Long Multiplication

Long multiplication is considered a special method of multiplying large numbers that are 2-digits and more. The method to multiply numbers more than 10 is known as the long multiplication method. For this method, knowledge of the multiplication table from 1-10 is needed. In this section, we will learn about long multiplication by understanding the multiplication of large numbers, the column method of multiplication, and how to apply them while solving problems.

Long multiplication is a method of multiplying 2 or more numbers together. Consider that we have to multiply any two numbers greater than 10 or 100, we usually perform long multiplication. The other name for long multiplication is the column method of multiplication as numbers can be multiplied in a column as well. Usually, finding the product of two numbers cannot be simple, that is when we use the long multiplication method.

Let us look at this example, consider 31 × 49. Here, we are multiplying 31 with 49 directly by writing one of these numbers in expanded form i.e. 31 = 30 + 1. 30 is the tenth part and 1 is the unit part. So 31 × 49 can be written as 30 × 49 + 1 × 49. First, we are multiplying 49 with 30 and then we are multiplying 49 with 1, and then we are adding them. So, instead of direct multiplication, we have performed long multiplication, which makes the process simple and accurate.

The column method of multiplication is almost as same as the long multiplication method, the only difference is here in the long multiplication method we perform multiplication horizontally, and in the column method of multiplication, we perform multiplication vertically. Just like the long multiplication, the column method also has a step-wise procedure. They are:

• Step 1: Arrange the numbers in a column format according to their place value. The larger number is usually written on top.
• Step 2: Once arranged, start by multiplying the bottom number in the one's place with the top number.
• Step 3: Always remember to move right to left, hence when the result is obtained arrange it below the two numbers. Start multiplying the ten's number in the bottom number with the top number. Place the result by leaving the one's place empty or placing a zero.
• Step 4: Once the numbers are obtained, using the method of addition to arrive at the final solution.

Let us look at an example for a better understanding. Multiply 52 × 11.

Step 1: Vertically arrange the numbers as shown below.

Step 2: First multiply 52 with 1.

Step 3: Now multiply 52 with the 1 at the tenth place, here we are actually multiplying 52 with 10.

Step 4: Now add 52 and 520.

Therefore, 52 × 11 = 572.

The long multiplication method can be used in decimal numbers as well. Let us look at an example: Multiply 4.1 × 2.7.

While performing the multiplication, keep the smaller number to the right-hand side.

Step 1: Remove the decimal and convert the decimal number into a fraction.

4.1 × 2.7 = 41/10 × 27/10

Step 2: Now keep the 10s in the denominator.

41/10 × 27/10 = (41 × 27) / (10 × 10)

Step 3: Perform the long multiplication on numerators and keep the denominator aside for a while.

Step 4: Now divide the result of the multiplication by the denominator that we kept aside. To derive at a decimal number we convert 1107 by considering the two zeros from the denominator and counting the decimal point from the last number i.e. 7 towards the next number i.e. 0. Hence, the decimal point is placed two numbers after the last number.

(1107) / (10 × 10) = 1107 / 100 = 11.07

Therefore, 4.1 × 2.7 = 11.07

In long multiplication, one of the methods apart from the column method is the horizontal method. This method is mostly used among single-digit numbers and 2-digit numbers. Let us look at the step-wise procedure of solving long multiplication in the horizontal method:

• Step 1: Arrange the numbers horizontally next to each other in the normal multiplication format.
• Step 2: Start by multiplying the first number at the one's place with the other number.
• Step 3: Always move from right to left in long multiplication. Once the first number is done, multiply the number in the ten's place with the other number.
• Step 4: Once the result is obtained and written in a column format, use the method of addition to find the final solution.

Let us use the above example for a better understanding of multiplying 2-digits. Multiply 31 × 49

Step 1: Arrange the numbers horizontally and begin with multiplying 49 with 1.

Step 2: Now, multiply 49 with 3, and put a cross sign just below 9 (the unit place of the number 49), this cross sign represents a 0.

Step 3: Write zeros before the number 49 so that it will cover till the number 1 of 147, write the numbers just below another number so that the addition becomes easy.

Step 4: Add these two numbers 0049 and 1470.

Hence, 31 × 49 = 1519. In long multiplication, the cross represents a zero. Also, while multiplying numbers up to 3-digits, we have to add two crosses while we multiply to a hundred place number.

The long multiplication on negative numbers follows the same rule as positive numbers, the only difference is the sign convention. We have to remember the sign conventions while multiplying the numbers. When a positive number is multiplied by a negative number, the solution is a negative number. Whereas, when two negative numbers are multiplied with each other, the solution is a positive number.

What is Meant by Long Multiplication?

Long multiplication is a method of multiplication used for multiplying numbers up to 2-digits and more. Long multiplication can be done in two ways - numbers written in a horizontal manner and numbers written in a column manner. Large numbers with more than 3-digits are multiplied using the long multiplication method. Multiplication of two numbers is also called the product of two numbers. Long multiplication can also be known as the column method of multiplication.

What are the Steps for Long Multiplication Horizontal Method?

There 4 steps required to perform long multiplication on large numbers, they are:

• Arrange the numbers in a horizontal manner.
• Start by multiplying the first number at the one's place with the other number.
• Always move from right to left in long multiplication. Once the first number is completed, multiply the number in the ten's place with the other number.
• Once the result is obtained and written in a column format, use the method of addition to find the final solution.

What are the Steps for Long Multiplication Column Method?

The steps for column-wise long multiplication are very similar to the normal long multiplication. The column method is done vertically and the 4 steps are:

• Arrange the numbers in a column format according to their place value.
• Start by multiplying the bottom number in the one's place with the top number.
• Once the result is obtained arrange it below the two numbers. Start multiplying the ten's number in the bottom number with the top number.
• Once the numbers are obtained, using the method of addition to arrive at the final solution.

Can Long Multiplication be Used for Decimal Numbers?

Yes, the long multiplication method can be used for decimal numbers as well. While multiplying decimal numbers, always keep the smaller number to the right-hand side. For decimal numbers, both the horizontal and the vertical long multiplication methods can be used.

Can Long Multiplication be Used for Negative Numbers?

Yes, the long multiplication method can be used for negative numbers with the help of a few rules such as:

• A positive number multiplied with a positive number will result in a positive number.
• A negative number multiplied with a positive number or vice versa will result in a negative number.
• A negative number multiplied by a negative number will result in a positive number.

How to Solve Long Multiplication Problems?

Long multiplication problems can be solved using two methods - the long multiplication method and the column method. The long multiplication method requires the numbers to be written horizontally whereas the column method requires the numbers to be written vertically. Both methods help in solving problems with large numbers.

How to do Long Multiplication Easily?

The long multiplication method can be performed easily and in a quick manner if we keep a few things in mind. For multiplication, it is always better to relate to addition as it is used to arrive at the final solution. Always remember the multiplication table from the numbers 1-10. For long multiplication, it is always helpful in breaking down the problem into steps for better understanding and arriving at the result faster.