Multiplication of Binary Numbers

Binary multiplication is the process of multiplying binary numbers. The process of multiplying binary numbers is the same as the arithmetic multiplication having the decimal numbers. The only difference which you can see there is that binary multiplication involves numbers that consist of 0s and 1s. But the decimal multiplication contains numbers that comprise digits from 0 to 9. Let us learn more about the method of binary multiplication step by step.

In the binary system, a number is expressed by two digits, 0 and 1. The base of the binary system is thus two. In this system, the individual digits, 0 and 1, represent the coefficients of powers of 2 (two). For example, the decimal number 6 is written as:

6=4+2=1×2²+1×2¹+0×2⁰. The binary representation of the decimal number 6 is thus 110 (read as one-one-zero and not one hundred ten).

Similarly, the binary equivalent of the decimal number 17 is 10001 (read as one - zeo - zero - zero - one).  To avoid any possible confusion with decimal numbers, binary numbers are sometimes embraced by brackets with the base 2 written as a subscript. Thus, the binary number 10101 is written as (10101)₂. The brackets are optional: the number can also be written as 10101₂.

A binary digit (1 or 0) is also known as a bit. A group of bits with a significance is called a bite (or byte), word or, code. Usually, ‘byte’ represents a group of eight bits: the word ‘byte’ is derived from ‘by eight’. For instance, the binary number 10010101 has eight digits or eight bits. The number itself is a byte. A group of four bits makes a nibble. Thus, 1101 is a nibble.

Decimal-To-Binary Conversion

To convert a decimal number into its binary equivalent, the decimal number is expressed as a sum of ascending power of 2. The successive coefficients of the power of 2 represent the number in the binary system. Thus, to convert the number 7 to its binary form, we write

7=4+2+1=1×2²+1×2¹+1×2⁰.

The coefficients of 2², 2¹, and 2⁰ are 1, 1, and 1 respectively. Hence the binary representation of 7 is 111 (one-one-one). An alternative method of converting from the decimal to the binary system is to divide the decimal number progressively by 2 until the quotient is zero. The remainders of the successive divisions, written in reverse order, give the binary number.

The Decimal number system is the number that we use in our daily day-to-day operations. The decimal system consists of 10 digits 0, 1, 2,3,4,5,6,7,8,9 and the binary number mentioned as base 2 number as discussed above.

Steps to Convert

Now that you simply understand the representation bottom of the base 2 number, you can convert from decimal to binary using successive division by 2. There should be the remainder of 0 and 1 and write that to the side of the division problem.

Take the result of the 1st division problem and divide that by 2 again. Like before there should be the remainder of 0 or 1, which you must write to the side of the problem.

Continue this method until the result is 0.

The reminder that you wrote to the side of the division problems is the resulting binary number. The number will be read from the bottom to the top. The least important digit should be at the top and the most important at the bottom.

 For example, convert 25 from base 10 to base 2.

25 ÷ 2 = 12 R¹

12 ÷ 2 = 6 R⁰

6 ÷ 2 = 3 R⁰

3 ÷ 2 = 1 R¹

1 ÷ 2 = 0 R¹

The remainder from the bottom up is 11001.

Binary-To-Decimal Conversion

The conversion from binary to decimal is done to represent a number given in the binary number system to its equivalent in the decimal number system. A number system is extremely essential to represent numbers.

A binary number can be converted into its decimal equivalent by the successive digits from the extreme right of a binary number are coefficients of ascending powers of 2, beginning with the zeroth power of 2 at the extreme right. For example, the binary number 10110 is written as

10110=1×2⁴+0×2³+1×2²+1×2¹+0×2⁰

= 16 + 0 + 4 + 2 + 0 = 22 (twenty two)

Thus the decimal equivalent of the binary number 10110 is 22.

Binary to decimal conversion can be done simply by summing the process of binary digits with their respective weight in the form of

- binary digit × 2 raised to a power of the position of the digit starting with the rightmost digit.

Conversion the binary number into a decimal number.

Solution:

Given binary number =  

Let’s multiply each digit from( MSB to LSB) by reducing the power of the base number 2.

1 × 2³ + 1 × 2² + 0 × 2¹  + 1 × 2⁰

= 8 + 4 + 0 + 1

= 13

Thus, the equal decimal number for the given binary number   is

How to Do Binary Multiplication

Multiplication of binary numbers obeys the following four binary multiplication rules:

1. 0 × 0 = 0;

2. 1 × 0 = 0;

3. 0 × 0 = 0; and

4. 1 × 1 = 1.

Multiplication of binary numbers (two large numbers) consisting of several bits (i.e., digits) is performed in a manner similar to decimal multiplication. According to the binary multiplication rules, the numbers in the bracket give the decimal equivalents of the binary numbers.

Steps to Multiply Binary Numbers

The process of multiplying binary numbers is easier to do than decimal multiplication as binary numbers contain two digits which are either 0 or 1. The method of multiplying binary numbers is given here. These rules are also applicable for binary numbers with a percentage point.

Let us now take an example of multiplying and. Here the decimal equivalent of the first number would be 29 and the decimal equivalent is 9. Now let us multiply these numbers.

Step 1: Write down the multiplicand and therefore the multiplier in the position of one below the other.

Step 2: Now Multiply the rightmost digit that is also called the least significant bit (LSB) of the multiplier (1),  with all of the digits of the multiplicand which is here

Step 3:  Now Add a placeholder of  'X' before multiplying the digit of the next order of the multiplier with the multiplicand.

Step 4: Repeat the equivalent process for all of the digits of the next order until you reach the leftmost digit of the multiplicand that is also called the most significant bit ( MSB).

Step 5: The product obtained in each of the rows will be named as the partial product. Now, in the end, add all of the partial products. To add them use the basic principles of binary addition

Given below are the binary multiplication examples:

   1001.11   

 × 100.1       

-----------

100111

000000

000000

100111

----------------

101011.111

Fun Facts

● In India, the binary system was developed by an Indian scholar Pingala (c. 2nd century BC) for describing prosody.

● The Binary system was actually discovered by a person called Gottfried Leibniz and is used in most modern computers for its easy use.

● Normally there are units, tens, hundreds, and thousands but this is not the same in the binary system. It consists of 1, 2, 4, 8. The unit column doubles itself no matter how big the number is.

● If the last digit of a binary number is 1, then the number is odd but if it’s 0, then the number is even.