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Walk-through

How to Use This Calculator

4 steps

1

Pick a base, bit width, and operation

Choose whether A and B are typed in decimal, binary, or hex, set the bit width (8, 16, or 32 bits), and pick an operation — AND, OR, XOR, NOT, left shift, or right shift.

2

Enter Value A (and Value B)

Type Value A in the base you selected. For AND/OR/XOR it also uses Value B; for a shift, Value B becomes the shift amount in bits; NOT ignores Value B entirely and works on A alone. The result updates instantly — there's no Calculate button.

3

Read the result

The result card shows the answer in decimal, binary, and hex at once. If the top bit is set, the calculator also shows the signed two's-complement decimal value alongside the unsigned one.

4

Explore the Operation, Bit View, and Shifts tabs

The Operation tab shows the equation plus a truth table (or the ×2ⁿ/÷2ⁿ shift explanation), highlighting the rows your numbers actually use. Bit View lines up A, B, and the result bit-by-bit in nibble groups. Shifts is a standalone visualizer for any left or right shift amount.

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Reference

Formula & Methodology

3 formulas

Bitwise AND, OR, XOR, NOT

AND: 1 if both bits are 1 · OR: 1 if either bit is 1 · XOR: 1 if the bits differ · NOT: flips every bit

Each operation is applied independently to every bit position, comparing (or flipping) the bits of A and B at that position. AND is 1 only when both input bits are 1; OR is 1 when at least one is 1; XOR is 1 exactly when the two bits differ; NOT simply flips every bit of A (0 becomes 1, 1 becomes 0).

Left and right shift

A << n = A × 2ⁿ · A >> n = ⌊A ÷ 2ⁿ⌋

A left shift by n bits moves every bit n places toward the most significant bit and fills the vacated positions on the right with 0 — the same as multiplying by 2ⁿ. A right shift moves every bit n places toward the least significant bit, discarding the bits that fall off the right edge — the same as an integer (floor) division by 2ⁿ.

Two's complement for negative numbers

negative(x) = 2ⁿ − |x| (at n-bit width)

Computers store negative integers using two's complement: at an n-bit width, a negative value x is stored as 2ⁿ − |x|. That's why the same bit pattern can be read two ways — as an unsigned value from 0 to 2ⁿ−1, or as a signed value from −2ⁿ⁻¹ to 2ⁿ⁻1−1 when the top (most significant) bit is treated as a sign bit.

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Glossary

Key Terms Explained

7 terms

Bitwise operation A calculation performed independently on each bit (binary digit) of a number's binary representation, rather than treating the number as a whole — AND, OR, XOR, NOT, and the shifts are all bitwise operations.

AND (&) Produces a 1 bit only where both input bits are 1; every other position becomes 0. Commonly used to mask off specific bits.

OR (|) Produces a 1 bit wherever at least one of the two input bits is 1. Commonly used to set specific bits.

XOR (^) Produces a 1 bit wherever the two input bits differ (one is 0 and the other is 1), and 0 where they match. Used for toggling bits and simple checksums.

NOT (~) A unary operation that flips every bit of its single input — every 0 becomes 1 and every 1 becomes 0. At a fixed bit width this is equivalent to computing −x−1.

Bit shift Moves every bit in a number a fixed number of places left or right. A left shift by n multiplies by 2ⁿ; a right (logical) shift by n divides by 2ⁿ, rounding down, discarding the bits shifted off the end.

Two's complement The standard way computers represent negative integers in binary: a negative number is stored as the bit pattern that, added to its positive counterpart, rolls over to zero at the chosen bit width. The most significant bit acts as a sign bit.

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Scenarios

Real-World Examples

3 worked examples

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Bitmask check

12 AND 10

Value A 12 (00001100) Value B 10 (00001010)

Lining up the bits: 00001100 AND 00001010 keeps a 1 only where both rows have a 1 — just bit position 3 (worth 8). So 12 AND 10 = 8 (binary 00001000, hex 0x08). The Bit View tab shows exactly which position survived the AND.

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XOR toggle

12 XOR 10

Value A 12 (00001100) Value B 10 (00001010)

XOR flips on wherever the two bit patterns disagree: 00001100 vs 00001010 differ at positions 2 and 1, so 12 XOR 10 = 6 (binary 00000110). Notice it's a different answer from AND (8) and OR (14) on the same two numbers — XOR is the only one of the three that's 0 when A equals B.

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Fast multiply by shifting

5 left-shifted by 1

Value A 5 (00000101) Shift amount 1

Left shift by 1 moves every bit one place left and fills 0 on the right: 00000101 becomes 00001010. That's 5 × 2¹ = 10 — a left shift by n is always the same as multiplying by 2ⁿ, which is why bit shifts are a common fast-multiply trick in low-level code.

