Multiplication: Definition, Symbol, Table

Multiplication is a mathematical operation that involves two or more numbers to produce a third number called the product. The numbers that are multiplied are called the factors. Multiplication is a basic math operation, along with addition, subtraction, and division.

Multiplication can also be thought of as scaling quantities. For example, 3 being multiplied by 2 gives 6 as a result.

Let’s learn about the methods, formulas, and examples of multiplication in maths.

Multiplication in Maths

Multiplication is defined as the fundamental operation of mathematics that is used to perform repetitive addition. This is also calculated when we have to find the sum of repetitive numbers, then the multiplication is carried out. Thus, multiplication is also called repeated addition.

This is explained by the example added below,

Simplify: 5 + 5 + 5 + 5 + 5

5 + 5 + 5 + 5 + 5

= 5 × 5

= 25

Thus, repetitive addition is also called multiplication. Thus, we say that multiplication is the equivalent process of additive addition and it is used to quickly solve repetitive addition.

Multiplication Definition

Multiplication is a fundamental arithmetic operation that represents the process of repeatedly adding a number to itself or another number a specified number of times.

Multiplication Symbol

The symbol that is used to find the product of two numbers is called the multiplication symbol. It is represented using a cross and written as (×).

If we have to find the product of two numbers we use the “×” symbol. For example,

For Example: Simplify 12 × 3

12 × 3 = 36

Image added below shows the same,

In above example, we use × to represent multiplication operation. We also use the dot operator (⋅), and asterisk sign (*) sign to represent multiplication.

Parts of Multiplication

There are various parts of multiplication. The various parts of multiplication are,

• Multiplicand
• Multiplier
• Product

Let’s learn about the same in detail.

Multiplicand

In multiplication operation, the first numbers that is multiplied is called the multiplicand. For example in the multiplication of 11 and 3, (11 × 3), 11 is multiplicand.

Multiplier

In multiplication operation, the second numbers that is multiplied is called the multiplier. For example in the multiplication of 11 and 3, (11 × 3), 3 is multiplier.

Product

In multiplication operation, the final result after multiplying multiplicand and multiplier is called product. For example in the multiplication of 11 and 3, (11 × 3 = 33) 33 is product.

Multiplication Formula

Formula that is used to find the multiplication of two numbers is called the multiplication formula. The multiplication formula used in mathematics is,

Multiplicand × Multiplier = Product

We use this multiplication formula to find product of two numbers.

How to Solve Multiplication Problems

To solve multiplication problems we have various methods and some of the methods used to solve multiplication problems are,

• Multiplication without Regrouping
• Multiplication with Regrouping
• Multiplication on Number Line

Now, let’s learn about these in detail.

Multiplication without Regrouping

We can multiply the number without regrouping as it don not have carry. Multiplication is carried by regrouping method by the following methods,

Step 1: Multiply the number with the unit digit of the number.

Step 2: As it does not have any carry write the result of multiplication below unit digit, and repeat the step for all the higher digits such as, tens digit, hundred digits, etc.

Step 3: Write the result of all the multiplication below each digit to get the final result of the multiplication.

The image added below shows the multiplication without regrouping. Here, we multiply 2123 by 3 to get the answer as 6369.

Multiplication with Regrouping

We multiply two number by regrouping numbers using the concept of the carry. This is explained by the example added below.

Example: Multiply 4075 and 4.

Solution:

We multiply 4075 with 4 by using the concept of carry. For this we multiply 4 with 5 that gives 20 in which 2 is carry to the next number(7). Then we multiply 7 with 4 that gives 28 in which add 2 to get 30 here also we have 3 as carry and 0 is written in the product. This process is carried to get the required product. Here, this is explained as,

• 5 × 4 = 20 (2 Carry)
• 7 × 4 = 28 + 2(carry value) = 30 (3 Carry)
• 0 × 4 = 0 + 3(carry value) = 3
• 4 × 4 = 16 (No Carry as Last Number)

Thus, we can multiply two numbers by regrouping them.

Multiplication On the Number Line

A number line is a way to represent numbers on a straight horizontal line such that they can be extended in both directions. On a number line one can easily find the product of two numbers and thus multiplication operation is easily carried on the number line.

On a number line, suppose if we have to find the product of 3 and 4 then we first mark 0 on the number line and then take a jump of 3 four times. The number on which we land at the end is the product of 3 and 4. (here, in this case, the product of 3 and 4 is 12). The same is explained in the image added below,

But for large numbers finding multiplication using a number line is not convenient.

