How to convert binary to decimal?

Multiply each bit by 2 raised to its position (counting from 0 on the right) and add the results. For example, binary 1101 = 1x8 + 1x4 + 0x2 + 1x1 = 13 in decimal.

How to add binary numbers step by step?

Add column by column from right to left, just like decimal addition. 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0 carry 1), 1+1+1=11 (write 1 carry 1). This calculator shows each carry bit.

What is two's complement?

Two's complement is a method for representing signed integers in binary. To negate a number, invert all bits and add 1. In 8-bit two's complement, values range from -128 to 127.

How to convert decimal to binary?

Repeatedly divide by 2 and record remainders. Read the remainders bottom-to-top. For example, 13: 13/2=6 R1, 6/2=3 R0, 3/2=1 R1, 1/2=0 R1, so 13 = 1101 in binary.

How do bitwise AND, OR, and XOR work?

These operations compare corresponding bits of two numbers. AND yields 1 only if both bits are 1. OR yields 1 if either bit is 1. XOR yields 1 if the bits differ. They are essential in programming for masks, flags, and encryption.

How to convert binary to hexadecimal?

Group binary digits into sets of 4 from right to left, then convert each group: 0000=0, 0001=1, ..., 1001=9, 1010=A, ..., 1111=F. For example, 11111111 = FF in hex.

Binary Calculator

Binary is the fundamental language of computers, encoding every piece of data as sequences of 0s and 1s. This calculator handles two essential tasks: base conversion (binary, decimal, hex, octal with ASCII display) and binary arithmetic (addition, subtraction, multiplication, division, plus bitwise AND, OR, XOR, and NOT). It supports unsigned integers and two's complement signed representation in 8, 16, and 32-bit widths, and shows carry-bit details for addition so you can follow each step of the computation.

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New

Bit Width

Signed

Input

Binary (Base 2)

11111111

Decimal (Base 10)

255

Hex (Base 16)

Octal (Base 8)

377

Bit Visualization

Operand A

Operation

Operand B

Result (Binary)

11000

Decimal

Hex

Octal

school Step-by-Step

How to Use This Calculator

Choose a mode

Select Conversion to translate between number bases, or Arithmetic to perform binary operations.

Enter your number

Type a number and select its base (binary, decimal, hex, or octal).

View results

See the converted value in all bases, or the arithmetic result with step-by-step binary working.

The Formula

To convert binary to decimal, multiply each bit by 2 raised to its position and sum the results. For example, 1011 = 1x8 + 0x4 + 1x2 + 1x1 = 11.

Value = sum(digit * base^position)

lightbulb Variables Explained

digit Each digit in the number (0 or 1 for binary)
base The number system base (2 for binary)
position Position from right, starting at 0

tips_and_updates Pro Tips

Each hex digit maps to exactly 4 binary bits, making hex-to-binary conversion straightforward

Two's complement flips all bits and adds 1 to represent negative numbers

Binary addition follows the same column rules as decimal: 1+1 = 10 (carry the 1)

Use bit widths (8/16/32) to see how numbers are stored in actual computer memory

AND, OR, XOR operate bit-by-bit and are fundamental to digital logic design

Binary Calculator - Conversion & Arithmetic with Steps

Convert numbers between binary, decimal, hexadecimal, and octal instantly. Perform binary addition, subtraction, multiplication, division, and bitwise operations with full step-by-step explanations. Supports unsigned and two's complement signed representation in 8, 16, and 32-bit widths.

Binary to Decimal Conversion

Binary (base 2) uses only digits 0 and 1. To convert to decimal, multiply each bit by 2 raised to its position power and sum the results. Example: 10110 = 16+4+2 = 22.

Binary Arithmetic Operations

Binary addition, subtraction, multiplication, and division follow the same principles as decimal arithmetic but with only two digits. This calculator shows carries, borrows, and partial products step by step.

Bitwise Operations & Two's Complement

Bitwise AND, OR, XOR, and NOT operate on individual bits and are fundamental to computing. Two's complement representation allows negative numbers in binary, with the most significant bit indicating the sign.

Frequently Asked Questions

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