Multiplication of Integers: Multiplication is one of the fundamental operations in mathematics. Integers form a significant collection of numbers that contains whole numbers and negative numbers. The multiplication of integers is the process of repetitive addition that follows specific multiplication of integers rules and properties.

The multiplying integers rules will help you in solving the various mathematical problems efficiently. The properties of multiplication like commutative, associative, distributive, identity and zero properties help solve complex mathematical tasks. The multiplying integers rules are there to solve various problems. Continue reading to understand what multiplication of integers and how to solve the problems related to this concept.

Definition of Integers

Integers can be defined as the set of natural numbers and their additive inverse, including zero. The set of integers is {….-3,-2,-1, 0, 1, 2,3…}. Integers are numbers that cannot be a fraction.

Define Multiplication of Integers

Multiplying integers is the process of repetitive addition, including positive or negative integers. The product of two integers is called Multiplication (denoted by the symbol ‘\(\times\)’. It is a method of finding the product of two or more integers.

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

Examples:

1. \(5 \times 9 = 45\)
2. \(8 \times 3 \times 4 = 96\)
3. \((6) \cdot (10) = 60\)
4. \((2) \cdot (9) = 18\)

If \(m\) is multiplied by \(n\), either ‘\(m\)’ is added to itself ‘\(n\)’ number of times or vice versa.

For example, \(3 \times 4\) means \(4\) times of \(3\), such as:
\(3 + 3 + 3 + 3 = 12\)

KNOW ABOUT NUMBER SYSTEM

Multiplication of Integers Examples

Below are some examples that show the repeated addition of integers and the multiplication of integers:

Multiplying Integers Through Number Line

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

Example: \(2 \times 7\) means \(7\) times \(2\). We need to skip \(2\) seven times to arrive at result \(14\).
Thus, \(2 \times 7 = 14\)

Multiplication of Integers Rules

Students can check the multiplying integers rules below:

1. The product of a Positive Integer and a Negative Integer is negative.
   Examples: \(4 \times \left( { – 6} \right) = \left( { – 24} \right)\); \(9 \times \left({ – 5} \right) = \left({ – 45} \right)\)
2. The product of a Negative Integer and a Positive Integer is negative.
   Examples: \(\left({ – 4} \right) \times 8 = \left({ – 32} \right)\); \(\left({ – 6} \right) \times 7 = \left({ – 42} \right)\)
3. The product of two Positive Integers is positive.
   Examples: \(5 \times 10 = 50\); \(20 \times 12 = 240\)
4. The product of two Negative Integers is always positive.
   Example: \(\left({ – 9} \right) \times \left({ – 5} \right) = + 45\); \(\left({ – 6} \right) \times \left({ – 8} \right) = + 48\)

Multiplication Signs

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

Type of Integers	Integers signs(Result)
\((+\rm{Positive})×(+\rm{Positive})\)	\(+\rm{Positive}\)
\((+\rm{positive})×(-\rm{Negative})\)	\(-\rm{Negative}\)
\((-\rm{Negative})×(+\rm{Positive}\))	\(-\rm{Negative}\)
\((-\rm{Negative})×(-\rm{Negative}\))	\(+\rm{Positive}\)

Properties of Multiplication

The following are the properties of multiplying integers:

1. Closure property of Multiplication
2. Commutative Property of Multiplication
3. Associative Property of Multiplication
4. Distributive Property of Multiplication
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, 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:
1. \(2 \times \left( { – 3} \right) = \, – 6\)
2. \(8 \times 6 = 48\)

Commutative Property of Multiplication

Commutative originated from the French word ‘commute or commuter’, which means switching or moving around, combined with the suffix ‘ative’ means ‘tend to’. Therefore, the literal meaning of the word is ‘tending to change 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 \(p\) and \(q\) are any two integers, then,
\(p \times q = q \times p\)

Examples:

1. \(5 \times 4 = 4 \times 5 \Rightarrow 20\)
2. \(7 \times \left( { – 9} \right) = \,\left( { – 9} \right) \times 7 \Rightarrow \, – 63\)

Let us understand it by the below picture:

Associative Property of Multiplication

The result of the product of three or more integers is not dependent on the grouping of these integers.

