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 commonly represented in 4-bit format for small numbers, but larger numbers use more bits. The table given below gives the binary equivalent of base 10 numbers: 

The binary numbers for 16 (10000), 17 (10001), 18 (10010), 19 (10011), and 20 (10100) are written by using 5 bits, since 4 bits can only represent numbers up to 15 (1111).

The methods to convert binary to decimal and vice versa are mentioned in detail below: 

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:

\(D = a_{n-1} \cdot 2^{n-1} + \dots + a_3 \cdot 2^3 + a_2 \cdot 2^2 + a_1 \cdot 2^1+a_0 \cdot 2^0 \)


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

\((10011)_2 = (1 \times 2^4) + (0 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) \)

\( = 16 + 0 + 0 + 2 + 1 = (19)_{10} \)

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.

• Ensure that all remainders are recorded during the division 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:

Convert (28)₁₀ to binary:
 

Divide 28 ÷ 2 = 14, remainder = 0
 

Divide 14 ÷ 2 = 7, remainder = 0
 

Divide 7 ÷ 2 = 3, remainder = 1
 

Divide 3 ÷ 2 = 1, remainder = 1
 

Divide 1 ÷ 2 = 0, remainder = 1

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

Step 1: Align the bits by place value from the right. 
 

Step 2: Add column by column from right to left, including any carry from the previous column. The addition rules are 0 + 0 = 0 (carry 0), 0 + 1 = 1 (carry 0), 1 + 1 = 0 (carry 1), 1 + 1 + 1 = 1 (carry 1). 
 

Step 3: Write the sum bit for that column and pass the carry to the next column on the left. 
 

Step 4: After the left most column, write any final carry as a new leftmost bit. 
 

Given below is a table showing an example of binary addition. 

Binary Subtraction

Step 1: Align the numbers to the right, according to the place value. 
 

Step 2: Subtract each column from right to left and use borrowing if needed. That is 0 - 0 = 0, 1 - 0 = 1, 0 - 1 requires a borrow. 
 

Step 3: When you need to subtract 1 from 0, the rule is, find the nearest 1 to the left, change it to 0 and change all the zeroes in between to 1. Now the current 0 will become 2 and you can subtract. 
 

Step 4: Calculate the difference and continue. 
 

The table below shows the subtraction of two given numbers.

Binary Multiplication
 

Step 1: Write the multiplicand (top) and multiplier (bottom), aligned to the right. 
 

Step 2:  Multiply the multiplicand by each bit of the multiplier (0 or 1). A partial product is either all zeroes (if multiplier bit is 0) or the multiplicand ( if bit is 1). 
 

Step 3: Shift each partial product left by the bit position, just like how decimal multiplication shifts by ten. 
 

Step 4: Add all partial products together using binary addition rules. 

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 widely used 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 1s and 1s to 0s.

This is how we find the 1’s complement of a given binary. Let’s understand this better with an example.

Let’s find the 1’s complement of, 101010
Flipping the bits, the 1’s complement of 101010 is 010101, interchanging all 1’s and 0’s.
 

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 in representing negative numbers.
   
• In 1’s complement, for 4-bit numbers, zero has two representations: 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, assuming a 6-bit representation, adding 1 gives 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.
   

Binary number system are easy to learn, and here are some tips and tricks for students to master the concept of binary number system. 
 

• Understand the place values. Just like the decimal numbers have place values of 10, 100, 1000, etc., binary numbers have place values as powers of 2: 1,2,4,8,16,....
   
• Memorize small binary equivalents. Learn binary equivalents for decimal numbers 0 to 15. This helps in quick conversions and reduces calculation time. 
   
• Practice decimal to binary conversion step-by-step. Use repeated division by 2 method, divide the decimal number by 2, write down the remainder and continue with the quotient until it is 0. The binary number will be the remainders read from bottom to top. 
   
• Use the subtraction method for binary conversion. For converting decimal to binary, subtract the largest power of 2 less than or equal to the number and mark 1 in that place. Repeat for the remainder until it becomes 0. 
   
• Practice binary arithmetic. Students can start with simple addition and subtraction in binary. This helps to provide an understanding of place values and binary logic. 

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 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 example, video streams and audio signals are converted into binary before transmitted as digital signals. 
   
• Communication systems: When you connect your device to the internet or pair your phone with Bluetooth devices, binary numbers are at work. Data transmission protocols, such as TCP/IP, rely on binary encoding to send and receive information. 
   
• Data storage and file management: All digital files, whether it be a photo album, movie or a school project, will be stored as binary data. Formats like JPEG for images, MP4 for videos and DOCX for documents are all encoded in binary.