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Clear bit shifting. Add notes on floor/ceiling functions.

c-declarations
Joshua Potter 2024-02-16 13:38:34 -07:00
parent 30c6fb97b5
commit 5fdd0b718e
8 changed files with 307 additions and 185 deletions
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notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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@ -128,7 +128,7 @@
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---
title: "2024-02-16"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
* Read "Queries, Modeling, and Transformation" of "Fundamentals of Data Engineering".
* Deleted many notes on shifting. Will wait until I look deeper into the actual representations of integral types before re-introducing.
* Spent time deriving properties of floor/ceiling functions and analogues in C. Added many more notes on partitioning arrays in halves.

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---
title: Floors & Ceilings
TARGET DECK: Obsidian::STEM
FILE TAGS: algebra algorithm
tags:
- algebra
---
## Overview
The **floor** of $x$ is the greatest integer less than $x$. The **ceiling** of $x$ is the least integer greater than $x$. These values are denoted $\lfloor x \rfloor$ and $\lceil x \rceil$ respectively.
%%ANKI
Basic
How is the floor of $x$ denoted?
Back: $\lfloor x \rfloor$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779607-->
END%%
%%ANKI
Basic
What is the floor of $x$?
Back: The greatest integer less than $x$.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779649-->
END%%
%%ANKI
Basic
How is the ceiling of $x$ denoted?
Back: $\lceil x \rceil$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779656-->
END%%
%%ANKI
Basic
What is the ceiling of $x$?
Back: The least integer greater than $x$.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779663-->
END%%
%%ANKI
Basic
When does $\lfloor x / 2 \rfloor = \lceil x / 2 \rceil$?
Back: When $x$ is even.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779668-->
END%%
%%ANKI
Basic
When does $\lfloor x / 2 \rfloor \neq \lceil x / 2 \rceil$?
Back: When $x$ is odd.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779674-->
END%%
%%ANKI
Basic
What does $\lceil x \rceil - \lfloor x \rceil$ equal?
Back: Either $0$ or $1$.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779681-->
END%%
%%ANKI
Basic
What can be said about $x$ if $\lceil x \rceil - \lfloor x \rceil = 0$?
Back: $x$ is even.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779687-->
END%%
%%ANKI
Basic
What can be said about $x$ if $\lceil x \rceil - \lfloor x \rceil = 1$?
Back: $x$ is odd.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779693-->
END%%
%%ANKI
Basic
What C operator corresponds to floor division?
Back: None.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779699-->
END%%
%%ANKI
Basic
When does C operator `/` behave like floor division?
Back: When the result is a nonnegative value.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779705-->
END%%
%%ANKI
Basic
When does C operator `/` behave like ceiling division?
Back: When the result is a nonpositive value.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779710-->
END%%
%%ANKI
Basic
What C operator corresponds to ceiling division?
Back: None.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779716-->
END%%
%%ANKI
Basic
How does C evaluate `10 / 3`?
Back: `3`
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c
<!--ID: 1708110779720-->
END%%
%%ANKI
Basic
How does C evaluate `floor(10.f / 3)`?
Back: `3`
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c
<!--ID: 1708110779725-->
END%%
%%ANKI
Basic
How does C evaluate `ceil(10.f / 3)`?
