 # Binary Multiplication

## Multiplication of Binary Numbers

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 - zero - 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 represents 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 coefficient 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.

### Binary-To-Decimal Conversion

A binary number can be converted into its decimal equivalent by noting that 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.

### How to do Binary Multiplication

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

1. × 0 = 0;

2. × 0 = 0;

3. × 0 = 0; and

4. × 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.

Given below are the binary multiplication examples:

1001.11        (9.75)

×100.1            (4.5)

-----------

1001 11

00000 0

000000

100111

-----------

101011.111    (43.875)

### Fun Facts

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

• 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.

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

• 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.

### Solved Examples

Multiply: (i) 10111 by 1101

Solution:

1 0 1 1 1

1 1 0 1

1 0 1 1 1           ← 1st partial product

1 0 1 1 1

1 1 1 0 0 1 1           ← 1st intermediate sum

1 0 1 1 1

1 0 0 1 0 1 0 1 1           ← Final sum.

The binary multiplication example shown above gives the binary product as 100101011.

(ii) 11011.101 by 101.111

Solution:                                        1 1 0 1 1 . 1 0 1

1 0 1 . 1 1 1

1 1 0 1 1 . 1 0 1

1 1 0 1 1 1 . 0 1          ← 1st partial product

1 0 1 0 0 1 0   1 1 1        ← 1st intermediate sum

1 1 0 1 1 1 0   1

1 1 0 0 0 0 0 1   0 1 1    ← 2nd intermediate sum

1 1 0 1 1 1 0 1

1 1 0 0 1 1 1 1 0   0 1 1        ← 3rd intermediate sum

1 1 0 1 1 1 0 1

1 0 1 0 0 0 1 0 0 1 0   0 1 1

Therefore, the example of binary multiplication gives the product as 10100010.010011.

Question 1: What are the Advantages of a Binary System?

1. implementation. Any system that has either "on" or "off" or "high" or "low" state can be used for encoding and/or manipulating data.

2. Any higher counting system can easily be encoded as the binary system is the lowest "base" possible (base 2).

3. The Binary logic is too easy to understand and can be used to build any type of logic gates (AND, OR, NAND, XOR). It can be used to build higher-order components (counters, multiplexers, adders, etc.) and ultimately the computers and other tech devices that our world now relies upon.

4. Binary data is very robust in transmission as any noise tends to be neither fully "on" nor "off" and thus it is easy to reject.

5. The binary system is the most effective way to communicate with any type of alien civilization. Just as "math" is a type of universal language, binary is a universal alphabet  (any alien civilization can understand a sequence of prime numbers)

6. Binary signals are very unambiguous.