Multiplication is one of the basic arithmetic operations in mathematics. Multiplication is the process of finding the product of any two numbers. In another way, we can say that multiplication is the process of repeated addition of a number with respect to the other number. Multiplication of \(p\) and \(q\) (\(p\) is multiplied to \(q\)) means \(p\) is added to itself \(q\) number of times or \(q\) is added to \(p\) number of times.

In this article, we will discuss about multiplication, its definition, concepts, facts, and properties. Continue reading to find out more about multiplication!

Learn the Concept of Multiplication and Division

What is Multiplication?

Multiplication is the fundamental arithmetic operation used in mathematics. Multiplication is the simple operation used to multiply or find the product of any numbers. The number obtained after the multiplication is called the “product”.

In another way, we can say that multiplication is the process of repeated addition of a number with respect to the other number. Multiplication is the process of combining groups of equal sizes to get the product.

Example: Ramu has three ice-creams with him, and his father has given him three more ice-creams. Now, the total number of ice-creams can be obtained by using the multiplication: \(2 \times 3 = 6.\)
Here, the total number of ice-creams can be found by using addition also, i.e. \(3 + 3 = 6.\) Because of this reason, we can say that multiplication is the successive (repeated) addition of the number.

Multiplication of \(p\) and \(q\) (\(p\) is multiplied to \(q\)) means \(p\) is added to itself \(q\) number of times or \(q\) is added to \(p\) number of times.

Example: Multiplication of \(2\) and \(3\) is nothing but adding \(2\) to itself three times or adding \(3\) to itself two times: \(2 \times 3 = 2 + 2 + 2 = 3 + 3\)

Multiplication Symbol

Multiplication is the simple operation used to multiply or find the product of any numbers. The number obtained after the multiplication is called the “product”. In mathematics, we use different types of symbols of various operations. Similarly, the multiplication symbol is the most widely used symbol in mathematics, which we will use in multiplication.

Multiplication is represented by using the symbols cross-sign \(\left( \times \right),\) an asterisk \(\left( * \right)\) and dot operator \(\left( . \right)\) between the numbers.
Example: \(5 \times 5 = 25\)

Multiplication Formula

Multiplication is the simple operation used to multiply or find the product of any numbers. Multiplication is the process of combining groups of equal sizes to get the product. The multiplication formula is given below:
\({\rm{Multiplier}} \times {\rm{Multiplicand}} = {\rm{Product}}\)
Here,
Multiplier – The number of equal groups to be multiplied.
Multiplicand – The number of items in each equal group.
Product – The result obtained after multiplication.
Example:
\(15 \times 2 = 30;\,15\) is the multiplicand, \(2\) is the multiplier, and \(30\) is the product.

Properties of Multiplication

Multiplication is the simple operation used to multiply or find the product of any numbers. The properties associated with the multiplication are given below:

Commutative Property

The commutative property of two integers, \(A\) and \(B,\) is given by \(A \times B = B \times A.\)
Example: \(2 \times 3 = 3 \times 2 = 6\)

Associative Property

The associative property of three integers \(A,\,B\) and \(C\) is given by
\(A \times \left( {B \times C} \right) = \left( {A \times B} \right) \times C\)
Example: \(2 \times \left( {3 \times 4} \right) = \left( {2 \times 3} \right) \times 4 = 24\)

Distributive Property

The distributive property of the multiplication on addition is given by
\(A \times \left( {B + C} \right) = \left( {A \times B} \right) + \left( {A \times C} \right)\)
Example: \(2 \times \left( {4 + 5} \right) = \left( {2 \times 4} \right) + \left( {2 \times 5} \right) = 18\)

Identity Property

If we multiply any number with one gives the same number as the product, and one is called the multiplicative identity.
\(a \times 1 = a\)
Example: \(2 \times 1 = 2\)

Zero Property

If we multiply any number with zero, we will get a zero as the product.
\(a \times 0 = 0\)
Example: \(5 \times 0 = 0\)

Solving Multiplication Problems

One-digit numbers can be multiplied in a simple way with the help of multiplication tables. But multiplying the larger numbers is difficult. To multiply the larger numbers, divide the numbers into columns and multiply according to their place values like ones, tens, hundreds etc.
The two types of multiplication problems are
1. Multiplication without grouping
2. Multiplication with grouping

Multiplication Without Regrouping

This involves the multiplication with the smaller numbers, which gives a single-digit product. In this process, there is no need to carry over to the next number in a multiplication. Let us understand this process by taking a simple example: \(2022 \times 4\)

