428832-->
END%%
%%ANKI
Basic
*Why* is $x \cdot 13$ equal to `(x << 3) + (x << 2) + (x << 0)`?
Back: Because the binary representation of $13$ is $1101_2$.
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: 1709570428836-->
END%%
%%ANKI
Basic
How can we rewrite $x \cdot 1100_2$ as an expression of *only* `<<` and `-`?
Back: `(x << 4) - (x << 2)`
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: 1709570428839-->
END%%
%%ANKI
Basic
Convert $x \cdot 11011100_2$ to an expression containing `-`. How many `-` operators are there?
Back: $2$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428844-->
END%%
%%ANKI
Basic
Convert $x \cdot K$ to an expression excluding `-`. The number of `+` operators correspond to what?
Back: One less than the number of `1`s in $K$'s binary representation.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428848-->
END%%
%%ANKI
Basic
Convert $x \cdot K$ to an expression containing `-`. The number of `-` operators correspond to what?
Back: The number of runs of `1`s in $K$'s binary representation.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709570428851-->
END%%
### Division
Integer division divides the result and discards any fractional result. This has the same effect as rounding toward zero.
%%ANKI
Basic
What is integer division?
Back: Division of two numbers that returns the integer part of the result.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709831032392-->
END%%
%%ANKI
Cloze
Integer division $x / y$ is $\lfloor x / y \rfloor$ when $x \geq 0$ and {1:$y > 0$} or $x \leq 0$ and {1:$y < 0$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709831032396-->
END%%
%%ANKI
Cloze
Integer division $x / y$ is $\lceil x / y \rceil$ when $x \geq 0$ and {1:$y < 0$} or $x \leq 0$ and {1:$y > 0$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709831032399-->
END%%
%%ANKI
Basic
What distinguishes integer division from floor division?
Back: The latter does not round towards $0$ with negative results.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709831032402-->
END%%
%%ANKI
Basic
What distinguishes integer division from ceiling division?
Back: The latter does not round towards $0$ with positive results.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709831032406-->
END%%
%%ANKI
Cloze
Integer division is often called "truncation {toward zero}".
Reference: dirkgently, “Answer to ‘What Is the Behavior of Integer Division?,’” _Stack Overflow_, August 30, 2010, [https://stackoverflow.com/a/3602857](https://stackoverflow.com/a/3602857).
<!--ID: 1709831032412-->
END%%
%%ANKI
Cloze
Unsigned division is to {logical} right shifts. Two's-complement division is to {arithmetic} right shifts.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709831032417-->
END%%
%%ANKI
Basic
What is the result of logical right-shifting unsigned $x$ by $k$ bits?
Back: $\lfloor x / 2^k \rfloor$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709831032421-->
END%%
%%ANKI
Basic
In unsigned encoding, *why* is floor a part of expression $x \mathop{\texttt{>>}} k = \lfloor x / 2^k \rfloor$?
Back: Because the least significant bit, which may have value `1`, is dropped.
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: 1709831032424-->
END%%
%%ANKI
Basic
In unsigned encoding, how is `x` equivalently modified using bitwise operators?
```c
x = floor(x / pow(2, k));
```
Back:
```c
x = (x >> k);
```
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: 1709831032428-->
END%%
%%ANKI
Basic
In unsigned encoding, how is `x` equivalently modified using arithmetic operations?
```c
x = (x >> k);
```
Back:
```c
x = floor(x / pow(2, k));
```
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: 1709831032432-->
END%%
%%ANKI
Basic
What is the result of arithmetic right-shifting two's-complement $x$ by $k$ bits?
Back: $\lfloor x / 2^k \rfloor$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709831032435-->
END%%
%%ANKI
Basic
In two's-complement, *why* is floor a part of expression $x \mathop{\texttt{>>}} k = \lfloor x / 2^k \rfloor$?
Back: Because the least significant bit, which may have value `1`, is dropped.
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: 1709831032440-->
END%%
%%ANKI
Basic
In two's-complement, what is `-1 >> 1`?
Back: `-1`
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: 1709831032444-->
END%%
%%ANKI
Basic
Why is division by a power of two using arithmetic right-shift `x >> k` considered incorrect?
Back: This right shift performs floor division, not integer division.
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: 1709831032449-->
END%%
%%ANKI
Basic
In two's-complement, how is `x` equivalently modified using bitwise operators?
```c
x = floor(x / pow(2, k));
```
Back:
```c
x = (x >> k);
```
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: 1709831032455-->
END%%
%%ANKI
Basic
In two's-complement, how is `x` equivalently modified using arithmetic operations?
```c
x = (x >> k);
```
Back:
```c
x = floor(x / pow(2, k));
```
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: 1709831032461-->
END%%
%%ANKI
Basic
Assuming no overflow, rewrite expression `x >> k` to instead yield $\lceil x / 2^k \rceil$.
Back: `(x + (1 << k) - 1) >> k`
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: 1714184300343-->
END%%
%%ANKI
Basic
Assuming no overflow, what is the result of `(x + (1 << k) - 1) >> k`?
Back: $\lceil x / 2^k \rceil$
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: 1714184300349-->
END%%
%%ANKI
Basic
What value of $Bias$ satisfies the following identity? $$\left\lceil \frac{x}{2^k} \right\rceil = \left\lfloor \frac{x}{2^k} + Bias \right\rfloor$$
Back: $(2^k - 1) / 2^k$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714184300352-->
END%%
%%ANKI
Basic
What value of $Bias$ satisfies the following identity? $$\left\lceil \frac{x}{2^k} \right\rceil = \left\lfloor \frac{x + Bias}{2^k} \right\rfloor$$
Back: $2^k - 1$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714184300355-->
END%%
%%ANKI
Basic
What floor/ceiling identity does expression `(x + (1 << k) - 1) >> k` exploit?
Back: $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x + y - 1}{y} \right\rfloor$$
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: 1714184300359-->
END%%
%%ANKI
Basic
In two's-complement, how do we use `>>` to perform integer division of `x > 0` by $2^k$?
Back: `x >> k`
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: 1714184300362-->
END%%
%%ANKI
Basic
In two's-complement, how do we use `>>` to perform integer division of `x < 0` by $2^k$?
Back: `(x + (1 << k) - 1) >> k`
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: 1714184300364-->
END%%
## Bibliography
* 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).