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Long Multiplication? Definition, Methods, Steps, Examples, Facts

What Is Long Multiplication?

Long multiplication is a method of multiplying large numbers easily. It simplifies the process of multiplication of two numbers that are not otherwise easy to multiply.

For example, we can easily find the product of $55 \times 20$ by multiplying 55 by 2 and then adding a 0 at the rightmost place of the answer.

$55 \times 2 = 110$ and $55 \times 20 = 1100$

However, sometimes, finding the product is not this easy. In such cases, we use the method of long multiplication. It is often termed as the long hand multiplication (long multiplication by hand).

Long Multiplication: Definition

Long multiplication is a method of multiplication using which multiplying two large numbers that have two or more digits becomes easier. 

For example, we can easily find the product of $55 \times 20$ by multiplying 55 by 2 and then adding a 0 at the rightmost place of the answer.

$55 \times 2 = 110$ and $55 \times 20 = 1100$

But how to multiply large numbers? Many times, finding the product is not this easy. In such cases, we use the long method of multiplication. 

For example: $47 \times 63 =$ ?

We can find the answer in simple steps using long multiplication. Let’s discuss the method in detail in the next section. 

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How to Multiply Using Long Multiplication

Let’s discuss the steps for multiplying a two-digit number by a two-digit number using the long multiplication method.

Let us multiply 47 by 63 using the long multiplication method.

Step 1: First, write the two numbers, one below the other, such that their place values are aligned. We generally write the bigger number on top and a multiplication sign on the left and draw a line below the numbers as shown below.

Important Note: 

  • In the long multiplication method, the number on the top (63) is called the multiplicand. The number by which it is multiplied, that is, the number at the bottom (43), is called the multiplier.
  • If you write the small number on the top and perform the multiplication, the answer will still be the same since the order does not matter while multiplying two numbers.

Step 2: Multiply the ones digit of the top number by the ones digit of the bottom number.

Write the product as shown in the image. Don’t forget the carryover that goes in the next place!

Step 3: Multiply the tens digit of the top number by the ones digit of the bottom number. Add the carryover.

This is our first partial product which we got by multiplying the top number by the ones digit of the bottom number.

Step 4: Now, we place a 0 below the ones digit as shown. This is because we will now be multiplying the digits of the top number by the tens digit of the bottom number. 

Step 5: Multiply the ones digit of the top number by the tens digit of the bottom number.

Step 6: Multiply the tens digit of the top number by the tens digit of the bottom number.

This is the second partial product obtained by multiplying the top number by the tens digit of the bottom number.

Step 7: Now, add the two partial products.

Let’s summarize.

We follow the above long multiplication steps only for multiplying numbers greater than two digits.

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Long Multiplication Column Method

Long multiplication is also known as the column method of multiplication since we perform the multiplication vertically or column wise.

Let’s take another example to understand this better. 

Multiply 321 by 23.

Take a look at the long multiplication chart that shows the long multiplication step by step. You can refer to it to avoid mistakes when solving long multiplication problems. 

Multiplying Decimals Using Long Multiplication

Let’s understand how to multiply decimals with the help of the long multiplication method.

Example: Multiply $3.6 \times 5.5$

Step 1: First, we place the smaller number out of the two on the right-hand side and change the decimal number to a fraction.

$5.5 \times 3.6 = \frac{55}{10} \times \frac{36}{10}$

Step 2: Then, we multiply the numerators using the steps of the long multiplication method. We leave the denominator as it is for now.

Step 3: Now, we divide the answer (that we got in step 2) by the denominator to get the final answer. 

$\frac{1980}{100} = 19.80$

Long Multiplication Using the Horizontal Method

Let’s now understand how to use the horizontal method with the help of a long multiplication example. As the name suggests, we multiply the numbers horizontally but follow the same steps as we discussed earlier.

Example: Multiply 48 by 24.

Step 1: First, we write the numbers beside each other and place them horizontally.

$48 \times 24$

Step 2: Now, we multiply the second number (number on the right side) with the ones digit of the first number.

Here, we multiply 24 with 8, since 8 is at the ones place in 48.

$24 \times 8 = 192$

Step 3: Now, we multiply the second number with the tens digit of the first number.