1. Multiply the one’s place digit \(\left( 2 \right)\) with \(4,\,2 \times 4 = 8\)
2. Next, multiply the ten’s place digit \(\left( 2 \right)\) with \(4,\) gives \(2 \times 4 = 8\)
3. Multiply the hundred’s place digit \(\left( 0 \right)\) with \(4,\) gives \(0 \times 4 = 0\)
4. Multiply the thousand’s place digit \(\left( 2 \right)\) with \(4,\) gives \(2 \times 4 = 8\)
Therefore, \(2022 \times 4 = 8088.\)

Multiplication With Regrouping

This type of multiplication involves the numbers with two-digit products. In this process, there is a carry over taken forward to the next number. Let us understand this process by taking a simple example: \(2468 \times 8\)

1. Multiply the one’s place digit \(\left( 8 \right)\) with \(8,\) gives \(8 \times 8 = 64.\) Keeping the \(4\) at one’s place and taking the carry \(6\) to the ten’s place.
2. Multiply the ten’s place digit \(\left( 6 \right)\) with \(8,\) gives \(6 \times 8 = 48.\) Add the carry to this product \(48 + 6 = 54.\) Keeping the \(4\) at ten’s place and taking carry \(\left( 5 \right)\) to hundred’s place.
3. Multiply the hundred’s place digit \(\left( 4 \right)\) with \(8,\) gives \(4 \times 8 = 32.\) Add the carry \(\left( 5 \right)\) to this, \(32 + 5 = 37.\) Keeping the \(7\) at hundred’s place and taking the carry \(\left( 3 \right)\) to the thousand’s place.
4. Multiply the thousand’s place digit \(\left( 2 \right)\) with \(8,\) gives \(2 \times 8 = 16.\) Add the carry \(\left( 3 \right)\) to this product \(16 + 3 = 19.\)
Therefore, \(2468 \times 8 = 19744.\)

Multiplication Using Number Line

Multiplication is the process of repeated addition of a number with respect to the other number. The number line is the best way of teaching the multiplication of numbers to the primary classes. Multiplication using the number line can be done by jumping the number starting from zero. Let us explore the above process by using the simple example: \(6 \times 3 = 18.\)

Here, we need to multiply \(6\) with \(3,\) which can be done by jumping on the number line from zero three times. Each jump equals \(6,\) that is \(0\) to \(6,\,6\) to \(12\) and \(12\) to \(18.\)
Thus, the final position on the number line gives the product of \(6 \times 3 = 18.\)

Multiplication Signs

When two numbers are multiplied, the sign of the product also changes depending on the multiplier and multiplicand sign.

Sign of multiplier	Sign of multiplicand	Sign of the product
\( + \,{\rm{ve}}\)	\( – \,{\rm{ve}}\)	\( – \,{\rm{ve}}\)
\( + \,{\rm{ve}}\)	\( + \,{\rm{ve}}\)	\( + \,{\rm{ve}}\)
\( – \,{\rm{ve}}\)	\( + \,{\rm{ve}}\)	\( – \,{\rm{ve}}\)
\( – \,{\rm{ve}}\)	\( – \,{\rm{ve}}\)	\( + \,{\rm{ve}}\)

Multiplication of Fractions

Two or more fractions are multiplied by multiplying the numerators and the denominators together separately.
\(\frac{a}{b} \times \frac{c}{d} = \frac{{a \times c}}{{b \times d}}\)
Example: \(\frac{2}{3} \times \frac{4}{5} = \frac{{2 \times 4}}{{3 \times 5}} = \frac{8}{{15}}\)

Multiplication of Decimal Numbers

Multiplying the two or more decimal numbers is the same as the multiplication of integers. Follow the below steps to multiply the decimal numbers.
1. First, multiply the given decimals removing the decimal point.
2. Now, add the decimal places of the given numbers.
3. Place the decimal point in the product after total decimal places obtained in the above step from right to left.

