24 KiB
| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Integer Encoding | Obsidian::STEM | binary |
|
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: c
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: c
END%%
%%ANKI Basic What integer values does unsigned encoding represent? Back: Nonnegative values. 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 integer values does two's-complement represent? Back: Negative, zero, and positive values. 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 0 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 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 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
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
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%%
Unsigned Encoding
Always represents nonnegative numbers. Given an integral type \vec{x} of w bits, we convert binary to its unsigned encoding with: $B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$
%%ANKI
Basic
What is the largest unsigned value an integral type of w bits can encode?
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 is the smallest unsigned value an integral type of w bits can encode?
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 half-open interval represents the possible w-bit unsigned 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 type 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}) = \sum_{i=0}^{w-1} 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.
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 vectors 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 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 hexadecimal representation of UMin_4?
Back: 0x0000
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
What is the hexadecimal representation of UMax_4?
Back: 0xFFFF
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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
With 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 is the largest two's-complement value an integral type of w bits can encode?
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
What is the smallest two's-complement value an integral type of w bits can encode?
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 two's-complement 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}
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 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.
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 vectors 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 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 hexadecimal representation of TMin_4?
Back: 0x8000
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
What is the hexadecimal representation of TMax_4?
Back: 0x7FFF
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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 encoding's 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
How does UMax relate to TMax?
Back: |UMax| = 2|TMax| + 1
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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%%
The "two's-complement" of an integer often refers to the binary representation of the integer's additive inverse. For example, 0110_2 = 6 has two's-complement 1010_2 = -6. In contrast, the "complement" of an integer is the other integer such that together add up to 2^{w-1}. For example, 0110_2 = 6 has complement 0010_2 = 2 (with respect to 2^3 = 8).
%%ANKI
Basic
What are the median values of two's-complement's encoding range?
Back: -1 and 0
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
%%ANKI Basic In two's-complement, how many negative values can be encoded compared to nonnegative values? Back: The same amount. Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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
How does n relate to ~n in two's-complement?
Back: ~n = -n - 1
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
%%ANKI
Basic
What value does not have a two's-complement?
Back: TMin
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
%%ANKI
Basic
Why doesn't TMin have a two's-complement?
Back: Because there is one less representable positive number than negative number.
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
%%ANKI
Basic
What is the result of applying the two's-complement operation on TMin?
Back: TMin
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
%%ANKI
Basic
What does the two's-complement of n refer to?
Back: The binary representation of -n.
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
%%ANKI
Basic
Using two's-complement, what is the complement of 0101_2?
Back: 0011_2
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
%%ANKI
Basic
Using two's-complement, the complement of 0101_2 is found with respect to what value?
Back: 2^3 = 8
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
%%ANKI
Basic
What bitwise operations yield the two's-complement of n?
Back: ~n + 1
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
%%ANKI
Basic
Why is two's-complement named the way it is?
Back: For x \geq 0, -x's w-bit representation is found using 2^w - x.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
What identity show 3 and 5 are complements with respect to 2^3?
Back: 3 + 5 = 8
Reference: “Two’s-Complement.” In Wikipedia, January 9, 2024. https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561.
END%%
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.