Binary Multiplication

Binary multiplication is similar to decimal multiplication but uses only the digits 0 and 1. It follows simple rules and operates bit by bit using logical operations. Though it may seem tedious, it is highly efficient for computers and widely used in digital electronics and programming.
 

Binary multiplication is based on these basic rules:

0 × 0 = 0

0 × 1 = 0

1 × 0 = 0

1 × 1 = 1

It is inclined to shift and add, just like in long multiplication in the decimal system, making it an integral part of how computers perform arithmetic operations inside them.
 

Multiplication is one of the fundamental arithmetic operations, where a number is repeatedly added to itself. So, it is also known as repeated addition.

For example, to multiply 5 and 3, we can add 5 three times, that is \(5 + 5 + 5 = 15\), so \(5 × 3 = 15\). In our everyday lives, we use multiplication to measure areas, manage money, calculate bills, and so on. It is represented by the symbol ‘×’ or ‘.’. 

Binary multiplication uses simple rules since it has only two digits, i.e., 0 and 1. The basic rules are:

 

Step-by-step explanation:
 

Step 1: Multiply Bit by Bit

To multiply binary numbers, we multiply each bit starting from the rightmost digit of the multiplier. The number written at the bottom is the multiplier, and the number written on the top is the multiplicand. 

For example, 1011 × 110
 

Step 2: Add the Results
Now we add the partial products to find the final product. This is done through binary addition rules, including carrying values when needed.

The product of multiplying 1011 × 110 is 110010, which is 66 in decimal.

Binary multiplication is a step-by-step process of long multiplication in the decimal system. The approach is to multiply every bit of the multiplier by the multiplicand, and then add the products to find the final product.
 

Step-by-Step Process:

1. Arrange the Numbers: Write the multiplicand on top and the multiplier below, aligning digits.
   
    
2. Multiply Each Bit:  If the multiplier bit is 1 → write the multiplicand. If it’s 0 → write zeros.
   
    
3. Shift Left: Shift each new line one place left for each step.
   
    
4. Add Results: Add all shifted values using binary addition rules.
   
    
5. Final Product: The sum gives the binary multiplication result.

Example:

Multiply: 101 × 11

101 in decimal is 5, and 11 in decimal is 3. 

   101 (Multiplicand)
× 011 (Multiplier)
--------
     101 (1 × 101)
+ 1010 (1 × 101, shifted left)
---------
1111 (Final answer)

Result: 101 × 11 = 1111 (5 × 3 = 15 in decimal)

Multiplying binary numbers is a rational, systematic process—not much different from decimal multiplication, but more manageable because it involves only two figures: 0 and 1.

Here's how you can do it:

Step 1: Write the Numbers
Write the multiplicand (the number being multiplied) above and the multiplier below.

Step 2: Multiply Bit by Bit

Multiply bit by bit, starting with the rightmost bit of the multiplier. 

If it is 1, write down the multiplicand.

If it is 0, then write a row of zeros.

Step 3: Shift Left

With each new bit, just like adding zero in decimal multiplication, here we shift the result one place to the left. 

Step 4: Add the Rows
Add all the rows using binary addition. 

Step 5: The Final Result
The sum of all the partial products is the resulting binary product.

Example: 101 × 10
         101
×         10
    ----------
         000           (0 × 101)
+     1010           (1 × 101, shifted left)
   -----------
       1010
 

Thus, 101 × 10 = 1010 (which is 5 × 2 = 10 in decimal).

Binary multiplication plays a vital role in enabling digital technology to be powered. Whether in electronics and computing, networking, or image processing, it enables fast, accurate, and effective operations. In this section, we will see how we use it in our real world.

1. Computers and Processors: Binary multiplication is a core function in the Arithmetic Logic Unit (ALU) of CPUs, powering arithmetic operations, logical decisions, and graphics rendering. Every software calculation, from spreadsheet formulas to 3D game rendering, relies on efficient binary computation.
   
    
2. Data Transmission and Networking: Binary multiplication is essential for encryption, error detection, and IP addressing. Packet checksums, subnet masks, and cryptographic algorithms depend on these operations to ensure the accuracy, speed, and secure data transmission.
   
    
3. Image and Video Processing: In image processing, binary multiplication is used in tasks such as image scaling, filtering, encoding, and video compression (e.g., MP4, H.264). It enables real-time rendering, motion detection, and video analytics while reducing data size without compromising quality.
   
    
4. Robotics and Automation: In robotics, binary multiplication is used in motion control, sensor data processing, and industrial automation. Robots calculate positions, speeds, and trajectories efficiently using binary arithmetic at the hardware level.
   
    
5. Signal Processing and Telecommunications: Digital signal processing (DSP) uses binary multiplication for filtering, modulation, and Fourier transforms. Telecommunications systems rely on these calculations for encoding, decoding, and transmitting data over networks.