Here, we multiply 24 with 40, since 4 is at the tens place in 48.

$24 \times 40 = 960$

Step 4: Finally, we add the partial products obtained from steps 2 and 3 to get the final answer.

Thus, $48 \times 24 = 1152$

Long Multiplication with Negative Numbers

The multiplication of negative numbers using the method of long multiplication follows the same steps as mentioned above. However, we have to carefully consider the signs of the two numbers in order to decide the sign of the product.

For instance, when we multiply two negative numbers with each other, the sign of the result remains positive. On the other hand, it becomes negative when a negative number and a positive number are multiplied.

The figure below shows how the sign changes upon multiplication with negative numbers:

Fun Facts about Long Multiplication

  • Long multiplication is also known as the column method of multiplication.
  • Whenever any number is multiplied by zero, the answer is also zero.
  • When the number 9 is multiplied by any other number, the sum of digits in the product is always 9.

For example, $9 \times 25 = 225$ and $2 + 2 + 5 = 9$; 

                               $9 \times 9 = 81$ and $8 + 1 = 9$.

  • Long multiplication can also be referred to as repeated addition because multiplying any number is just an alternative to adding it repeatedly.
  • Whenever any even number from 0 to 9 is multiplied by 6, the product has the same even number in the ones digit. 

For example, $6 \times 4 = 24,\;  6 \times 2 = 12$

  • Whenever two large numbers (let’s say three-digit or four-digit numbers) are multiplied, the process of long multiplication remains the same. For example:

Conclusion

Long multiplication is a method that simplifies the multiplication of large numbers, including two-digit, three-digit numbers, etc., with each other. Let’s solve a few examples and practice problems based on long multiplication!

Solved Examples on Long Multiplication

1. Multiply 38 by 91 using the column method.


Solution: 

2. Multiply 72 by 44 using the column method.


Solution

3. Multiply 58 by 30.

Solution: 

The sign of the product when we calculate $(\;-\;58) \times (30)$ will be negative.

Since 58 has a negative sign, the final answer will be $1740$.

$(\;-\;58) \times (30) = \;-\;1740$

4. Alia bought 42 notebooks, each worth $\$9. 6$. How much did she spend?

Solution:

To find the amount Alia spent, multiply 9.6 by 42.

$9.6 \times 42 = \frac{96}{10} \times 42$

Now, we divide 4032 by 10 to get the final answer, which is $403.2$.

$9.6 \times 42 = 403.2$

Thus, Alia spent $403.2 on notebooks.

5. Multiply $-70$ and $-15$.

Solution: 

Let’s find $(70) \times (15)$. 

Since both the numbers are negative, the final answer will remain positive, $1050$.

$(70) \times (15) = 1050$

Practice Problems on Long Multiplication

1

Multiply 95 by 13 using the column method.

1125

1235

1545

1425

Correct answer is: 1235
$95\times13 = 1235$

2

Multiply the decimals 1.2 and 6.7.

9.05

13.5

8.04

6.08

Correct answer is: 8.04
$6.7\times1.2=\frac{67}{10}\times\frac{12}{10}$
First, we multiply the numerators. Then, we divide their product with the denominators to reach the final answer.

3

Multiply the negative numbers $(-\;89)$ and $(-\;61)$.

5429

3489

5299

4529

Correct answer is: 5429
Since both the numbers are negative, the final answer will remain positive.
$(-\;89)\times(-\;61) = 5429$.

4

Multiply 45 by 21 using the horizontal method.

645

745

845

945

Correct answer is: 945
$21\times5=105$ __(i)
$21\times4=84$ __(ii)
Adding (i) and (ii), we get
$105\;+\;84=945$.

5

Multiply 530 by 22.

12600

13550

11660

14650

Correct answer is: 11660
$530\times22 = 11660$

Frequently Asked Questions on Long Multiplication

What are negative numbers?

Negative numbers are those whose value is less than zero and thus have a negative sign (-) before them.

What are positive numbers?

Positive numbers are those whose value is greater than zero.

What is a decimal?

A decimal number represents the number that separates a whole number from a fractional number. It is represented by a dot (.) sign.

What is the meaning of the horizontal method of multiplication?

The horizontal method refers to a method wherein numbers are arranged in a horizontal line from left to right.

