273 lines
10 KiB
Markdown
273 lines
10 KiB
Markdown
---
|
|
title: Radices
|
|
TARGET DECK: Obsidian::STEM
|
|
FILE TAGS: algebra
|
|
tags:
|
|
- algebra
|
|
---
|
|
|
|
## Overview
|
|
|
|
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).
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the process of subtracting a larger digit from a smaller one in radix $r$?
|
|
Back: Decrement the next non-zero and add $r$ to the smaller digit in question.
|
|
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).
|
|
<!--ID: 1708534662981-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does the first step in the subtraction process of $100_2 - 10_2$ *look* like?
|
|
Back: $020_2 - 10_2$
|
|
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).
|
|
<!--ID: 1708534662989-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
In a positional numeral system, what does "radix" refer to?
|
|
Back: The number of unique digits used to represent numbers.
|
|
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).
|
|
<!--ID: 1708534662993-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the radix of the decimal system?
|
|
Back: $10$
|
|
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).
|
|
<!--ID: 1708534662997-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the radix of the octal system?
|
|
Back: $8$
|
|
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).
|
|
Tags: binary
|
|
<!--ID: 1708534663001-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the radix of the hexadecimal system?
|
|
Back: $16$
|
|
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).
|
|
Tags: binary::hex
|
|
<!--ID: 1708534663005-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the radix of the binary system?
|
|
Back: $2$
|
|
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).
|
|
Tags: binary
|
|
<!--ID: 1708534663009-->
|
|
END%%
|
|
|
|
## Hexadecimal
|
|
|
|
Hexadecimal is a 16-base numeral system, usually represented with digits `0` to `9` and `a` to `f` or `A` to `F`.
|
|
|
|
%%ANKI
|
|
Cloze
|
|
A hexadecimal digit represents {4} bits.
|
|
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).
|
|
Tags: binary::hex
|
|
<!--ID: 1708534663013-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
An octal digit represents {3} bits.
|
|
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).
|
|
Tags: binary
|
|
<!--ID: 1708534663018-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
A byte consists of {2} hexadecimal digits.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641563-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Cloze
|
|
A nibble consists of {1} hexadecimal digits.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1708534663022-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Hexadecimal digits are represented by what characters?
|
|
Back: `a` to `f`, `A` to `F`, and `0` to `9`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641565-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How does C denote a hexadecimal numeric constant?
|
|
Back: With `0x` or `0X`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex c
|
|
<!--ID: 1707432641567-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the decimal equivalent of hex `A`, `C`, and `F`?
|
|
Back: `10`, `12`, and `15` respectively.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641568-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What is the hexadecimal equivalent of decimal `11`, `12`, and `14`?
|
|
Back: `B`, `C`, and `E` respectively.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641570-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*When* should padding be introduced in binary to hexadecimal conversion?
|
|
Back: When the number of bits is not a multiple of `4`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641571-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
*Where* is padding introduced in binary to hexadecimal conversion?
|
|
Back: To the left of the binary sequence.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641573-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What are the possible hex values the first digit of $2^n$ can take on?
|
|
Back: `1`, `2`, `4`, and `8`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641579-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What are the possible values in binary that the first nibble of $2^n$ can take on?
|
|
Back: `0001`, `0010`, `0100`, and `1000`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641580-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $j$ interpreted in the hex representation of $2^{i + 4j}$?
|
|
Back: As the number of `0`s in the encoding.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641582-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the $0$ in $2^{0 + 4j}$ translated to hex?
|
|
Back: As hex digit `1`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641583-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the $1$ in $2^{1 + 4j}$ translated to hex?
|
|
Back: As hex digit `2`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641585-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the $2$ (power) in $2^{2 + 4j}$ translated to hex?
|
|
Back: As hex digit `4`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641586-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the $3$ in $2^{3 + 4j}$ translated to hex?
|
|
Back: As hex digit `8`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641587-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is $n$ in $2^n$ factored to quickly write the decimal value's hex representation?
|
|
Back: $n = i + 4j$ where $0 \leq i \leq 3$.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641589-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the *remainder* of e.g. `158 / 16` managed in decimal to hex conversion?
|
|
Back: As the next least significant bit of our conversion.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641594-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the *quotient* of e.g. `158 / 16` managed in decimal to hex conversion?
|
|
Back: As the next value to divide by `16`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641595-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
When does repeated division in decimal to hex conversion end?
|
|
Back: When the quotient (*not* the remainder) is `0`.
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707919792632-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is e.g. `0xAC32` expressed as a sum of decimal values?
|
|
Back: $(16^3 \times 10) + (16^2 \times 12) + (16^1 \times 3) + (16^0 \times 2)$
|
|
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
Tags: binary::hex
|
|
<!--ID: 1707432641596-->
|
|
END%%
|
|
|
|
## References
|
|
|
|
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
|
|
* “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). |