Multiplication Properties

The repetitive process of addition is called Multiplication and Multiplication has various properties. Some of the important properties of multiplication are,

Closure Property of Multiplication

According to the closure property of multiplication,

“Product of two real numbers is always a real number.”

Commutative Property of Multiplication

According to the Commutative property of multiplication,

For any two real numbers p and q, their product is always commutative, i.e. a × b = b × c.

Associative Property of Multiplication

According to Associative property of multiplication,

For three real numbers a, b, and c their product in any order is equal, i.e. (a × b) × c = a × (b × c).

Identity Property of Multiplication

According to the Identity property of multiplication,

For any real number a, a × 1 = a.

In the case of dealing with a real number, we have a number which on multiplying with any real number gives the original number itself.

Inverse Property of Multiplication (Multiplicative Inverse)

For real numbers, we have an inverse element of each element such that the product of the two elements is one. Suppose the product of a and b gives the result as one(1), then a and b are called multiplicative inverse of one another.

According to the Inverse Property of Multiplication,

a × (1/a) = 1

For any real number, its multiplicative number is always its reciprocal

Read more about Properties of Real Numbers

Multiplication Chart

The table that contains the multiplication of the first ten natural numbers with one another is called the multiplication table. This table is used to solve multiplication problems quickly. The multiplication table is added below,

Multiplication Table

Here’s the multiplication table for numbers 1 to 10:

Multiplication Table 1 to 10
Number	x1	x2	x3	x4	x5	x6	x7	x8	x9	x10
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

This table is a basic tool for learning and practicing multiplication, commonly used in elementary mathematics education.

Multiplication of Exponents

Exponents are numbers represented in power form.

To find the multiplication of exponents the formulas used are,

• aⁿ × a^m = a^(n+m)
• aⁿ × bⁿ = (ab)ⁿ

Multiplication of Fractions

Fractions are numbers that are represented in the numerator and denominator format. The fractions are easily multiplied by following the rule. Suppose we have to find the product of two numbers(fractions) then we multiply the numerator of the first fraction with the numerator of the second fraction and the denominator of the first fraction to the denominator of the second fraction individually to get the product of two fractions.

Suppose we have two fractions, a/b and c/d and we have to find the product of these two fractions then multiplication of fraction is achieved as,

(a/b) × (c/d) = (a × c)/(b × d)

For example, find the product of 2/3 and 7/11

2/3 × 7/11

= (2×7)/(3×11)

= 14/33

Multiplication of Integers

Multiplication for two integers is the same as normal multiplication but the sign of the result depends upon the Multiplicand and Multiplier. The following table can be used to find the sign of the product of any two integers.

Multiplication of Integers: Rules
Multiplication Rule	Examples
(-) × (-) = (+)	(-3) × (-5) = 15
(-) × (+) = (-)	(-3) × (5) = -15
(+) × (+) = (+)	3 × 5 = 15
(+) × (-) = (-)	3 × (-5) = -15

Multiplication of Decimals

Decimals can be easily multiplied by following the steps added below,

Step 1: Note the digits after the decimal in both the number of the multiplication.

Step 2: Multiply both the numbers without worrying about decimals.

Step 3: Now put the decimal in the place after counting the same from the right side.

This is explained by the example added below,

Example: Find the product of 2.34 and 1.2

Solution:

For 2.34 × 1.2

Here we have total 3 digits after decimal from both the numbers.

Multiplying the same without worrying about decimal, we get 2808.

Thus, putting decimal at third place from the right,

We get, 2.808

Multiplication of 2 Digits

Multiplication of two digits number can be easily achieved by the following steps:

Step 1: Multiply the ones digit of the second number by the first number and write down the result.

Step 2: Multiply the tens digit of the second number by the first number and write down the result shifted one position to the left.

Step 3: Add the two partial products from steps 1 and 2 to obtain the final result.

Let’s see following illustration, where 23 and 17 are multiplied together to get the result of 391.

Multiplication of 3 Digits

Multiplication of three digits number can be easily achieved using the following steps:

Step 1: Multiply the ones digit of the second number by the first number and write down the result.

Step 2: Multiply the tens digit of the second number by the first number and write down the result shifted one position to the left.

Step 3: Multiply the hundreds digit of the second number by the first number and write down the result shifted two positions to the left.

Step 4: Add the three partial products from steps 1, 2, and 3 to obtain the final result.

Let’s see following illustration, where 234 and 114 are multiplied together to get the result of 26676.

