--- 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 nonzero 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 process of "borrowing" in subtraction $100_2 - 10_2$ *look* like? Back: ``` 2 000 - 10 ----- 010 ``` 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 c17 <!--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: Assuming big-endian, as the next most 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%% %%ANKI Basic Which hexadecimal digits are encoded in binary with a leading `1` bit? Back: `8` through `F`. 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: 1708631918825--> END%% %%ANKI Basic Which hexadecimal digits are encoded in binary with a leading `0` bit? Back: `0` through `7`. 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: 1708631918829--> END%% ## Bibliography * 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).