298 lines
11 KiB
Markdown
298 lines
11 KiB
Markdown
---
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title: Radices
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra
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tags:
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- algebra
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---
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## Overview
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The **radix** is the number of unique digits used to represent numbers in a positional numeral system. Most commonly used systems tend to be binary ($2$-base), octal ($8$-base), decimal ($10$-base), and [[#Hexadecimal|hexadecimal]] ($16$-base).
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%%ANKI
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Basic
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What is the process of subtracting a larger digit from a smaller one in radix $r$?
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Back: Decrement the next non-zero and add $r$ to the smaller digit in question.
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Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
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<!--ID: 1708534662981-->
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END%%
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%%ANKI
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Basic
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What does the process of "borrowing" in subtraction $100_2 - 10_2$ *look* like?
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Back:
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```
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2
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000
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- 10
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-----
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010
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```
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Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
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<!--ID: 1708534662989-->
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END%%
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%%ANKI
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Basic
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In a positional numeral system, what does "radix" refer to?
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Back: The number of unique digits used to represent numbers.
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Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
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<!--ID: 1708534662993-->
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END%%
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%%ANKI
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Basic
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What is the radix of the decimal system?
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Back: $10$
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Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
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<!--ID: 1708534662997-->
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END%%
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%%ANKI
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Basic
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What is the radix of the octal system?
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Back: $8$
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Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
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Tags: binary
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<!--ID: 1708534663001-->
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END%%
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%%ANKI
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Basic
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What is the radix of the hexadecimal system?
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Back: $16$
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Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
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Tags: binary::hex
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<!--ID: 1708534663005-->
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END%%
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%%ANKI
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Basic
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What is the radix of the binary system?
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Back: $2$
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Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
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Tags: binary
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<!--ID: 1708534663009-->
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END%%
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## Hexadecimal
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Hexadecimal is a 16-base numeral system, usually represented with digits `0` to `9` and `a` to `f` or `A` to `F`.
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%%ANKI
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Cloze
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A hexadecimal digit represents {4} bits.
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Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
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Tags: binary::hex
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<!--ID: 1708534663013-->
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END%%
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%%ANKI
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Cloze
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An octal digit represents {3} bits.
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Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
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Tags: binary
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<!--ID: 1708534663018-->
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END%%
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%%ANKI
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Cloze
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A byte consists of {2} hexadecimal digits.
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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: binary::hex
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<!--ID: 1707432641563-->
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END%%
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%%ANKI
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Cloze
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A nibble consists of {1} hexadecimal digits.
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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: binary::hex
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<!--ID: 1708534663022-->
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END%%
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%%ANKI
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Basic
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Hexadecimal digits are represented by what characters?
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Back: `a` to `f`, `A` to `F`, and `0` to `9`.
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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: binary::hex
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<!--ID: 1707432641565-->
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END%%
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%%ANKI
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Basic
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How does C denote a hexadecimal numeric constant?
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Back: With `0x` or `0X`.
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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: binary::hex c
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<!--ID: 1707432641567-->
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END%%
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%%ANKI
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Basic
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What is the decimal equivalent of hex `A`, `C`, and `F`?
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Back: `10`, `12`, and `15` respectively.
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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: binary::hex
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<!--ID: 1707432641568-->
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END%%
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%%ANKI
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Basic
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What is the hexadecimal equivalent of decimal `11`, `12`, and `14`?
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Back: `B`, `C`, and `E` respectively.
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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: binary::hex
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<!--ID: 1707432641570-->
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END%%
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%%ANKI
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Basic
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*When* should padding be introduced in binary to hexadecimal conversion?
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Back: When the number of bits is not a multiple of `4`.
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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: binary::hex
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<!--ID: 1707432641571-->
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END%%
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%%ANKI
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Basic
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*Where* is padding introduced in binary to hexadecimal conversion?
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Back: To the left of the binary sequence.
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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: binary::hex
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<!--ID: 1707432641573-->
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END%%
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%%ANKI
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Basic
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What are the possible hex values the first digit of $2^n$ can take on?
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Back: `1`, `2`, `4`, and `8`.
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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: binary::hex
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<!--ID: 1707432641579-->
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END%%
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%%ANKI
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Basic
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What are the possible values in binary that the first nibble of $2^n$ can take on?
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Back: `0001`, `0010`, `0100`, and `1000`.
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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: binary::hex
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<!--ID: 1707432641580-->
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END%%
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%%ANKI
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Basic
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How is $j$ interpreted in the hex representation of $2^{i + 4j}$?
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Back: As the number of `0`s in the encoding.
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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: binary::hex
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<!--ID: 1707432641582-->
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END%%
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%%ANKI
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Basic
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How is the $0$ in $2^{0 + 4j}$ translated to hex?
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Back: As hex digit `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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Tags: binary::hex
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<!--ID: 1707432641583-->
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END%%
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%%ANKI
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Basic
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How is the $1$ in $2^{1 + 4j}$ translated to hex?
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Back: As hex digit `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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Tags: binary::hex
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<!--ID: 1707432641585-->
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END%%
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%%ANKI
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Basic
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How is the $2$ (power) in $2^{2 + 4j}$ translated to hex?
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Back: As hex digit `4`.
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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: binary::hex
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<!--ID: 1707432641586-->
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END%%
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%%ANKI
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Basic
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How is the $3$ in $2^{3 + 4j}$ translated to hex?
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Back: As hex digit `8`.
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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: binary::hex
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<!--ID: 1707432641587-->
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END%%
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%%ANKI
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Basic
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How is $n$ in $2^n$ factored to quickly write the decimal value's hex representation?
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Back: $n = i + 4j$ where $0 \leq i \leq 3$.
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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: binary::hex
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<!--ID: 1707432641589-->
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END%%
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%%ANKI
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Basic
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How is the *remainder* of e.g. `158 / 16` managed in decimal to hex conversion?
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Back: As the next least significant bit of our conversion.
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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: binary::hex
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<!--ID: 1707432641594-->
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END%%
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%%ANKI
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Basic
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How is the *quotient* of e.g. `158 / 16` managed in decimal to hex conversion?
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Back: As the next value to divide by `16`.
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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: binary::hex
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<!--ID: 1707432641595-->
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END%%
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%%ANKI
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Basic
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When does repeated division in decimal to hex conversion end?
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Back: When the quotient (*not* the remainder) is `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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Tags: binary::hex
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<!--ID: 1707919792632-->
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END%%
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%%ANKI
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Basic
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How is e.g. `0xAC32` expressed as a sum of decimal values?
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Back: $(16^3 \times 10) + (16^2 \times 12) + (16^1 \times 3) + (16^0 \times 2)$
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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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Tags: binary::hex
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<!--ID: 1707432641596-->
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END%%
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%%ANKI
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Basic
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Which hexadecimal digits encode binary with a leading `1` bit?
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Back: `8` through `F`
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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: binary::hex
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<!--ID: 1708631918825-->
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END%%
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%%ANKI
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Basic
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Which hexadecimal digits encode binary with a leading `0` bit?
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Back: `0` through `7`
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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: binary::hex
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<!--ID: 1708631918829-->
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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.
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* “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173). |