Back: `4`
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c
<!--ID: 1708110779729-->
END%%
%%ANKI
Basic
How does C evaluate `-10 / 3`?
Back: `-3`
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c
<!--ID: 1708110779734-->
END%%
%%ANKI
Basic
How does C evaluate `floor(-10.f / 3)`?
Back: `-4`
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c
<!--ID: 1708110779738-->
END%%
%%ANKI
Basic
How does C evaluate `ceil(-10.f / 3)`?
Back: `-3`
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
Tags: c
<!--ID: 1708110779742-->
END%%
%%ANKI
Basic
Given $r = \lfloor (p + q) / 2 \rfloor$, fair partitioning requires `A[r]` to be included in which of `A[p..r-1]` or `A[r+1..q]`?
Back: `A[p..r-1]`
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708115109770-->
END%%
%%ANKI
Basic
Given $r = \lfloor (p + q) / 2 \rfloor$, when is `A[p..r]` or `A[r+1..q]` equally sized?
Back: When `A[p..q]` has even size.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708115745322-->
END%%
%%ANKI
Basic
Given $r = \lceil (p + q) / 2 \rceil$, fair partitioning requires `A[r]` to be included in which of `A[p..r-1]` or `A[r+1..q]`?
Back: `A[r+1..q]`
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708115109777-->
END%%
%%ANKI
Basic
If `A[p..q]` has odd size, what `r` most fairly allows partitions `A[p..r]` and `A[r+1..q]`?
Back: $r = \lfloor (p + q) / 2 \rfloor$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708114757958-->
END%%
%%ANKI
Basic
If `A[p..q]` has odd size, what `r` most fairly allows partitions `A[p..r-1]` and `A[r..q]`?
Back: $r = \lceil (p + q) / 2 \rceil$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708114757961-->
END%%
%%ANKI
Basic
If `A[p..q]` has odd size, what `r` ensures `A[p..r-1]` has same size as `A[r+1..q]`?
Back: $r = \lfloor (p + q) / 2 \rfloor = \lceil (p + q) / 2 \rceil$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708114757964-->
END%%
%%ANKI
Basic
If `A[p..q]` has even size, what `r` most fairly allows partitions `A[p..r]` and `A[r+1..q]`?
Back: $r = \lfloor (p + q) / 2 \rfloor$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708114757968-->
END%%
%%ANKI
Basic
If `A[p..q]` has even size, what `r` most fairly allows partitions `A[p..r-1]` and `A[r..q]`?
Back: $r = \lceil (p + q) / 2 \rceil$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708114757971-->
END%%
%%ANKI
Basic
Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, how does the size of `A[p..r]` compare to `A[r+1..q]`?
Back: It either has zero or one more members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708115683351-->
END%%
%%ANKI
Basic
Given `A[p..q]` and $r = \lceil (p + q) / 2 \rceil$, how does the size of `A[p..r-1]` compare to `A[r..q]`?
Back: It either has zero or one fewer members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708115683354-->
END%%
%%ANKI
Basic
Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, how does the size of `A[p..r-1]` compare to `A[r..q]`?
Back: It either has one or two fewer members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708115683358-->
END%%
%%ANKI
Basic
Given `A[p..q]` and $r = \lceil (p + q) / 2 \rceil$, how does the size of `A[p..r]` compare to `A[r+1..q]`?
Back: It either has one or two more members.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708115683362-->
END%%
%%ANKI
Basic
Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, *why* is the size of `A[p..r]` potentially larger than `A[r+1..q]`?
Back: If `A[p..q]` has odd size, `A[p..r]` contains the midpoint.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708115683366-->
END%%
## References
* Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).

