Properties of Multiplication: Multiplication is one of the four basic operations in arithmetic. So, it is very crucial for young minds to grasp the concepts thoroughly for further mathematics. 

Multiplication properties will help students in building upon a clear concept of multiplication in solving a variety of numbers. Commutative, associative, distributive, identity and zero properties are some of the most important properties for multiplication in mathematics.

What is Multiplication?

In mathematics, multiplication (denoted by the symbol \(‘ \times ‘\)) is a method of finding the product of two or more values. In arithmetic, the multiplication of two numbers represents the repeated addition of one number with respect to another number.

Symbol:

The symbol of multiplication is denoted by a cross sign \((×)\) and also sometimes by a dot \(\left( \cdot \right)\).

Examples:

1. \(6 \times 9=54\)
2. \(8 \times 2 \times 4=64\)
3. \((8) \cdot(10)=80\)
4. \((2) \cdot(4)=8\)

If \(a\) is multiplied by \(b\), then it means either \(a\) is added to itself \(‘b’\) number of times or vice versa.

For example, \(2×4\) means \(4\) times of \(2\), such as:

\(2+2+2+2=8\)

Let us understand it by some pictures with some more examples.

In the above example, we have learnt to multiply whole numbers. Similarly, we can also multiply fractions and find the product of decimals.

Multiplication is one of the major fundamental topics in Maths, apart from addition, subtraction and division. Students learn the four basic arithmetic operations in their primary classes themself.

Multiplication Through Number Line

On a number line, one can skip count to add repeatedly to multiply.

Example: \(2×3\) means \(3\) times \(2\), we need to skip \(2\) for three times. So, finally we will reach at \(6\), which is the product.

Thus, \(2×3=6\)

The above example shows \(2\) is multiplied \(7\) times on the number line. \(2×7=14\).

Definition of Properties of Multiplication

Multiplication plays a very important role in mathematics as well as in our daily life. As multiplication is the process of repeated addition, it saves our time to get the solution through multiplication instead of adding the same number repeatedly. There are some properties of multiplication. They are the commutative property (also called order property), associative property (also called grouping property), distributive property, property of one and zero property etc.

Types of Multiplication Property

1. Closure property of multiplication
2. Commutative property of multiplication
3. Associative property of multiplication
4. Distributive property of multiplication over addition
5. Identity property of multiplication
6. Zero property of multiplication

Closure Property of Multiplication

According to the closure property, if two integers \(a\) and \(b\) are multiplied, then their product \(a×b\) is also an integer. Therefore, integers are closed under multiplication.

\(a×b\) is an integer, for every integer \(a\) and \(b\).

Examples:

\(2×(-3)=-6\)

\(8×6=48\)

Commutative Property of Multiplication

The word, Commutative, originated from the French word ‘commute or commuter’ means to switch or move around, combined with the suffix ‘ative’ means ‘tend to’. Therefore, the literal meaning of the word is tending to switch or move around.

It states that if we change the orders of the integers, the result will remain the same.

The commutative property of multiplication states, if \(a\) and \(b\) are any two integers, then, \(a×b=b×a\)

Examples: 

\(3×4=4×3=12\)

\(7×(-8)=-8×7=-56\)

Let us understand it by the below picture,

Associative Property of Multiplication

The result of the product of three or more integers is irrespective of the grouping of these integers.

In general, if \(a, b\) and \(c\) are three integers then,
\(a \times(b \times c)=(a \times b) \times c\)
Examples:
\(2 \times(4 \times 3)=(2 \times 4) \times 3=24\)

\(-2 \times(-5 \times-3)=(-2 \times(-5)) \times-3=-30\)

Let us understand it by the below picture,

Distributive Property of Multiplication Over Addition

According to the distributive property of multiplication over Addition, if \(a, b\) and \(c\) are three integers then,

\(a×(b+c)=(a×b)+(a×c)\)

Example:

\(3 \times \left( {2 + 3} \right) = \left( {3 \times 2} \right) + \left( {3 \times 3} \right)\)
\(3 \times 5 = 6 + 9\)
\(15 = 15\)

Let us understand it by the below picture,

Identity Property of Multiplication

On multiplying any integer by \(1\) the result obtained is the integer itself. In general, if \(a\) and \(b\) are two numbers then,

\(a×1=1×a=a\)

Therefore \(1\) is the multiplicative identity of Integers.

Examples:

\(25 \times 1 = 25\)
\( – 799 \times 1 = – 799\)
\(8 \times 1 = 8\)

Let us understand it by the below picture,

Zero Property of Multiplication

On multiplying any number by zero, the result is always zero. It is called the zero property.

If \(a\) and \(b\) are two integers then,

\(a×0=0×a=0\)

Examples:

\(12×0=0\)

\(-102×0=0\)

Thus, we can see, any integer, whether it is the smallest or the largest when multiplied by zero, results in zero only.

Other Properties of Multiplication

1. If \(a, b\) and \(c\) are the integers and a>b, then,
\(a×c>b×c\)

Example:

If \(6 > 4\)
\(6 \times 2 = 12\)
\(4 \times 2 = 8.\)

Therefore,
\(6 \times 2 > 4 \times 2\)

From the above example, we get that when we multiply any number on both LHS and RHS, we will get the same comparison symbol as before.

2. Change of Sign Property.

Multiplication of two positive integers is always positive.

Examples:

\((+4)×(+ 5)=+20\)

Multiplication of two negative integers is always positive.

Example: 

\((-3)×(-5)=+15\)

When multiplying one positive integer and one negative integer, the product is a negative integer.

