Binary Number System

The binary number system is a base 2 numeral system that uses only two digits: 0 and 1.

It is the foundation of digital computing and electronic systems, as computers process and store data in binary form.

Each binary digit (bit) represents a power of 2, with the rightmost bit being the least significant.

Binary is used in arithmetic operations, data encoding, logic circuits, and memory storage.

Binary numbers are always represented using a 4-bit format. The table given below gives the binary equivalent of base 10 numbers: 

Binary to Decimal Conversion:

 

To convert a binary to a decimal number, multiply each binary digit by 1, 2, 4, 8, and so on from right to left, and then add the answers. You can also use a formula to get the conversion right. The formula used is:


D = (a_n-1 x 2^n-1) +...+ (a₃ x 2³) + (a₂ x 2²) + (a₁ x 2¹) + (a₀ x 2⁰)

For example, convert (10011)2 to a decimal number:

(10011)₂ = (1 x 2⁴) + (0 x 2³) + (0 x 2²) + (1 x 2¹) + (1 x 2⁰)

= 16 + 0 + 0 + 2 + 1 = (19)₁₀

Hence, the binary number (10011)2 is expressed as (19)₁₀.

 

Decimal to Binary Conversion:

 

To convert a decimal number into binary, keep dividing the number by 2 until the quotient becomes 0. We need to ensure we note down all the remainders that we get during this process. When the quotient becomes 0, just write down the remainders (either 1 or 0) upside down to get the final answer.

For example, convert (28)₁₀ into a binary number:

Using the above division method, we find that (28)₁₀ = (11100)₂

Basic mathematical operations like addition, subtraction, and multiplication are also applicable to the binary number system. 

Let us see the operations we use for the binary number system:



Binary Addition:


In binary addition, we add each digit one by one, and carry over when needed. The table below shows the addition of two given numbers.

Binary Subtraction:


The binary numbers are subtracted digit by digit and the answer is obtained. The table below shows the subtraction of two given numbers.

 

Binary Multiplication:


The multiplication of two binary numbers is shown below:

 

1’s complement and 2’s complement are simple methods to represent negative numbers in binary and help with binary subtraction. These methods are used widely in computers and digital devices. Let us now see the steps involved in finding the 1’s complement and 2’s complement .
 

Let’s look at the steps to find 1’s complement:


Step 1: We must write down the binary number for which we need the 1’s complement.


Step 2: Change all 0s to 1 and 1s to 0.

 

This is how we find 1’s complement of a given binary. Let’s understand this better with an example.
Let’s find the 1’s complement of 101010
Now, all we have to do is flip the bits. So, the 1’s complement of 101010 is 010101. As you can see, we have only interchanged the 1’s and 0’s in the binary.  

 

Now, there is some vital information about 1’s complement that we need to know:
 

• In older computers, 1’s complement played and still plays a significant role representing negative numbers.
  
   
• In 1’s complement, zero can be represented in two different ways: 0000 (positive zero) and 1111 (negative zero). This causes confusion and creates a problem. 
  
   
• Due to this drawback, many modern systems use 2’s complement instead of 1’s complement.
   

These are the steps to find the 2’s complement of a binary: 


Step 1: First, find the 1’s complement. 


Step 2: Just add 1 to the 1’s complement. 


Here, let’s find the 2’s complement of 101010 

To do that, let’s first find its complement.

            So, 1’s complement of 101010 is 010101

Now adding 1 to the 1’s complement, we get, 010101 + 1 = 010110
 

Remember these points about 2’s complement:
 

• It is used in modern computers.
  
   
• It has only one representation of zero 
  
   
• The sign is determined by the most significant bit (MSB). If the MSB is 0, it means positive, if it’s 1, then the number is negative.
   

There are numerous real life applications of the binary number system. Some of them are discussed below:

 

• Computer Systems and Digital Electronics:
  
  Without the binary number system, we wouldn’t even have computers. From processors to the logic circuits, almost everything in a computer needs the binary number system to function. 
  
   
• Data Storage and Transmission:
  
  Electronic devices like USB flash drives, hard drives, CDs, SSDs, and DVDs use binary to store and transmit data. Every bit of data is stored as a binary. The number system is also used by digital communication services to transmit data via wired or wireless networks.  
  
   
• Digital Communication Systems:
  
  The modern communication system, which includes mobile phones and satellites, uses binary to transmit data. For e.g., video streams and audio signals are converted into binary before transmitted as digital signals.