Bit Manipulation and Bitwise Operations Cheat Sheet

Updated 2026-05-19

Next Topic: Blockchain Basics Cheat Sheet

🧠Study flashcards on this topic98 cards · spaced repetition→

Bitwise operations work directly on the binary representation of integers, enabling compact data encoding, high-performance algorithms, and low-level hardware control. Mastering the seven core operators—AND, OR, XOR, NOT, left shift, arithmetic right shift, and logical right shift—unlocks a toolkit of O(1) tricks: toggling flags, isolating bits, counting set bits in O(k), detecting power-of-two values, implementing branchless arithmetic, and solving classic problems (missing number, XOR swap, Gray code). Two's complement representation ties everything together, making signed-integer arithmetic consistent with unsigned bit patterns in nearly every modern language and CPU architecture.

What This Cheat Sheet Covers

This topic spans 12 focused tables and 156 indexed concepts, 98 flashcards. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.

Table 1 — Binary Representation FundamentalsTable 2 — Core Bitwise OperatorsTable 3 — Set / Clear / Toggle / Check BitsTable 4 — XOR Properties and Key IdentitiesTable 5 — Power-of-Two and Rightmost-Bit TricksTable 6 — Counting and Locating BitsTable 7 — Shift Operations Deep DiveTable 8 — Bitmask and Flag TechniquesTable 9 — Branchless Arithmetic TricksTable 10 — XOR-Based AlgorithmsTable 11 — Language-Specific Behavior and Built-insTable 12 — Real-World Applications

Quick IndexSubscribe to unlock

A jump-to index of every table row in this cheat sheet.

Mind MapSubscribe to unlock

An interactive map of every table and concept in this topic.

Table 1 — Binary Representation Fundamentals

Before any bit trick makes sense you need a firm grasp of how integers are actually laid out in memory. These rows cover the building blocks—unsigned versus two's complement, where the sign bit lives, what -1 and INT_MIN look like in raw bits, and how sign extension, hex notation, and endianness shape the bytes you'll be manipulating.

Concept	Example	Description
Unsigned binary	`0b1011` = 11	• Each bit $k$ contributes $2^k$ • value always ≥ 0
Two's complement (positive)	`0b0110` = +6	• MSB = 0 → positive • value = normal binary
Two's complement (negative)	`0b1010` = −6 (8-bit)	• MSB = 1 → negative • value = −(~n + 1)
Negate in two's complement	`~n + 1`	Flip all bits then add 1
One's complement	`~0b0110` = `0b1001`	• All bits flipped • has +0 and −0 (not used in modern CPUs)
Sign bit (MSB)	bit 31 in `int32`	0 = positive, 1 = negative
−1 representation	`0xFFFFFFFF` (32-bit)	• All bits set • universal across widths
INT_MIN	`0x80000000` (32-bit)	• Only the sign bit set • has no positive counterpart
INT_MAX	`0x7FFFFFFF` (32-bit)	All bits except sign bit set
Signed n-bit range	$-2^{n-1}$ to $2^{n-1}-1$	e.g., 8-bit: −128 to 127
Unsigned n-bit range	$0$ to $2^n - 1$	e.g., 8-bit: 0 to 255
Sign extension	`0b1010` (4-bit) → `0b11111010` (8-bit)	Replicate MSB into all new upper bits when widening
Hex notation	`0xFF` = 255	• Each hex digit = 4 bits • common for masks
Byte order (endianness)	`0x0A0B` in memory	• Little-endian: LSB first • big-endian: MSB first

Table 2 — Core Bitwise Operators

The seven operators here are the entire vocabulary—everything else in this cheat sheet is built by combining AND, OR, XOR, NOT, and the three shifts. Pay attention to the truth tables and the precedence rows: == binds tighter than &, which is the single most common source of silent bugs, so parenthesize religiously.

Concept	Example	Description
AND (&)	`5 & 3` → 1	• Output bit 1 only if both input bits are 1 • masks bits

The rest of this section is part of the full cheat sheet

Table 3 — Set / Clear / Toggle / Check Bits

The four everyday operations you reach for whenever a bit stands in for a flag or option. Each one is just a single-bit mask 1 << n combined with the right operator—OR to set, AND with the complement to clear, XOR to toggle, and a shift-and-mask to test—and the same patterns extend cleanly to whole bit fields and multi-bit masks.

Concept	Example	Description
Set bit n	`x \|= (1 << n)`	OR with a mask that has only bit n set

The rest of this section is part of the full cheat sheet

Table 4 — XOR Properties and Key Identities

XOR is the workhorse of clever bit algorithms, and the reason is the handful of algebraic identities collected here. Because it's commutative, associative, and self-inverse (x ^ x = 0), duplicate terms cancel out anywhere in a chain—the property that makes XOR feel almost magical in the algorithms further down.

Concept	Example	Description
Identity element	`x ^ 0` = x	XOR with 0 leaves value unchanged

The rest of this section is part of the full cheat sheet

Table 1 — Binary Representation Fundamentals

Concept	Example	Description
Unsigned binary	`0b1011` = 11	• Each bit $k$ contributes $2^k$ • value always ≥ 0
Two's complement (positive)	`0b0110` = +6	• MSB = 0 → positive • value = normal binary
Two's complement (negative)	`0b1010` = −6 (8-bit)	• MSB = 1 → negative • value = −(~n + 1)
Negate in two's complement	`~n + 1`	Flip all bits then add 1
One's complement	`~0b0110` = `0b1001`	• All bits flipped • has +0 and −0 (not used in modern CPUs)
Sign bit (MSB)	bit 31 in `int32`	0 = positive, 1 = negative
−1 representation	`0xFFFFFFFF` (32-bit)	• All bits set • universal across widths
INT_MIN	`0x80000000` (32-bit)	• Only the sign bit set • has no positive counterpart
INT_MAX	`0x7FFFFFFF` (32-bit)	All bits except sign bit set
Signed n-bit range	$-2^{n-1}$ to $2^{n-1}-1$	e.g., 8-bit: −128 to 127
Unsigned n-bit range	$0$ to $2^n - 1$	e.g., 8-bit: 0 to 255
Sign extension	`0b1010` (4-bit) → `0b11111010` (8-bit)	Replicate MSB into all new upper bits when widening
Hex notation	`0xFF` = 255	• Each hex digit = 4 bits • common for masks
Byte order (endianness)	`0x0A0B` in memory	• Little-endian: LSB first • big-endian: MSB first

Table 2 — Core Bitwise Operators

Concept	Example	Description
AND (&)	`5 & 3` → 1	• Output bit 1 only if both input bits are 1 • masks bits

Table 3 — Set / Clear / Toggle / Check Bits

Concept	Example	Description
Set bit n	`x \|= (1 << n)`	OR with a mask that has only bit n set

Table 4 — XOR Properties and Key Identities

Concept	Example	Description
Identity element	`x ^ 0` = x	XOR with 0 leaves value unchanged