Binary Calculator

How to use the binary calculator

1. Choose Calculate for arithmetic and bitwise operations, or Convert for binary ↔ decimal.
2. Type binary numbers using only 0 and 1. Spaces are ignored, so 1111 1011 is fine, and a 0b prefix is allowed. For +, − and × a leading minus sign is accepted.
3. Pick the operation. For shifts, enter how many places; for NOT, the bit width decides how many bits are flipped.
4. Choose a bit width to see the result in two's complement, the form computers use for signed integers.

Inputs can have up to 256 bits (or 78 decimal digits), and all arithmetic is exact. The column layout is drawn for numbers up to 64 bits; longer inputs still get exact answers.

Binary formulas

Binary is base 2: each digit (a bit, b₀ on the right) is 0 or 1, and each place is worth twice the place to its right. The arithmetic rules are the same as in decimal, but a column overflows at 2 instead of 10, so 1 + 1 is written 10 with a carry. Two's complement stores a negative number −x in w bits as 2^w − x, which lets a computer subtract by adding.

Worked examples

Addition: 1011 + 111

carry   1111
         1011
      +   111
        ─────
        10010

From the right: 1 + 1 = 10, write 0 and carry 1; 1 + 1 + 1 = 11, write 1 and carry 1; 0 + 1 + 1 = 10, write 0 and carry 1; 1 + 1 = 10. The answer is 10010, and in decimal 11 + 7 = 18.

Subtraction: 1011 − 111

borrow  1
        1011
      −  111
        ────
         100

The third column from the right needs a borrow: the leftmost 1 lends to it, which turns 0 − 1 into 10 − 1 = 1. The result, 100, is 11 − 7 = 4.

Converting 1101 to decimal, and 13 back to binary

1101₂ = 1 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰ = 8 + 4 + 0 + 1 = 13₁₀. Going the other way, divide by 2 and keep the remainders:

1. 13 ÷ 2 = 6 remainder 1
2. 6 ÷ 2 = 3 remainder 0
3. 3 ÷ 2 = 1 remainder 1
4. 1 ÷ 2 = 0 remainder 1

Reading the remainders from the last to the first gives 1101.

−5 in 8-bit two's complement

Write 5 in 8 bits, 0000 0101; flip every bit, 1111 1010; add 1: 1111 1011. Reading that pattern back as a signed 8-bit number gives −5, because the top bit counts as −128 instead of +128.

Bitwise: 1100 and 1010

1100 AND 1010 = 1000, 1100 OR 1010 = 1110, and 1100 XOR 1010 = 0110.

Powers of 2

8-bit two's complement

Tips and common mistakes

• Group bits in fours. Each group of 4 bits is one hexadecimal digit, which makes long binary numbers easier to read and check. The hex calculator converts between the two.
• Overflow. In 8 bits, 127 + 1 gives the pattern 1000 0000, which a signed 8-bit number reads as −128. The calculator flags results that only fit as unsigned.
• Right shifts lose bits. 1011 >> 2 is 10: the remainder of 11 ÷ 4 is thrown away.
• NOT depends on the width. NOT 1011 is 0100 in 4 bits but 1111 0100 in 8 bits, so always say how many bits you mean.