304 lines
11 KiB
Markdown
304 lines
11 KiB
Markdown
---
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title: Integer Encoding
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TARGET DECK: Obsidian::STEM
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FILE TAGS: binary
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tags:
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- binary
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---
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## Overview
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Integers are typically encoded using either **unsigned encoding** or **two's complement**.
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%%ANKI
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Basic
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What is a C integral type?
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Back: A type representing finite ranges of integers.
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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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Tags: c
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<!--ID: 1708177246087-->
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END%%
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%%ANKI
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Basic
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In what two ways are C integral types encoded?
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Back: Unsigned encoding or two's-complement.
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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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Tags: c
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<!--ID: 1708177246093-->
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END%%
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%%ANKI
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Basic
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What integer values does unsigned encoding represent?
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Back: Nonnegative values.
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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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<!--ID: 1708177246097-->
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END%%
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%%ANKI
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Basic
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What integer values does two's-complement represent?
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Back: Negative, zero, and positive values.
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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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<!--ID: 1708177246102-->
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END%%
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%%ANKI
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Basic
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An integral value of 0 has what encoding?
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Back: Either unsigned or two's-complement.
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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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<!--ID: 1708177246105-->
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END%%
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%%ANKI
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Basic
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An integral value of 100 has what encoding?
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Back: Either unsigned or two's-complement.
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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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<!--ID: 1708177246109-->
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END%%
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%%ANKI
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Basic
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An integral value of -100 has what encoding?
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Back: Two's-complement.
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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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<!--ID: 1708177246114-->
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END%%
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### Unsigned Encoding
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Always represents nonnegative numbers. Given an integral type $\vec{x}$ of $w$ bits, we convert binary to its unsigned encoding with: $$B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$$
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%%ANKI
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Basic
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What is the largest unsigned value an integral type of $w$ bits can encode?
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Back: $2^w - 1$
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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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<!--ID: 1708177246119-->
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END%%
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%%ANKI
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Basic
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What is the smallest unsigned value an integral type of $w$ bits can encode?
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Back: $0$
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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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<!--ID: 1708177246123-->
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END%%
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%%ANKI
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Basic
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What half-open interval represents the possible $w$-bit unsigned values?
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Back: $[0, 2^w)$
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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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<!--ID: 1708177246128-->
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END%%
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%%ANKI
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Basic
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What is the binary representation of the smallest $4$-bit unsigned number?
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Back: $0000_2$
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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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<!--ID: 1708177246133-->
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END%%
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%%ANKI
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Basic
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What is the binary representation of the largest $4$-bit unsigned number?
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Back: $1111_2$
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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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<!--ID: 1708177246138-->
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END%%
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%%ANKI
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Basic
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What is the decimal expansion of unsigned type $1010_2$?
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Back: $2^3 + 2^1 = 10$
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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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<!--ID: 1708177246143-->
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END%%
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%%ANKI
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Basic
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What does the "uniqueness" of unsigned encoding refer to?
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Back: The function used to convert integral types to their unsigned encoding is a bijection.
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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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<!--ID: 1708177246148-->
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END%%
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%%ANKI
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Basic
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How does Bryant et al. define $B2U_w$?
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Back: $B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$
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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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<!--ID: 1708179147785-->
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END%%
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%%ANKI
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Basic
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What is $B2U_w$ an acronym for?
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Back: **B**inary "to" **u**nsigned.
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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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<!--ID: 1708179147791-->
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END%%
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%%ANKI
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Basic
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What does $w$ in $B2U_w$ represent?
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Back: The number of bits in the integral type being interpreted.
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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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<!--ID: 1708179147795-->
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END%%
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%%ANKI
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Basic
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What is the domain of $B2U_w$?
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Back: Bit vectors of size $w$.
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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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<!--ID: 1708179147798-->
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END%%
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%%ANKI
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Basic
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What is the range of $B2U_w$?
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Back: $[0, 2^w)$
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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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<!--ID: 1708179147801-->
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END%%
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%%ANKI
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Basic
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Why is "$B2U$" insufficient for use?
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Back: We need to understand how many bits conversion is with respect to.
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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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<!--ID: 1708179147804-->
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END%%
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### Two's-Complement
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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: $$B2T_w(\vec{x}) = -2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$$
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%%ANKI
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Basic
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What is the largest two's-complement value an integral type of $w$ bits can encode?
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Back: $2^{w-1} - 1$
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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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<!--ID: 1708179147807-->
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END%%
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%%ANKI
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Basic
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What is the smallest two's-complement value an integral type of $w$ bits can encode?
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Back: $-2^{w-1}$
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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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<!--ID: 1708179147810-->
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END%%
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%%ANKI
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Basic
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What half-open interval represents the possible $w$-bit two's-complement values?
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Back: $[-2^{w-1}, 2^{w-1})$
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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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<!--ID: 1708177246128-->
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END%%
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%%ANKI
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Cloze
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$[${1:$0$}, {2:$2^w$}$)$ is to unsigned as $[${1:$-2^{w-1}$}, {2:$2^{w-1}$}$)$ is to two's-complement.
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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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<!--ID: 1708179147813-->
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END%%
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%%ANKI
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Basic
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What is the binary representation of the smallest $4$-bit two's-complement number?
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Back: $1000_2$
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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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<!--ID: 1708179649872-->
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END%%
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%%ANKI
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Basic
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What is the binary representation of the largest $4$-bit two's-complement number?
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Back: $0111_2$
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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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<!--ID: 1708179649876-->
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END%%
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%%ANKI
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Cloze
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The {sign bit} refers to the {most significant bit} in two's-complement.
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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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<!--ID: 1708179649881-->
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END%%
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%%ANKI
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Basic
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What is the weight of the sign bit in $w$-bit two's-complement?
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Back: $-2^{w-1}$
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The {sign bit} refers to the {most significant bit} in two's-complement.
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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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<!--ID: 1708179649887-->
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END%%
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%%ANKI
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Basic
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What does the "uniqueness" of two's-complement refer to?
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Back: The function used to convert integral types to two's-complement is a bijection.
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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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<!--ID: 1708179649894-->
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END%%
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%%ANKI
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Basic
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How does Bryant et al. define $B2T_w$?
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Back: $B2T_w(\vec{x}) = -2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$
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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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<!--ID: 1708179649901-->
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END%%
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%%ANKI
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Basic
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What is $B2T_w$ an acronym for?
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Back: **B**inary "to" **t**wo's-complement.
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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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<!--ID: 1708179649907-->
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END%%
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%%ANKI
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Basic
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What does $w$ in $B2T_w$ represent?
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Back: The number of bits in the integral type being interpreted.
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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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<!--ID: 1708179649913-->
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END%%
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%%ANKI
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Basic
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What is the domain of $B2T_w$?
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Back: Bit vectors of size $w$.
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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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<!--ID: 1708179649921-->
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END%%
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%%ANKI
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Basic
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What is the range of $B2T_w$?
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Back: $[-2^{w-1}, 2^{w-1})$
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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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<!--ID: 1708179649928-->
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END%%
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%%ANKI
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Basic
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Why is "$B2T$" insufficient for use?
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Back: We need to understand how many bits conversion is with respect to.
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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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<!--ID: 1708179649935-->
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END%%
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## References
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* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. |