How to multiply or divide integers in binary

Binary operations of multiplication, integer division, and remainder are widely used in programming to perform mathematical calculations and data processing.

Just like binary addition and subtraction, these operations are not too different from their decimal system equivalents. In fact, they are even simpler to perform.

The only thing we need to keep in mind is what we saw in the previous entry about sums and subtractions, and comparisons in binary systems.

Binary Integer Multiplication

Binary multiplication is a fundamental operation used to calculate the product of two binary numbers.

Just like in decimal multiplication:

For example, to multiply the binary numbers 1011 (11 in decimal) and 1101 (13 in decimal):

    1011
  × 1101
  _______
    1011  (1011 × 1)
+0000000  (1011 × 0, shifted one place to the left)
+0101100  (1011 × 1, shifted two places to the left)
+1011000  (1011 × 1, shifted three places to the left)
________
10001111

The result is 10001111, which is equal to 143 in decimal.

In fact, as we see, the operation depends on the digits of the second factor.

• If the digit is a 0, it will convert to all 0, so there is no need to include it
• If the digit is a 1, it contributes a sum with a left shift

Additionally, we can use that multiplication is commutative to put “as the second operator” the one with more 0s. This way, we will have fewer sums to perform.

So basically, binary multiplication becomes:

• Rearrange the numbers so that the one with more 0s is the second factor (it will simplify things)
• Now go through the number from right to left. Each 1 contributes a number to the sum, shifted to the left.
• Now do all the sums you need to.

Binary Integer Division

Binary integer division is an operation that calculates the quotient of integer division between two binary numbers (just like in decimal division).

For example, to divide the binary number 1101 (13 in decimal) by 101 (5 in decimal):

1101 (dividend) / 101 (divisor)
  __________
 1101  | 101 
- 101     10 (quotient)
______
   11   (remainder)

The quotient is 10 (2 in decimal), and the remainder is 11 (3 in decimal).

Let’s see the process of doing the division “by hand” easily.

• We put the dividend and divisor
• We shift the divisor until it no longer fits in the dividend
• We subtract, obtaining the remainder
• We repeat the operation until the remainder is smaller than the divisor
• The quotient will have a 1 in each shift we have made

Example

Let’s divide the binary number 1101 (13 in decimal) by 101 (5 in decimal):

  1101 
/  101

We start by shifting 101 to the left until it no longer fits.

   1101 
/  1010  (<< 1, fits)

   1101 
/ 10100  (<< 2, no longer fits)

So we know that the quotient has a 1 in position 2. Now we subtract to calculate the remainder.

   1101 
-  1010
--------
     11

And in this case, the divisor 101 no longer fits in the remainder 11, so the divisor remains 10 and the final remainder is 11.

If there were more digits and the divisor fit in the remainder, we would repeat the operation. Shift and subtract, shift and subtract… In each step, I would obtain a digit of the quotient until the divisor no longer fits in the remainder.