Long Multiplication

Long multiplication is a method of breaking down large numbers into smaller ones to simplify the multiplication process. It is used to multiply large numbers with two or more digits together to get an accurate product.

For instance, if we want to multiply 50 by 100, we use the long multiplication method. Understanding long multiplication is useful in real life for calculating prices, and solving complex mathematical problems. 
   
 

In the long multiplication method, we multiply large integers to find their products. The column method is the most common, but other types include multiplication with decimals, negative numbers, and the horizontal method. 
 

The column method is almost similar to the long multiplication method. The key difference between both methods is that the long method involves horizontal multiplication and the column method involves vertical multiplication. When we perform the column method, we have to adhere to certain steps. They are: 


Step 1: Based on the place value of each number, arrange the digits in a column format. On top, place the larger number. For example, if we need to multiply 50 × 12. We have to arrange the numbers vertically as shown below: 
   50 
× 12

Step 2: Once the numbers are arranged, start by multiplying the top number by the bottom number in the one's place. For example: 


Step 3: Keep in mind to move from right to left. After finding the result, place it below the two numbers. Next, start multiplying the top number with the tens in the bottom number. Then, write the answer by placing a zero in the ones place or leaving it empty. For instance, now we multiply 50 by 1 at the tens place. Here, we multiply 50 by 10, not just 1.
 

Here 100 =  50 × 2 
500 = 50 × 10 
 

Step 4: After finding the numbers, use the addition method to get the final result. For instance, now we have to add 100 and 500.

Therefore, 50 × 12 = 600. 
 
 

To multiply decimal numbers, we can use the long multiplication method. For example, if we need to multiply 5.1 × 1. 5
When performing long multiplication with decimals, the smaller number should be placed on the right-hand side. Here we follow several steps: 

 

Step 1: Convert the decimal numbers into fractions by removing the decimal. When we convert decimal numbers into fractions, we divide the numbers by 10.
5.1 × 1. 5 = 51/10 × 15/10

Step 2: Multiply the denominators and numerators separately. 
(51×15) / (10×10)

Step 3: Next, we can perform long multiplication on the numerators. Multiply 51×15:
 
So, 51×15 = 765

Step 4: We can convert the 765 to a decimal number. Since we removed two decimal places (one from each number), we must divide the result by 100 to get the correct decimal value.

765 ÷ 100 = 7.65

So, the final answer is 7.65

5.1 × 1. 5 = 7.65
 

Numbers are written side by side in the horizontal method of long multiplication. It is commonly used for single-digit and two-digit numbers. Here are the key steps to follow: 
 

Step 1: Write the numbers horizontally, like In a normal multiplication format. For example, multiply 16 × 22. 
 

Step 2: When we perform the horizontal method, multiply the digit in the ones place of the first number by the other number. For instance, multiply 6 by 22.
22 × 6 = 132   
 

Step 3: Remember, when performing long multiplication, always move from right to left. After multiplying the ones-place digit, move to the tens-place number (1, which represents 10) and multiply it by the other number (22). For example, multiply 22 by 10. 
22 × 10 = 220
 

Step 4: After finding the result, write it in the column format and use the addition method to get the final result. Write the two results, 132 and 220. 
132 + 220 = 352
Therefore, 16 × 22 = 352

When we perform long multiplication with negative numbers, keep in mind the following rules: 
 

• Negative × Positive = Negative
• Negative × Negative = Positive
   

The sign convention is the only difference between long multiplication on negative numbers and positive numbers.

Multiplying a negative number by a positive number, gives a negative number, whereas, multiplying two negative numbers gives a positive number.
 

To multiply large numbers easily we can use the long multiplication method. The real-world significance of this basic mathematical operation is countless. Here are a few real-life applications:  

• Business and sales professionals use long multiplication to calculate their profits, sales, and the prices of bulk items. For example, if a company needs to find the total salary of its thousand employees, it can use this technique to determine the overall wages.   
   
• In the field of architecture and construction, engineers use long multiplication to estimate the required materials for construction. 
   
• The long multiplication method can be used by researchers and astronomers to calculate the distance between comets or other celestial bodies. 
   
• Students and learners can utilize the approaches of long multiplication to solve complex mathematical problems, allowing them to focus on advanced topics like algebra and calculus.