Example:

\((+7)×(-4)=-28\)

Rules of Multiplication

There are various rules to multiply numbers. These are:

1. Multiplication of two integers is an integer.
2. Any number multiplied by \(0\) is \(0\).
3. Any number multiplied by \(1\) is equal to the number itself.
4. If an integer is multiplied by multiples of \(10\), then the same number of \(0s\) are added at the end of the given number.
Example: \(49×1000=49000\)
5. The order of the numbers, when multiplied together, does not change the product.
Example: \(5×6×7×8=8×7×6×5=6×5×8×7=1680\)
6. Multiplication of even numbers of negative integers is always positive.
Example: \((-2)×(-4)=(+8)\)

Multiplication Signs

When two or more numbers are multiplied with different sign \((+\) and \(-)\), then the output result varies, as per the sign rules given below:

Operation	Sign
\(\left( { + {\rm{ve}}} \right) \times \left( { + {\rm{ve}}} \right)\)	\( + {\rm{ve}}\)
\(\left( { + {\rm{ve}}} \right) \times \left( { – {\rm{ve}}} \right)\)	\( – {\rm{ve}}\)
\(\left( { – {\rm{ve}}} \right) \times \left( { + {\rm{ve}}} \right)\)	\( – {\rm{ve}}\)
\(\left( { – {\rm{ve}}} \right) \times \left( { – {\rm{ve}}} \right)\)	\( + {\rm{ve}}\)

Solved Examples – Properties of Multiplication

Q.1. Show that, \(12×15=15×12\).
Ans: 
\({\text{LHS}} : 12 \times 15 = 180\)
\({\text{RHS}} : 15 \times 12 = 180\)
From the above two equations, we get the same product, i.e. \(180=180\)
Hence, \(12×15=15×12\) (proved).

Q.2. Find the product of \(16780×0\).
Ans:
Given, \(16780×0\)
As we know, on multiplying any number by \(0\), the result is always \(0\) (called the zero property of multiplication)
Thus, \(16780×0=0\)
Hence, the required product is \(0\).

Q.3. Solve \(8945×1\).
Ans:
Given,
\(8945×1\)
As we know, on multiplying any number by \(1\) the result obtained is the number itself (called the identity property of multiplication).
Thus, \(8945×1=8945\)
Hence, the product is \(8945\).

Q.4. Find the product of \(25×(-48)+(-48)×(-36)\) using a suitable property.
Ans:
Given, \(25×(-48)+(-48)×(-36)\)
By rearranging the above expression, using commutative property, we get,
\((-48)×(25)+(-48)×(-36)\)
Again using distributive property, we get,
\(\left( { – 48} \right) \times \left[ {25 + \left( { – 36} \right)} \right]\)
\( = \left( { – 48} \right) \times \left[ {25 – 36} \right]\)
\( = \left( { – 48} \right) \times \left( { – 11} \right)\)
\( = 528\)
Hence, the product is \(528.\)

Q.5. Find the product of \(6 \times 23 \times (- 125)\) using a suitable property.
Ans:
Given, \(6 \times 23 \times(-125)\)
Using associative property, we can rearrange the given expression as:
\(23 \times 6 \times(-125)\)
\(=23 \times[6 \times(-125)]\)
\(=23 \times(-750)\)
\(=-17250\)
Hence, the product is \(-17250\).

Q.6. Find the product of \((-25) \times 103\) using a suitable property.
Ans:
Given, \((-25) \times 103\)
We can write the above expression as:
\((-25) \times(100+3)\)
Using distributive property, we get;
\((-25 \times 100)+(-25 \times 3)\)
\(=-2500+(-75)\)
\(=-2500-75\)
\(=-2575\)
Hence, the product is \(-2575.\)

Summary

In this article, we covered what multiplication is and what are the different properties of multiplication with examples, what is the use of properties of multiplication, multiplication rules, how the sign of product changes while the numbers are having different signs etc. This article will help the students identify the multiplication properties and get used to them by using them mentally for calculations in further mathematical studies, thus increasing their creative thinking, math skill and mental maths too.

Frequently Asked Questions (FAQ) – Properties of Multiplication

Q.1. What are examples of Properties of Multiplication?
Ans:
Example of commutative property:
\(3 \times 5=5 \times 3\)
Example of associative property:
\((2 \times 4) \times 5=2 \times(4 \times 5)\)
Example of distributive property:
\(5 \times(2+8)=(5 \times 2)+(5 \times 8)\)
Example of identity property:
\(10 \times 1=10\)
Example of zero property:
\(5 \times 0=0\)

Q.2. What are the types of Properties of Multiplication?
Ans: There are five types of properties of multiplication.
These are,
(a) Commutative Property
(b) Associative Property
(c) Distributive property
(d) Identity Property
(e) Zero Property

Q.3. What is the formula for the distributive property of multiplication?
Ans: The formula for the distributive property of multiplication is given by:
\(a \times(b+c)=(a \times b)+(a \times c)\)

Q.4. What is the commutative property of multiplication?
Ans: When we multiply two integers, the answer we get after multiplication will remain the same, even if the order of the integers are interchanged.
Let \(a\) and \(b\) be the two integers, then the formula for the commutative property is,
\(a \times b=b \times a\)
Example: \(5 \times 6=6 \times 5=30\)

Q.5. What is the meaning of ‘distributive’?
Ans: Sharing of something to each member of the group with a definite rule is called distributive.

Q.6. What are the uses of Properties of Multiplication?
Ans: The properties of multiplication helps us to solve the questions with brackets. It also speeds up our mental calculations. As in both cases, the answer we get is the same.
Example: \(2 \times(3 \times 4)=(2 \times 3) \times 4=24\)
The same product we got after multiplying it in two different way.

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\(5 \times(2+8)=(5 \times 2)+(5 \times 8)\)
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We hope you find this article on the Properties of Multiplication helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.