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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 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 underflow/overflow safe? Back: Yes Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.

END%%

%%ANKI Basic According to the C standard, Is signed underflow/overflow safe? Back: No Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.

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 Why is it UMax = 2 \cdot TMax + 1? Back: All bit patterns denoting negative numbers in two's-complement are positive in unsigned encoding. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.

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_{i=0}^{w-2} 2^ix_i Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.

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 ~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.

END%%

%%ANKI Basic Using unsigned encoding, why does n + ~n = UMax? Back: Because this 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_{i=0}^{w-2} 2^ix_i Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.

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%%

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%%

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:

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?
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
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: It actually does, though we technically call it zero extension.
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.
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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: $U2T_k(T2U_w(x) \bmod 2^k)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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%%ANKI
Cloze
Two's-complement $k$-truncation of $w$-bit $x$ is {$U2T_k$}$(${$T2U_w(x) \bmod 2^k$}$)$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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%%ANKI
Basic
What is the purpose of $U2T_k$ in two's-complement truncation expression $U2T_k(T2U_w(x) \bmod 2^k)$?
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.
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%%ANKI
Basic
What is the purpose of $T2U_w$ in two's-complement truncation expression $U2T_k(T2U_w(x) \bmod 2^k)$?
Back: To ensure $x$ is encoded with the right "type".
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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%%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.
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%%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.
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## References

* 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).

32 KiB Raw Blame History Unescape Escape

Overview

Unsigned Encoding

Two's-Complement

Casting

32 KiB

Raw Blame History