π’ Binary Calculator
Binary arithmetic with bitwise operations and decimal equivalents.
Binary result
How to Use the Binary Calculator
Enter two binary numbers (digits 0 and 1 only), pick an operation, and calculate. You will see the binary result plus both decimal equivalents and the decimal answer for verification.
Enter the first binary number using only 0s and 1s.
Choose arithmetic (+, β, Γ, Γ·) or a bitwise operation (AND, OR, XOR).
Enter the second number and calculate. Check the decimal row to verify your answer.
Binary Arithmetic Rules
Binary addition has four cases per column: 0+0=0, 1+0=1, 0+1=1, 1+1=10 (write 0, carry 1), and 1+1+carry=11 (write 1, carry 1). Subtraction borrows 2 instead of 10: 0β1 borrows to become 10β1=1. Multiplication works exactly like decimal long multiplication, which is simpler in binary because every partial product is either the number or zero. Division is repeated subtraction, same as long division.
Bitwise Operations Explained
Bitwise operators work on each bit position independently, so both numbers are aligned to the same width first. AND is used for masking: 1100 AND 1010 = 1000 keeps only bits set in both. OR sets a bit if either has it: 1100 OR 1010 = 1110. XOR detects differences: 1100 XOR 1010 = 0110, which is why it powers checksums and simple encryption. NOT flips every bit. Shifts (not included here) move bits left (multiply by 2) or right (divide by 2).
Reading the Decimal Equivalents
Each result shows the decimal value so you can sanity-check your binary work. If 1010 + 1100 gives 10110, the decimals confirm 10 + 12 = 22. When learning, cover the decimal column, work the problem by hand, then reveal it. For bitwise ops the decimal equivalents help you see patterns: AND can only clear bits (result β€ each input), OR can only set bits.
Frequently Asked Questions
How does binary addition work?
Binary addition follows four rules: 0+0=0, 0+1=1, 1+0=1, and 1+1=10 (write 0, carry 1). Work right to left like decimal addition, carrying the 1 whenever a column sums to 2 or 3. For example, 101 + 11: rightmost 1+1=10 (write 0 carry 1), next 0+1+carry1=10 (write 0 carry 1), next 1+carry1=10 (write 0 carry 1), final carry 1 gives 1000.
What are bitwise AND, OR, and XOR?
Bitwise operators compare two binary numbers bit by bit. AND outputs 1 only where both bits are 1. OR outputs 1 where at least one bit is 1. XOR outputs 1 where the bits differ. Example: 1010 AND 1100 = 1000, OR gives 1110, XOR gives 0110. Programmers use these for flags, masks, and low-level optimization.
How do I convert binary to decimal?
Number the bits from right to left starting at 0. Each 1-bit contributes 2 raised to its position. For 1011: 1x8 + 0x4 + 1x2 + 1x1 = 11. This calculator shows the decimal equivalent of every result automatically.
Why do computers use binary?
Hardware is built from transistors that are either on or off, so two states map perfectly to 1 and 0. Binary arithmetic needs only simple logic gates, making circuits fast, cheap, and reliable. Everything from text to video is ultimately encoded as binary numbers.
What is two's complement for negative numbers?
Two's complement is how computers store negatives: invert all bits of the positive number and add 1. In 8-bit, -5 is stored as 11111011. The leftmost bit acts as a sign flag. This system makes subtraction work as addition, which simplifies CPU design enormously.