Binary Addition, Multiplication, Subtraction, And Division

Basic mathematical operations with binary numbers works similar to the decimal system. However there are a few rules specific to the binary system. We’ll look at each of them individually.

Addition

There are 3 basic rules for adding binary numbers:

1.     0 + 0 = 0

2.     0 + 1 = 1

3.     1 + 1 = 10. If the sum of 2 bits is greater than 1, we need to shift a column on the left. In decimal system, 1 + 1 = 2. Binary notation of 2 is 10 (1 * 2^1 + 0 * 2^0). So we keep 0 in the 1's column and shift (carry over) 1 to the 2's column.

Other rules are same as the decimal system, i.e. we add from right to left and the carry over get’s added to the digits in the next column.

Now lets try adding 11 to 13. Binary for 11 is 1011 and that for 13 is 1101.

1011

+ 1101

1.     1's col = 1 + 1 = 10. We keep 0 in 1's col and carry over 1 to 2's col.

2.     2's col = 1 + 0 + 1 (carry over) = (1 + 0) + 1 = 1 + 1 = 10. Once again we keep 0 in 2's col and carry over 1 to 4's col.

3.     4's col = 0 + 1 + 1 (carry over) = (0 + 1) + 1 = 1 + 1 = 10. Keep 0 in 4's col and carry over 1 in 8's col.

4.     8's col = 1 + 1 + 1 (carry over) = (1 + 1) + 1 = 10 + 1 = 11. Keep 1 in 8's col and carry over 1 in 16's col.

The sum is 11000. 11000 = 1 * 2^4 + 1 * 2^3 + 0 * 2^2 + 0 * 2^1 + 0 * 2^0 = 16 + 8 + 0 + 0 + 0 = 24 = 11 + 13.

Multiplication

Multiplication in binary is exactly as it is in decimal, i.e. multiply numbers right to left and multiply each digit of one number to every digit of the other number, them sum them up. The 3 basic binary multiplication rules are also similar to decimal.

1.     1 * 1 = 1

2.     1 * 0 = 0 * 1 = 0

3.     0 * 0 = 0

Also, remember that for every left shift of digit of the multiplier, an extra zero needs to be appended to the product. This is similar to the decimal system as well.

1011

X 1101

1.     1011 * 1 (multiplier 1's col) = 1011

2.     1011 * 0 (multiplier 2's col) = 00000 (one zero appended at the end)

3.     1011 * 1 (multiplier 4's col) = 101100 (two zero’s appended at the end)

4.     1011 * 1 (multiplier 8's col) = 1011000 (three zero’s appended at the end)

5.     Sum up. 1011 + 00000 + 101100 + 1011000 = ((1011 + 00000) + 101100) + 1011000 = (01011 + 101100) + 1011000 = 110111 + 1011000 = 10001111

So the product is 10001111 which is = 1 * 2^7 + 1 * 2^3 + 1 * 2^2 + 1 * 2^1 + 1 * 2^0 = 128 + 8 + 4 + 2 + 1 = 143 = 11 * 13.

Subtraction

Before trying subtraction, we need to understand how negative numbers are represented in binary. Whatever system is used (i.e. 4-bit, 8-bit, 16-bit etc.), signed number must all have same number of bits. 0s are used to fill up empty bits. We’ll use 8-bit for this tutorial. There are 3 basic standards for notating negative numbers.

1. If you have to subtract small binary number from the large one then use the simple subtraction method as used in decimal number system.
2. While if you have to subtract large binary number from the small one then use the special method ( Complementary Method ) of subtraction.

Division

Division method for the binary numbers is clearly explained in the video below: