Binary Calculator — Add, Subtract, Multiply & Divide with Steps

How to use the binary calculator

This binary calculator adds, subtracts, multiplies and divides binary numbers, and it also does the bitwise operations AND, OR, XOR and NOT, two’s complement, and left and right shifts. Under every answer it shows the working step by step, the way you would write it on paper, so you can check homework or learn the method rather than just copy a result.

1. Choose how you type the numbers. Binary is the default. Switch to decimal, hex or octal and the numbers already in the boxes are converted for you, so nothing is lost.
2. Type the first number, pick an operation, type the second. Spaces and underscores are ignored (1011 0110 is fine), a 0b or 0x prefix is accepted, and a minus sign makes a number negative. NOT and two’s complement use only the first number.
3. Pick a bit width of 8, 16, 32 or 64 bits. It decides how a negative answer is written in two’s complement, how many bits NOT flips, and whether an answer overflows.
4. Read the answer in binary, decimal, hexadecimal and octal, each with a Copy button, plus the exact bit pattern a register of your chosen width would hold.
5. Follow the steps. Addition shows the carry row, subtraction the borrows and the two’s complement method, multiplication the shifted rows, and division a long division table, with a decimal check at the end.

The calculator works with whole numbers of any size. It uses exact integer arithmetic, so even a 64-bit number multiplied by itself comes out right to the last digit: 18446744073709551615 squared is 340282366920938463426481119284349108225. Everything runs in your browser and nothing you type is sent anywhere.

Binary addition

Binary addition works exactly like the column addition you learned at school, except that each column can only hold 0 or 1. As soon as a column adds up to 2, you write 0 and carry 1 to the next column, just as you carry at 10 in decimal. There are only four cases to remember:

Worked example: 1011 + 110

Line the numbers up on the right and add one column at a time, starting with the ones column. The carries are written above the next column:

carry 111  
       1011
    +  0110
      -----
      10001

1. Column 1: 1 + 0 = 1, so write 1.
2. Column 2: 1 + 1 = 10, so write 0 and carry 1.
3. Column 3: 0 + 1 + 1 = 10, so write 0 and carry 1.
4. Column 4: 1 + 0 + 1 = 10, so write 0 and carry 1.
5. The last carry has no column to go into, so it becomes a new 1 at the front.

A run of 1s makes a chain of carries. Adding 1 to 1111 carries all the way along and gives 10000: 15 + 1 = 16. That is why 1111…1 plus one is always a 1 followed by zeros.

Overflow in binary addition

On paper a sum can always grow by one digit. Inside a computer, numbers live in a fixed number of bits, and an answer that needs more bits than that overflows. With 8 bits, 11001000 (200) + 1100100 (100) should be 100101100 (300), which needs 9 bits. An 8-bit register keeps only the lowest 8 bits, 00101100, and reads it as 44. The calculator always gives the true answer, and the “In 8 bits” box shows what the register would hold and says when an answer overflows.

Binary subtraction

Subtraction also goes column by column from the right. Three of the four cases are easy: 0 − 0 = 0, 1 − 0 = 1 and 1 − 1 = 0. The fourth, 0 − 1, needs a borrow: take 1 from the next column to the left, which is worth 2 in this column, and work out 2 − 1 = 1. The column you borrowed from is then one smaller.

Worked example with borrowing: 1100 − 101

borrow  111
       1100
     −  101
       ----
       0111

The first column is 0 − 1, so it borrows. The second column now has to give 1 away but holds 0, so it borrows too, and so does the third. The fourth column pays the last borrow: 1 − 0 − 1 = 0. If the second number is bigger than the first, subtract the other way round and put a minus sign in front of the answer; the calculator does this for you and says so in the steps.

Binary subtraction using two’s complement

Computers don’t borrow. They subtract by adding the two’s complement of the second number: flip every bit, add 1, add that to the first number, and drop any carry out of the top bit. Here is 5 − 9 in 8 bits:

          9 = 00001001
flip the bits 11110110
        add 1 11110111
              --------
          5 = 00000101
     + (−9) = 11110111
              --------
              11111100

The result starts with a 1, so as a signed number it is negative: flip it and add 1 to find its size, 100, which is 4. So 5 − 9 = −4. The calculator shows this method under every subtraction, next to the borrowing method, for whichever bit width you pick.

Binary multiplication

Multiplying by a single binary digit is trivial: anything times 0 is 0 and anything times 1 is itself. So long multiplication in binary comes down to shift and add. For every 1 in the second number, write the first number shifted left by that digit’s position; for every 0, write a row of zeros. Then add the rows.

