17 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%%
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 0
s.
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 F
s.
Reference: Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.
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 0
s.
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 F
s.
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 1
s correspond to negatives but leading 0
s 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%%
References
- Bryant, Randal E., and David O'Hallaron. Computer Systems: A Programmer's Perspective. Third edition, Global edition. Always Learning. Pearson, 2016.