Multiplication Real life Applications

Here are some real-life applications of multiplication.

• Area Calculation: When measuring the area of a rectangular room, you multiply the length by the width (length × width) to find the total square footage.
• Volume Calculation: To determine the volume of a box or container, you multiply the length, width, and height (length × width × height).
• Time and Distance: When calculating speed, you multiply the rate (e.g., miles per hour) by the time (hours) to find the distance travelled.
• Cooking: Recipes often require multiplying ingredients. For example, if a recipe calls for doubling the ingredients, you multiply each quantity by 2.
• Shopping: When buying multiple items with the same price, you multiply the price by the quantity to find the total cost.
• Salary Calculation: Monthly or annual salaries are often calculated by multiplying the hourly wage or the daily rate by the number of hours worked or days worked.

Fast Multiplication Tricks and Tips

Use the following tips and tricks to calculate product of numbers fast:

Finger Multiplication

1. Assign each finger a number from 6 to 10.
2. To multiply, touch the corresponding fingers and count the touches to get the result.

Multiplying by 9

1. For multiplying any number by 9, use the “finger trick”.
2. For example, to multiply 7 by 9, fold down the 7th finger from the left. The result is 63.

Multiplying by 11

1. For two-digit numbers, add the digits together and place the sum in the middle.
2. For example, 23 multiplied by 11 becomes 2 (2+3) 3, so the result is 253.

Distributive Property

1. Break down multiplication into simpler parts.
2. For example, to multiply 6 by 17, you can break it down into (6 x 10) + (6 x 7).

Multiplying by 5

1. To multiply any number by 5, simply divide the number by 2 and add a zero.
2. For example, 8 multiplied by 5 is (8 / 2) with a zero, giving 40.

Squaring Numbers Ending in 5

• To square a number ending in 5, take the digit before 5, multiply it by the next number, and append 25.
• For example, 75 squared is (7 x 8) with 25, resulting in 5625.

Cross Multiplication

1. Useful for multiplying numbers with two digits.
2. Multiply the tens and units separately and add them together.
3. For example, 23 multiplied by 34 is (20 x 30) + (20 x 4) + (3 x 30) + (3 x 4).

Zero Property

1. Any number multiplied by zero is zero.

Using Powers of 10

1. To multiply by powers of 10, add the number of zeros to the original number.
2. For example, 6 multiplied by 100 is 600.

Breaking Down Large Numbers

1. Break down large numbers into more manageable parts for easier multiplication.

Multiplication Word Problems

We have provided you with some solved word problems on multiplication.

Multiplication Problem 1: If a box of pencils has 12 pencils. Then 5 box has how many pencils.

Solution:

No. of pencils in one box = 12

Total box = 5

Total pencils in 5 boxes = 12 × 5 = 60

Thus, 5 box has a total of 60 pencils.

Multiplication Problem 2: A class has 50 students and in school, there are 8 classes with an equal number of students. Now find the total number of students in the school.

Solution:

Number of Students in a Class = 50

Number of classes in the School = 8

Total Number of Students in School = 50 × 8 = 400

Thus, there are a total of 500 students in the school.

Multiplication Problem 3: A farmer has 7 baskets of apples, and each basket contains 5 apples. How many apples does the farmer have in total?

Solution:

Number of apples in a basket = 5

Number of basket = 7

Total Apples = 5 × 7 = 35

Thus, there are a total of 35 apples

Multiplication Practice Questions

Here are a few more solved examples on multiplication.

Multiplication Example 1: Isha wants to make 4 sets of bracelets, and each set requires 6 beads. How many beads does Emma need in total?

Solution:

Number of beads required for one bracelet = 6 beads

Number of bracelet to be made = 4

Total beads required = 6 × 4 = 24 beads

Thus, a total of 24 beads are required to make 4 bracelets.

Multiplication Example 2: A garden has 5 rows of sunflowers, with 9 sunflowers in each row. How many sunflowers are there in the garden?

Solution:

Number of Sunflowers in each row = 9

Number of Rows = 5

Total Number of Sunflower in the Garden = 9 × 5 = 45

Thus, there are a total of 45 sunflowers in the garden.

Multiplication Example 3: Find the product of 232 × 24

Solution:

232 × 24

= 232 × 20 + 232 × 4

= 4640 + 928

= 5568

Multiplication Example 4: Find the product of 123 × 45

Solution:

123 × 45

= 123 × 40 + 123 × 5

= 4920 + 655

= 5575

Practice Problems on Multiplication

Following are some practice problems on Multiplication for you to solve.