Example: \(2.3 \times 1.2\)
1. Here, multiplying the numbers removing the decimal places.
(a) \(23 \times 12 = 276\)
2. The Sum of decimals places in the given numbers are \(1 + 1 = 2\)
3. Place the decimal point after two places from right to left in the product.
Therefore, \(2.3 \times 1.2 = 2.76\)

Matrix Multiplication

Matrix is the rectangular arrangement of elements in the number of rows and columns. Two matrices multiplied of any order, but only one condition that the number of columns in the first matrix equals the number of rows in the second matrix.
\({a_{m \times n}} \times {b_{n \times p}} = a{b_{m \times p}}\)

Solved Examples on Multiplication

Q.1. Solve the given word problem of multiplication: What are \(4\) times \(784?\)
Ans: We know that ”times” means multiplication.
So, we need to find \(784 \times 4\)
1. Multiply the one’s digit \(4\) of \(784\) with \(4,\) gives the product \(4 \times 4 = 16.\)
2. Keeping the \(6\) at one’s place of the product and taking the number \(1\) as carrying to the ten’s place.
3. Multiply the ten’s digit \(8\) of \(784\) with \(4,\) gives the product \(8 \times 4 = 32.\)
4. Add the carry \(1\) to the above product gives \(32 + 1 = 33.\)
5. Keeping the \(3\) at ten’s place of the product and taking the number \(3\) as carrying to the Hundred’s place.
6. Multiply the hundred’s digit \(7\) with \(4,\) gives the product \(7 \times 4 = 28.\)
7. Add carry \(3\) to the above product gives \(29 + 3 = 31.\)
Therefore, the product obtained is \(3136.\)
Hence, the value of \(4\) times \(784\) is \(3136.\)

Q.2. Cat drinks around \(15\) litres of milk in a week. How much milk will it consume in three weeks?
Ans: Given a cat drinks \(15\) litres of milk in a week.
The quantity of milk consumed by a cat in three weeks was obtained by multiplying the total weeks by the quantity of milk consumed each week.
\( = 15 \times 3\)
\( = 45\)
Therefore, a cat consumes \(45\) litres of milk in three weeks.

Q.3. Find the value of \(\frac{3}{7} \times \frac{5}{2}.\)
Ans: To multiply the fractions, multiply their numerators and their denominator separately.
\( \Rightarrow \frac{3}{7} \times \frac{5}{2} = \frac{{3 \times 5}}{{7 \times 2}}\)
\( \Rightarrow \frac{3}{7} \times \frac{5}{2} = \frac{{15}}{{14}}\)
Hence the value of \(\frac{3}{7} \times \frac{5}{2} = \frac{{15}}{{14}}.\)

Q.4. Puppy went to the market to buy oranges, and each bag contains \(4\) oranges. She bought \(5\) such types of bags. How many oranges total she bought?
Ans: Given each bag contains four oranges.

In total, she bought five such bags. So, the total number of oranges she bought is calculated by multiplying a total bag by the number of oranges in each bag.
\( = 5 \times 4 = 20\)
Therefore, Puppy bought a total of \(20\) oranges from the market.

Q.5. The cost of one chewing gum is \({\rm{Rs}}.\,1.50.\) Then find the cost of \(10\) chewing gums.
Ans:
Given the cost of one chewing gum is \({\rm{Rs}}.\,1.50.\)
Then the total cost of \(10\) chewing gums is given by \({\rm{10}} \times {\rm{Rs}}.\,1.50. = {\rm{Rs}}.\,15\)
Hence, the cost of ten chewing gums is \({\rm{Rs}}.\,15.\)

Summary

In this article, we have discussed the definition of multiplication, which is the successive addition of one number with respect to the other number. We have discussed the symbols used for multiplication in mathematics. We have studied the properties of the multiplication such as commutative, associative, identity and the zero property.

We also studied the multiplication formula along with the parts of multiplication. Here, we have discussed how to multiply the numbers with regrouping and without regrouping. We also discussed how to do multiplication by using the number line. 

Learn About Multiplication of Fractions

FAQs on Multiplication

Q.1. What is multiplication?
Ans: Multiplication is the process of finding the product of any two numbers.

Q.2. What are the properties of multiplication?
Ans: The properties of the multiplication are commutative, associative, distributive and identity.

Q.3. How to do multiplication?
Ans: Multiplication of two numbers can be multiplied by doing repeated addition. Multiplication of \(p\) and \(q\) (\(p\) is multiplied to \(q\)) means \(p\) is added to itself \(q\) number of times or \(q\) is added to \(p\) number of times.

Q.4. What is the multiplication formula?
Ans: The multiplication formula is given by \({\rm{Multiplier}} \times {\rm{Multiplicand}} = {\rm{Product}}.\)

Q.5. What is the multiplication symbol?
Ans: The symbol used for multiplication is cross \(\left( \times \right).\)

Now you are provided with all the necessary information on the concept of multiplication and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.