What is the meaning of the column method of multiplication?

The column method refers to a method wherein numbers are placed one below the other from top to bottom.

What is short multiplication?

Short multiplication is the method of multiplication commonly used when multiplying a three-digit or larger number with a single-digit number.

Multiplication chart

A multiplication chart, also known as a multiplication table, or a times table, is a table that can be used as a reference for the 100 multiplication facts. It can be helpful for learning and memorizing multiplication facts, which is essential, since multiplication is used throughout all areas of mathematics in some form or another. Being familiar with all the multiplication facts enables a person to focus on more complex mathematical concepts that involve multiplication, without having to worry about the actual multiplication.

× 1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10
2 2 4 6 8 10 12 14 16 18 20
3 3 6 9 12 15 18 21 24 27 30
4 4 8 12 16 20 24 28 32 36 40
5 5 10 15 20 25 30 35 40 45 50
6 6 12 18 24 30 36 42 48 54 60
7 7 14 21 28 35 42 49 56 63 70
8 8 16 24 32 40 48 56 64 72 80
9 9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100

How to use a multiplication chart

To use a multiplication chart, first look at the rows and columns in grey in the figure above. Rows are read horizontally from left to right while columns are read vertically from top to bottom. The green diagonal on the chart represents the squares of the numbers, namely 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, etc.

Use the green diagonal to understand how the multiplication chart is read. The value in the green diagonal is the product of the column and row values that align with it in grey. For example, 25 is the product of 5 and 5. 20 is the product of 4 and 5 and 5 and 4, etc.

In other words, to use the multiplication chart, choose the two values that you want to multiply from the grey row and column, then determine what value an imaginary horizontal and vertical line (drawn from the grey column and row respectively) would intersect at to determine the product of the two values.

Although 100 facts may seem like a large number to memorize when just starting to learn multiplication, the number of facts that need to be memorized can be reduced by using certain properties of multiplication.

Commutative property of multiplication

The commutative property of multiplication states that the order of multiplication doesn’t matter. Given two numbers, a and b:

a × b = b × a

We can confirm this by looking at the multiplication chart and seeing that regardless of whether we look at the multiplication fact 2 × 8 = 16 or 8 × 2 = 16, the solution is still 16. This is true for anything being multiplied. Since the order doesn’t matter, we only really need to memorize numbers below or above (and including) the diagonal shown in green on the chart. This property almost halves the number of multiplication facts we need to memorize.

Identity property of multiplication

The identity property of multiplication states that any number a multiplied by 1, is equal to a:

1 × a = a

Since 1 multiplied by any number is that number, as long as we know this property, it is not necessary to memorize the first row or column of the multiplication chart.

Multiplication by 10

Due to the nature of the decimal numeral system, multiplying any integer by 10 results in that same integer with a 0 added to the end. For example, 2 × 10 = 20, 20 × 10 = 200, 200 × 10 = 2000, and so on. It doesn’t matter what the number is, multiplication by 10 results in a shift of the decimal place one position to the right. Since integers have no decimal values, this just means adding a 0 (For decimals, the decimal point is moved to the right one place). This allows us to remove the last row and column of the multiplication chart, leaving the following 36 multiplication facts that need to be memorized.

× 1 2 3 4 5 6 7 8 9 10
1                    
2 4                
3 6 9              
4 8 12 16            
5 10 15 20 25          
6 12 18 24 30 36        
7 14 21 28 35 42 49      
8 16 24 32 40 48 56 64    
9 18 27 36 45 54 63 72 81  
10                    

Read online “Multiplication table in 3 days.

How to memorize the multiplication table in 3 days without cramming it”, Stanislav Baranov — Litres

© Stanislav Baranov, 2021

ISBN 978-5-4483-3755-0

Created in the intellectual publishing system Ridero

9 0008 Introduction

The purpose of this book is to teach children the multiplication table in 3 days without tormenting them by cramming. The book is written both for parents who want their children to know the multiplication table, but do not memorize it, and for children who can already read and learn the multiplication table on their own.

In 3 conditional days after reading the book, the child will know the multiplication table and at the same time will acquire the initial skills of multiplying by single digits.

