math

Bitwise Calculator

Bitwise operations are fundamental to computer science, cryptography, and low-level programming. This calculator executes bitwise AND, OR, and XOR operations on numbers from binary, octal, and decimal systems, displaying results in all three formats. Whether you're working with embedded systems, debugging bit flags, or studying digital logic, this tool simplifies multi-base bitwise manipulation.

Last updated: May 5, 2026

Creators Mateusz Mucha, BEng

Reviewers Anna Szczepanek and Jasmine J. Mah

1,364 people find this calculator helpful

Bits and Number Systems Explained

A bit is the atomic unit of digital information, representing a single logical value: 1 or 0. Computers process information by grouping bits into larger structures—a byte contains 8 bits, allowing 2⁸ = 256 distinct states.

Most programming involves three number systems:

Binary (base 2): Digits are 0 and 1. Used directly by processors. Example: 1010₂ = 10₁₀
Decimal (base 10): Familiar everyday system. Example: 87₁₀
Octal (base 8): Digits 0–7, historically important in Unix file permissions. Example: 127₈ = 87₁₀

Converting between systems: 87 in decimal becomes 1010111 in binary and 127 in octal.

Understanding Bitwise Operators

Bitwise operators compare two numbers position-by-position in binary, generating a result based on logical rules applied to each bit pair.

Bitwise AND: Returns 1 only when both input bits are 1; otherwise returns 0. Useful for masking—extracting specific bits.

Bitwise OR: Returns 1 if either input bit is 1. Combines flags and sets multiple bits simultaneously.

Bitwise XOR (exclusive OR): Returns 1 only when bits differ. Toggles flags and detects changes between states.

Example with 87 (binary: 1010111) AND 101 (binary: 1100101):

Position 0: 1 AND 1 = 1
Position 1: 1 AND 0 = 0
Position 2: 1 AND 1 = 1
Result: 1010101 = 85 in decimal

Bitwise Operation Formulas

These are the core logical rules applied at each bit position. The calculator supports 8, 16, and 20-bit representations:

AND: result = input1 ∧ input2

OR: result = input1 ∨ input2

XOR: result = input1 ⊕ input2

input1, input2 — Numbers in binary, octal, or decimal format to be compared bit-by-bit
result — Output number displayed in binary, octal, and decimal after the operation

How to Use This Calculator

Select your bit width (8, 16, or 20 bits) to define the range of valid inputs—8 bits supports −128 to 127 (signed) or 0 to 255 (unsigned).

Choose your input datatype: binary (prefix with 0b or use 0 and 1), octal (prefix with 0o or use 0–7), or decimal. Enter two numbers and select your operation (AND, OR, or XOR).

The calculator immediately displays the result in all three numeral systems. Check the sign handling if using negative numbers—two's complement representation is applied automatically for signed datatypes.

Common Pitfalls and Tips

Bitwise operations are precise, but input errors and sign misunderstandings are frequent.

Watch Your Number System — A number entered as '101' in decimal is very different from 101 in binary (5 in decimal). Always explicitly specify or verify your input base to avoid off-by-one errors and incorrect results.
Bit Width Affects Range — Selecting 8 bits limits unsigned integers to 0–255, while 16 bits extends to 0–65535. Oversized inputs will trigger an error. Plan your bit width before entering large numbers.
Signed vs. Unsigned Matters — Negative numbers use two's complement in signed mode, which inverts bits and adds 1. This changes XOR and AND results dramatically. Confirm your datatype selection matches your use case.
XOR for Toggle, AND for Masking — Use XOR to flip specific bits and detect changes. Use AND to isolate bits by zeroing unwanted positions. Mixing operators often produces unexpected results; test with small examples first.

Frequently Asked Questions

What is the difference between bitwise AND and logical AND?

Bitwise AND compares each individual bit of two numbers and produces a number result. Logical AND evaluates two boolean values and returns a single true/false. For example, bitwise AND of 6 (binary 110) and 3 (binary 011) yields 2 (binary 010). Logical AND of true and false yields false. Bitwise operators work on integers at the bit level; logical operators work on truth values.

Can I use negative numbers in the bitwise calculator?

Yes, but only when using signed datatypes. Negative numbers are internally stored using two's complement representation. In 8-bit signed mode, −1 is represented as 11111111 in binary. The calculator validates that your number fits within the selected bit width and datatype. Unsigned modes reject negative inputs outright, so verify your datatype setting if you need to perform bitwise operations on negative integers.

Why would I need bitwise operations in real-world programming?

Bitwise operations are essential for hardware interaction, flag management, and performance-critical code. Device drivers use bitwise AND to check if specific bits are set in status registers. Graphics engines use bit shifting and masking for pixel manipulation. File permissions in Unix use octal bitwise representations. Cryptography relies heavily on XOR. Network programming uses bitwise AND to determine subnet masks. Understanding these operations is crucial for embedded systems and low-level optimization.

What is two's complement and how does it affect XOR?

Two's complement is the standard method for representing negative numbers in binary. To negate a number, invert all bits and add 1. In 8-bit two's complement, −5 becomes 11111011. This affects XOR results because −5 XOR −3 is not the same as 5 XOR 3. The calculator applies two's complement automatically for signed inputs, but you must understand that the bit patterns of negative numbers are different, leading to different XOR and AND outcomes.

What happens if I enter a number larger than the bit width allows?

The calculator will reject it with a validation error. For example, if you select 8 bits unsigned, the maximum allowed input is 255 in decimal or 377 in octal. The bit width fundamentally limits the range of representable numbers. Choose your bit width based on your data requirements before entering large numbers. Attempting to use more bits than allowed will prevent calculation and display an out-of-range warning.

How do I convert between binary, octal, and decimal?

The calculator displays results in all three formats automatically. To convert manually: binary to decimal, multiply each bit by powers of 2 and sum (e.g., 1010 = 1×8 + 0×4 + 1×2 + 0×1 = 10). Decimal to binary, repeatedly divide by 2 and collect remainders in reverse. Octal uses base 8, so 127 octal = 1×64 + 2×8 + 7×1 = 87 decimal. The calculator handles these conversions, so focus on understanding the operation rather than manual conversion arithmetic.

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