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---
title: Algebra
---

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%%ANKI
Basic
What is endianness?
Back: The ordering of bytes of a multibyte object.
Back: The ordering of bytes of a multi-byte object.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707661283771-->
END%%

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@ -20,30 +20,12 @@ Left shift operations (`<<`) drop the `k` most significant bits and fills the ri
%%ANKI
Basic
How is $2^n$ written in binary?
How is decimal value $2^n$ written in binary?
Back: As `1` followed by $n$ zeros.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707432641574-->
END%%
%%ANKI
Basic
How is $2^n$ written using bitwise shift operators?
Back: `1 << n`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tag: c
<!--ID: 1707432641576-->
END%%
%%ANKI
Basic
What decimal value does `1 << n` translate to?
Back: $2^n$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tag: c
<!--ID: 1707432641577-->
END%%
%%ANKI
Basic
What kinds of left shift operations are there?
@ -52,24 +34,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1707854589773-->
END%%
%%ANKI
Basic
How many most significant bits are dropped on a left shift by `k` (e.g. `<< k`)?
Back: `k`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1707854589778-->
END%%
%%ANKI
Basic
How many zeros exist in the result of a left shift by `k` (e.g. `<< k`)?
Back: At least `k`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1707854589781-->
END%%
%%ANKI
Basic
What kinds of right shift operations are there?
@ -94,39 +58,11 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1707854589789-->
END%%
%%ANKI
Basic
How many least significant bits are dropped on a right shift by `k` (e.g. `>> k`)?
Back: `k`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1707854589792-->
END%%
%%ANKI
Basic
How many zeros exist in the result of a logical right shift by `k` (e.g. `>> k`)?
Back: At least `k`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1707854589794-->
END%%
%%ANKI
Basic
How many zeros exist in the result of an arithmetic right shift by `k` (e.g. `>> k`)?
Back: Zero or more.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1707854589797-->
END%%
%%ANKI
Basic
What kind of right shift operation is *usually* applied to signed numbers?
Back: Arithmetic.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1707854589801-->
END%%
@ -135,7 +71,6 @@ Basic
What kind of right shift operation is applied to unsigned numbers?
Back: Logical.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1707854589804-->
END%%
@ -158,24 +93,6 @@ END%%
In C, it is undefined behavior to shift by more than the width $w$ of an integral type. Typically though, only the last $w$ bits are considered in the computation. For example, given `int32_t x`, `(x << 32) = (x << 0)`.
%%ANKI
Basic
According to the C standard, what is the result of shifting an `int32_t` value by `32`?
Back: It is undefined behavior.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1707873410765-->
END%%
%%ANKI
Basic
According to the C standard, what is the result of shifting an `int32_t` value by a negative value?
Back: It is undefined behavior.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1707873410773-->
END%%
%%ANKI
Basic
Ignoring UB, what *typically* happens when shifting an `int32_t` by `k ≥ 32` bits?
@ -194,101 +111,6 @@ Tags: c
<!--ID: 1707873410780-->
END%%
## Arithmetic
Shifting left by a single bit is equivalent to multiplication by $2$. Likewise, logically shifting right is equivalent to floor division by $2$. This is most clearly seen by examining the decimal expansion of a binary value. For example, consider $$001101_2 = 2^3 + 2^2 + 2^0$$
Multiplying by $2$ yields $$2 \cdot (2^3 + 2^2 + 2^0) = 2^4 + 2^3 + 2^1 = 011010_2$$
Dividing by $2$ yields $$(2^3 + 2^2 + 2^0) / 2 = 2^2 + 2^1 = 000110_2$$
%%ANKI
Basic
Multiplication by 2 is equivalent to what bitwise shift operation?
Back: Left shifting by 1 bit.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708009783552-->
END%%
%%ANKI
Basic
Floor division by 2 is equivalent to what bitwise shift operation?
Back: Logical right shifting by 1 bit.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708009783562-->
END%%
%%ANKI
Basic
Ceiling division by 2 is equivalent to what bitwise shift operation?
Back: None.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708009783565-->
END%%
%%ANKI
Cloze
{1:Multiplication} by 2 is to {2:left shifts} whereas {2:floor division} by 2 is to {1:logical right shifts}.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708009783568-->
END%%
%%ANKI
Basic
What arithmetic operation is equivalent to shifting left by $n$ bits?
Back: Multiplication by $2^n$.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708009783571-->
END%%
%%ANKI
Basic
When is $2n + 1$ equivalent to a one-bit cyclic shift left?
Back: When $n$'s leading bit is `1`.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708012138055-->
END%%
%%ANKI
Basic
What arithmetic operation is equivalent to a one-bit cyclic shift left?
Back: Given $n$ with no leading zeros, $2n + 1$.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708012138062-->
END%%
%%ANKI
Basic
What arithmetic operation is equivalent to logically shifting right by $n$ bits?
Back: Floor division by $2^n$.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708009783574-->
END%%
%%ANKI
Basic
Given $n = 0110_2$, what is the binary value of $2n$?
Back: $1100_2$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
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END%%
%%ANKI
Basic
Given $n = 0110_2$, what is the binary value of $2n + 1$?
Back: $1101_2$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708012138068-->
END%%
%%ANKI
Basic
Given $n = 0110_2$, what is the binary value of $\lfloor n / 2 \rfloor$?
Back: $0011_2$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708012138071-->
END%%
## References
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

2
notes/c/declarations.md
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@ -62,7 +62,7 @@ END%%
%%ANKI
Basic
What word sizes are typically nowadays?
What word sizes are typical nowadays?
Back: 32- and 64-bit word sizes.
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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