Problem 1: Multiply the following:

• 23 × 17 = ?
• 46 × 29 = ?
• 87 × 63 = ?
• 124 × 58 = ?
• 315 × 42 = ?

Problem 2: Sarah bought 5 crates of apples, with each crate containing 24 apples. How many apples did she buy in total?

Problem 3: A library has 8 rows of bookshelves, and each row has 32 bookshelves. How many bookshelves are there in total? Problem 4: A sports store sells soccer balls in packs of 6. If a school buys 9 packs of soccer balls, how many soccer balls do they get?

Problem 5: Emily has a garden with 6 rows of sunflowers, and each row has 7 sunflowers. How many sunflowers are there in her garden?

Multiplication Practice Questions

Answer the following MCQs (Here only one answer is correct)

1. What is the product of 8 multiplied by 9?

1. 72
2. 64
3. 81
4. 17

2. If you multiply 6 by 7, what is the result?

1. 49
2. 42
3. 36
4. 14

3. What is the product of 5 multiplied by 3?

1. 8
2. 15
3. 18
4. 23

4. If you have 4 groups of 6, what is the total?

1. 12
2. 24
3. 30
4. 36

5. Calculate the product of 9 multiplied by 4.

1. 27
2. 36
3. 45
4. 54

6. How much is 7 times 8?

1. 56
2. 64
3. 72
4. 80

7. If you multiply 11 by 3, what is the result?

1. 33
2. 22
3. 44
4. 5

8. What is the product of 2 multiplied by 10?

1. 12
2. 15
3. 20
4. 25

9. If you have 3 groups of 7, what is the total?

1. 10
2. 17
3. 21
4. 24

10. Calculate the product of 12 multiplied by 5.

1. 48
2. 60
3. 72
4. 84

Multiplication in Summary

Here are some key notes on Multiplication in Maths:

1. Definition: Multiplication is a fundamental arithmetic operation.
2. Symbol: Denoted by “X” or “*”.
3. Result: The outcome of multiplying two or more numbers is called the product.
4. Commutative: Order of numbers doesn’t affect the result .
5. Associative: Grouping of numbers doesn’t affect the result
6. Example:

Articles Related to Multiplication:

• Mathematical Concepts for Beginners
• Basic Arithmetic Operations

Multiplication- FAQs

What is Multiplication in Maths?

Multiplication is the fundamental operation of mathematics that is equal to the repetitive addition of any number. Suppose if we have to find the sum of 7 added to itself 5 times then we simply find the product of 7 × 5 and that gives the required answer.

What is the Symbol of Multiplication?

The symbol used to represent multiplication in mathematics is the cross (×) symbol. Sometimes we also use an asterisk (*) or a dot(.) to represent multiplication operations.

What are the terms involved in Multiplication?

In a multiplication equation (a × b), “a” and “b” are called the factors. “a” is the multiplicand, and “b” is the multiplier. The result is called the product.

What are properties of Multiplication?

Various properties of the multiplication operation are,

• Multiplication of any number with zero is always zero.
• Multiplication of any number with one is always the number itself.
• Multiplication of either two positive numbers or two negative numbers is always positive.
• Multiplication of a positive and a negative number is always a negative number, etc.

How is multiplication represented?

Multiplication is often represented using the “×” symbol or an asterisk “*”, as well as in mathematical notation, like “2 × 3” or “2 * 3”, which means multiplying 2 by 3.

What is relation between Multiplication and Addition?

Multiplication is called the process of repetitive addition. Suppose we have to find the sum of a number added to itself multiple times. Then this is easily achieved by taking the product of the number with how many times it is repeated.

How do I Multiply 2 Numbers?

To multiply any 2 numbers, we can take any number and start multiplying its one’s digit and write in the first row, and then with ten’s digit but shifted one place away from the decimal position. This proceeds until all the numbers are used. Then finally add all the rows, to get the desired result.

How do you Multiply a number by 0?

When any number is multiplied by 0, the result is always 0. This is known as the zero property of multiplication.

Can you Multiply Negative Numbers?

Yes, we can multiply negative numbers. When we multiply two negative numbers, the result is positive and when you multiply a positive number by a negative number (or vice versa), the result is negative.

What is the Multiplication Identity?

Multiplication identity is a number multiplied by any number, resulting in the original number and multiplication identity is 1.

How is Multiplication related to Exponential Notation?

Exponential notation is a shorthand for writing repeated multiplications. For example, 2³ means 2 × 2 × 2.