notebook/notes/binary/shifts.md

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---
title: Shifts
TARGET DECK: Obsidian::STEM
FILE TAGS: binary
tags:
- binary
---
## Overview
Left shift operations (`<<`) drop the `k` most significant bits and fills the right end of the result with `k` zeros. Right shift operations (`>>`) are classified in two ways:
* **Logical**
* Drops the `k` least significant bits and fills the left end of the result with `k` zeros.
* This mode is always used when calling `>>` on unsigned data.
* Sometimes denoted as `>>>` to disambiguate from arithmetic right shifts.
* **Arithmetic**
* Drops the `k` least significant bits and fills the left end of the result with `k` copies of the most significant bit.
* This mode is usually used when calling `>>` on signed data.
%%ANKI
Basic
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.
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%%ANKI
Basic
What kinds of left shift operations are there?
Back: Just logical.
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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END%%
%%ANKI
Basic
What kinds of right shift operations are there?
Back: Logical and arithmetic
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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END%%
%%ANKI
Basic
What is a logical right shift operation?
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Back: One that fills the left end of the result with zeros.
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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END%%
%%ANKI
Basic
What is an arithmetic right shift operation?
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Back: One that fills the left end of the result with copies of the most significant bit.
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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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.
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END%%
%%ANKI
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.
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END%%
%%ANKI
Basic
What portability issue do shift operations introduce?
Back: There is no standard on whether right shifts of signed numbers are logical or arithmetic.
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: c17
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END%%
%%ANKI
Cloze
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{1:Arithmetic} right shifts are to {1:signed} numbers whereas {2:logical} right shifts are to {2:unsigned} numbers.
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: c17
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END%%
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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
Ignoring UB, what *typically* happens when shifting an `int32_t` by `k ≥ 32` bits?
Back: The shift value is interpreted as `k mod 32`.
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: c17
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END%%
%%ANKI
Basic
How is $x \bmod 2^k$ equivalently written as a bit mask?
Back: `x & ((1 << k) - 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.
2024-02-23 14:40:31 +00:00
Tags: c17
2024-02-14 01:23:48 +00:00
<!--ID: 1707873410780-->
END%%
## 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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* Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).