From the experience of our education 1 children learn the multiplication table depending on their abilities on average from 6 months to 1 year. The presented technique allows you to reduce the learning of the multiplication table in 3 days, start learning on Friday evening and finish on Sunday evening. At the same time, there are no psychological traumas: “Mathematics is a very difficult subject — mathematics is not for me.”

In my practice, I met different children — both with a mathematical mindset (these learned the multiplication table according to the method in 1 day), and «lyricists» — they joyfully learned that mathematics is a pleasure and learned the table in three days. There are only 10 examples to memorize in the technique.

I do not attribute the authorship of this method to myself — I just wrote down what is already known in printed form.

What the multiplication table looks like and how to use it

Never use a multiplication table that looks like many lines. Nothing in it is clear to the child and only scares him away: he will subconsciously seek to postpone memorization for another period.

Figure 1. Wrong multiplication table — don’t use one

«Which table should I use?» the reader will ask. The multiplication table should be used in this form, the so-called Pythagorean table:

The author has slightly colored the table to further emphasize its simplicity and genius.

Figure 3. Colorized Pythagorean table.

How to use this table? To find out the result of multiplying two numbers, you need to find the intersection cell of the row of the first number with the column of the second number (you can also do the opposite, since the result will not change).

Figure 4. The result of multiplication 5*6=30

09

Multiply by 1

Figure 5. Multiply by 1.

Rule.

The product of any number by 1 will equal the same number.

In order to multiply by 1, you just need to name the number that was multiplied by 1.

in as examples for solving give large numbers.

Thus, multiplication by 1 does not need to be taught. Therefore, from our table of 10 by 10 (it will be 100), we need to subtract 19 examples of multiplication by 1. In the table, the works that “learned” are highlighted in green. It remains to learn 81 examples (in the table they are highlighted in black).

Figure 7. Multiplication table after learning to multiply by 1.

1. education — I call it «cutting» — it cuts off children’s creativity and makes them cogs in our system

experts told how to learn the multiplication table

What should you do if your child brings home a triple on the test and “floats” in the world of multipliers?

November, gray sky, dreary, and at school they demand to know the multiplication table by heart. PD learned how to help an elementary school student learn it without discouragement from math and turning the process into meaningless cramming.

The multiplication table is the foundation of the basics, without it, further study of mathematics threatens to turn into a nightmare. But many children experience great difficulties in mastering the irreplaceable brainchild of Pythagoras. What to do if your child brings home a three for the control and “floats” in the world of multipliers?

As children’s clinical psychologist Viktoria Korpacheva told PD, the biggest mistake parents make when studying the multiplication table is the requirement to memorize the entire table in the shortest possible time. However, with this task, haste will not lead to anything good. As a result, both the child and the parents will be nervous. Moreover, the mechanical memorization of the results will not help the child on the math test.

To understand the principle of the multiplication table is the main condition. A visual table of Pythagoras will help with this, print it out and hang it in a conspicuous place. The child will see the result of the intersection of two numbers. Thus, through visual-figurative thinking, causal relationships will appear.

Teach in parts, take your time. For example, take three days to multiply 1, 2 and 3, then three days to multiply 4 and 5, in the second week go to multiply by 6, 7, 8, 9, and getting close to ten, the student will quickly and easily understand what the essence of . This is an important principle of memorization sequence.

Game methods will help to cope with the table. There is nothing complicated here. “Write the results of the addition on the pieces of paper. The child takes a number from the pile and must give an example of which numbers need to be multiplied to get such a result. Through gaming activity, cognitive motivation will appear, ”advises Victoria Korpacheva.

To help your mechanical memory remember the examples from the table, come up with funny and interesting associations for the multiplication answers together with your child. For example: «Five by five — twenty-five, soon I’ll go for a walk.» Thus, associative thinking will stimulate memorization.

Don’t be afraid to use materials at hand: counting sticks, toys, candy, pencils, cars or car wheels — anything your child has fun playing with. Create a mathematical fairy tale story with these items.

It is important to always praise the child and encourage the results. Even if he memorized a small part of the table, but did it well, be sure to admire the child’s work. It is possible at certain intervals, for example, 3-4 columns of the table per week, to reward with a trip to the cinema or to the zoo.

By alexxlab

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