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Deep Dive

Understanding the Bitwise Calculator

3 chapters · 4 min read

Bitwise operators — AND, OR, XOR, NOT, and the two shifts — work directly on the binary representation of a number, one bit at a time, instead of on its decimal value. They show up constantly in programming: bitmasks, flags, permissions, hashing, and low-level performance tricks all lean on them. This calculator lets you enter two integers in decimal, binary, or hex, apply any of the six operations, and see exactly which bits changed.

How each operation works

AND, OR, and XOR each compare A and B one bit position at a time and combine them according to a simple rule: AND keeps a 1 only where both inputs are 1, OR keeps a 1 where either input is 1, and XOR keeps a 1 only where the inputs disagree. NOT is different — it takes a single input and flips every bit, which at a fixed bit width is the same as computing −A−1.

The two shift operations move bits sideways instead of combining two numbers. A left shift by n multiplies the value by 2ⁿ, shifting every bit toward the high end and filling the low end with zeros. A right (logical) shift by n divides by 2ⁿ and rounds down, shifting every bit toward the low end and discarding whatever falls off the right edge.

Inputs and what they mean

Pick the base your numbers are written in — decimal, binary, or hex — and the bit width (8, 16, or 32 bits) that the calculation is performed at. The bit width matters because it determines how NOT and negative numbers behave: everything is computed using two's complement at that width, so the same operation on the same numbers can give a different bit pattern at 8-bit versus 16-bit width.

For AND, OR, and XOR, both Value A and Value B are full operands. For a shift, Value B is reinterpreted as the shift amount in bits (always read as a plain decimal number, regardless of the selected base) rather than as a second operand. NOT only uses Value A.

Limits and edge cases

Negative decimal input is supported and stored via two's complement at the chosen bit width — for example, −1 at 8-bit width is the all-ones pattern 11111111. Shifting by an amount at or beyond the configured bit width simply clears every bit, since there's nothing left to shift. This calculator focuses on integer bitwise operations; for converting a number between number systems without applying an operation, use the Number Base Calculator instead.

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Questions

Frequently Asked Questions

6 questions

What does AND actually do?+

Bitwise AND compares A and B one bit at a time and outputs a 1 only where both bits are 1 — every other position becomes 0. It's commonly used to check or clear specific bits (a "bitmask").

How is XOR different from OR?+

OR outputs a 1 wherever at least one input bit is 1, including when both are 1. XOR outputs a 1 only where the two bits differ — if both bits are 1, XOR gives 0 while OR still gives 1. That's the key difference: XOR is 0 whenever A and B match.

Why does a left shift equal multiplying by 2ⁿ?+

Each bit position in binary represents a power of two, just like each digit in decimal represents a power of ten. Shifting every bit one place to the left is the binary equivalent of moving every decimal digit one place left (multiplying by 10) — except in binary, one place left means multiplying by 2. Shifting by n places multiplies by 2ⁿ.

What does NOT do to a number?+

NOT flips every bit of its input — every 0 becomes a 1 and every 1 becomes a 0. At a fixed bit width, the result is the two's-complement representation of −(A + 1), which is why NOT of a positive number comes out negative when read as a signed value.

What's the difference between the decimal, binary, and hex bases?+

They're just three different ways of writing the same number: decimal (base 10) uses digits 0-9, binary (base 2) uses only 0 and 1, and hexadecimal (base 16) uses 0-9 and A-F, where each hex digit represents exactly 4 binary bits. Switching the Input Base only changes how you type A and B — the calculator always shows the result in all three.

How are negative numbers handled?+

Negative integers are stored using two's complement at the bit width you've selected — at 8-bit width, −1 is the bit pattern 11111111. Because the calculation happens at a fixed width, the same negative decimal value can produce a different bit pattern at 8-bit versus 16-bit or 32-bit width, so check that your bit width matches what you're modeling.

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Bitwise Calculator

Result

How to Use This Calculator

Pick a base, bit width, and operation

Enter Value A (and Value B)

Read the result

Explore the Operation, Bit View, and Shifts tabs

Formula & Methodology

Key Terms Explained

Real-World Examples

Bitmask check

XOR toggle

Fast multiply by shifting

Understanding the Bitwise Calculator

How each operation works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Result

How to Use This Calculator

Pick a base, bit width, and operation

Enter Value A (and Value B)

Read the result

Explore the Operation, Bit View, and Shifts tabs

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Bitwise Calculator

How each operation works

Inputs and what they mean

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Keep Exploring

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