    1011
×    101
  ------
    1011
+  00000
+ 101100
  ------
  110111

This is exactly how simple processors multiply, one shift and one addition per bit. The calculator draws these rows for any product where the second number has up to 32 bits, and grays out the rows of zeros.

Binary division with remainder

Binary long division is easier than decimal long division, because at each step the divisor either fits once or not at all. Bring down the next bit of the dividend; if the running value is at least the divisor, write 1 in the quotient and subtract; otherwise write 0. Here is 11011 ÷ 101 (27 ÷ 5):

The quotient bits read 00101, which is 101 (5), and the remainder is 10 (2). Check: 5 × 5 + 2 = 27. The calculator always gives both the quotient and the remainder. With negative numbers it rounds the quotient toward zero and gives the remainder the sign of the first number, as most programming languages do.

Negative binary numbers and two’s complement

Binary on paper can simply put a minus sign in front: −101 is minus five. Computers have no minus sign, only bits, so they use two’s complement. In a fixed number of bits, the top bit counts as negative: in 8 bits it is worth −128 instead of +128. To write a negative number, take the positive one, flip every bit and add 1:

          5 = 00000101
flip the bits 11111010
        add 1 11111011

Some 8-bit values, computed by the calculator:

A neat property: −1 is always all ones, whatever the width. The bit width sets the range of numbers you can store:

One’s complement and two’s complement

Flipping every bit on its own gives the one’s complement, which is what the NOT button does. Adding 1 to that gives the two’s complement, which is what the 2’s button does. In 8 bits, NOT 101 is 11111010, which reads as 250 unsigned or −6 signed, while the two’s complement of 5 is 11111011, which is −5. Most computers use two’s complement because the same adding circuit then works for positive and negative numbers alike, and there is only one zero.

Bitwise AND, OR, XOR and NOT

Bitwise operations compare two numbers one bit at a time, with no carrying between columns. They are the basic tools for masks, flags, permissions and colors in programming. The rules for each pair of bits:

Applied to 1100 (12) and 1010 (10): AND gives 1000 (8), OR gives 1110 (14), XOR gives 0110 (6). AND keeps the bits that are on in both, so it is used to test or clear bits with a mask. OR switches bits on. XOR flips the bits where the mask has a 1, and XOR-ing twice with the same mask gives back the original number. NOT flips every bit within the chosen width.

Left shift and right shift

A shift moves every bit sideways. Shifting left by one place adds a 0 on the right and doubles the number: 101 << 3 = 101000, which is 5 × 8 = 40. Shifting right drops bits off the end and halves the number, rounding down: 1101 >> 1 = 110, which is 13 ÷ 2 = 6 with the remainder lost. For negative numbers the calculator does an arithmetic shift, which keeps the sign.

Binary, decimal, hex and octal

Every answer is shown in four number systems. Hexadecimal (base 16) is the most compact way to write binary, because each hex digit stands for exactly four bits; octal (base 8) uses three bits per digit. That makes it easy to convert by hand: split the binary number into groups of four from the right and replace each group with its hex digit. The first sixteen numbers:

For conversions on their own, without a calculation, use the Binary Converter, which also shows the division-by-2 method. To turn words into binary code, use the Binary Translator, and for hex codes of letters see the Hex to Text converter.

Tips for checking binary arithmetic

• Convert to decimal and back. The last line of the steps repeats the sum in decimal. If the decimal sum is right, the binary one is too.
• Look at the last bit. An odd number ends in 1 and an even number in 0, so odd + odd must end in 0, and anything times an even number ends in 0.
• Count the digits. A sum is at most one digit longer than the longer number. A product has at most as many digits as the two numbers together.
• Line up on the right. Most mistakes on paper come from columns that slipped. Pad the shorter number with zeros on the left, as the calculator does.
• Group in fours. Long binary numbers are easier to read as 1011 0110; the result box does this for you, and copying gives the digits without spaces.

Want to get quicker at reading binary in your head? The Binary Game turns place values into a card game, with levels from 4 to 8 bits and a 60-second challenge.

A short history of binary arithmetic

Gottfried Wilhelm Leibniz described arithmetic with only the digits 0 and 1 in a paper published in 1703. In the 1840s and 1850s George Boole worked out an algebra of logic, the system of true and false values that AND, OR and NOT come from. In 1937 Claude Shannon showed in his master’s thesis that Boolean algebra could describe electrical switching circuits, and the computers that followed did their arithmetic in binary with exactly the carries, borrows and two’s complement shown above.