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# Binary Multiplication

Binary multiplication is one of the mathematical operations performed on binary numbers or binary digits. It is similar to the process of arithmetic multiplication which is operated upon decimal numbers. However, only the digits 0 and 1 are used in binary multiplication as these are the only components of the binary number system. The procedure of binary multiplication is carried forward step by step by using 0s and 1s only. The solution of multiplication of binary numbers is known as a binary product. The binary product also comprises of digits 0 and 1 only.

## Points to remember while multiplication

1. The product of binary digit 0 with itself is 0.
2. The product of binary digit 1 with itself is 1.
3. The product of binary digit 0 with the binary digit 1 is 0.
4. The product of binary digit 1 with the binary digit 0 is 0.
5. The sum of binary digit 0 with itself is 0.
6. The sum of binary digit 1 with itself is 10.
7. The sum of binary digit 0 with 1 is 1.
8. The sum of binary digit 1 with 0 is 1.

## Procedure of binary multiplication

Step 1: Write the given binary numbers as in the conventional method of multiplication i.e. one below the other. The number on the upper position is known as multiplicand and the number placed below the multiplicand is called the multiplier. For example, for multiplication of the binary numbers (1101)₂ and (1010)₂

(1101)₂

X (1010)₂

Step 2: To begin with multiplication, we consider the corner most digit from the right side. Taking the digit from the extreme right, first, multiply it with the extreme right digit of the multiplicand and proceed in the same way towards the left of the multiplicand.

1101

X 1010

___________

= 0000

Here, as the product of binary digit 0 with 0 and 1 is 0 so place 0s in the first row.

Step 3: Proceeding the same way for the rest of the digits of the multiplicand and multiplier, we get

1101

X1010

___________

= 0000

1101X

0000XX

1101XXX

___________

The product obtained in each row by multiplying a digit of the multiplier with all the digits of the multiplicand is called an intermediate product.

Step 4: To obtain the final product, add up all the numbers obtained till now. Adding all the intermediate products, we get

1101

X1010

___________

= 0000

1101X

0000XX

+1101XXX

___________

Step 5: Before adding all the numbers, always remember to apply the addition rule for binary digits i.e.

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10

For binary addition of 1 with itself, the sum becomes 10 and is written as 0 in the place and 1 carried to the next digit.

1101

X1010

___________

= 0000

1101X

0000XX

+1101XXX

___________

10000010

___________

Hence, the binary multiplication of (1101)₂ and (1010)₂ is (10000010)₂.

The same process is applicable to the multiplication of all kinds of binary numbers.

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