In general, if \(p\),\(q\) and \(r\) are three integers then,
\(p \times (q \times r) = (p \times q) \times r\)

Examples:

1. \(3 \times (4 \times 5) = (3 \times 4) \times 5 \Rightarrow 60\)
2. \( – 2 \times \left( { – 4 \times \left( { – 3} \right)} \right) = \left( { – 2 \times \left( { – 4} \right)} \right) \times – 3 \Rightarrow \, – 24\)

Let us understand it by the below picture:

Distributive Property of Multiplication

According to the distributive property of multiplication, if \(a,b\) and \(c\) are three integers then,
\(p \times (q + r) = (p \times q) + (p \times r)\)

Examples:
\(4 \times (2 + 3) = (4 \times 2) + (4 \times 3)\)
\( \Rightarrow 4 \times 5 = 8 + 12\)
\( \Rightarrow 20 = 20\)

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.
\({{p}} \times 1 = 1 \times {{p}} = {{p}}\)

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

Examples:

1. \(25 \times 1 = 25\)
2. \(( – 799) \times 1 = ( – 799)\)
3. \(8 \times 1 = 8\)
4. \(1 \times ( – 34) = ( – 34)\)
5. \(1 \times 100 = 100\)

Let us understand it by the below picture.

Zero Property of Multiplication

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

Then, \({{p}} \times 0 = 0 \times {{p}} = 0\)

Examples:

1. \(13 \times 0 = 0\)
2. \(( – 105) \times 0 = 0\)
3. \(0 \times 38 = 0\)
4. \(0 \times 25 = 0\)

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

Let us understand this by using the figure.

Let us understand the Multiplication of Integers facts by solving some problems.

Solved Examples on Multiplication of Integers

Multiplication of Integers practices problem will help in remembering the properties of multiplication.

Q.1: Find the product of \(250 \times 0\)
Ans: Given, \(250 \times 0\)
As we know, on multiplying any number with \(0\), the result is always \(0\) (called the zero property of multiplication).
Thus, \(250 \times 0 = 0\)
Therefore, the obtained product is \(0\).

Q.2: Show that, \(10 \times 15 = 15 \times 10.\)
Ans:
L.H.S: \(10 \times 15 = 150\)
R.H.S: \(15 \times 10 = 150\)
From the above two equations, we get the same product, i.e., \(150 = 150\)
Therefore, \(10 \times 15 = 15 \times 10\) is proved.

Q.3: Solve \(894 \times 1.\)
Ans: As we know, on multiplying any number by \(1\), the result obtained is the number itself (called the identity property of multiplication).
Thus, \(894 \times 1 = 894\)
Therefore, the obtained product is \(894\).

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

Q.5: Find the product of \(5 \times 23 \times ( – 125)\) using a suitable property.
Ans: From the given, \(5 \times 23 \times ( – 125)\)
Using associative property, we can rearrange the given expression as:
\(23 \times 5 \times ( – 125)\)
\( = 23 \times [5 \times ( – 125)]\)
\( = 23 \times ( – 625)\)
\(= -14375\)
Hence, the product is \(-14375\).

Q.6: Find the product of \(( – 24) \times 103\) using a suitable property.
Ans: Given, \(( – 24) \times 103\)
We can write the above expression as:
\(( – 24) \times (103 + 3)\)
Using distributive property, we get;
\(( – 24 \times 100) + ( – 24 \times 3)\)
\(= \, – {\rm{ }}2400{\rm{ }} + {\rm{ }}\left( {{\rm{ }} – {\rm{ }}72} \right)\)
\(= \, – {\rm{ }}2400{\rm{ }} + {\rm{ }} – {\rm{ }}72\)
\(= \, – {\rm{ }}2472\)
Hence, the product is \(- 2472\).

Summary

The Multiplication of Integers is the process of repetitive addition, including positive or negative Integers. This article covered the meaning of multiplication, multiplication of Integers on a number line, multiplication rules, and the different properties of multiplication with examples. Students can use the properties discussed in the articles to solve questions quickly, and with practice, they can start doing calculations mentally.

FAQs

Check frequently asked questions related to multiplying integers below:

Q.1: How to solve the multiplication of integers?
Ans: The Multiplication of Integers is the process of repetitive addition, including positive or negative integers.
If \(m\) is multiplied by \(n\), either ‘\(m\)’ is added to itself ‘\(n\)’ number of times or vice versa. This method can be used to solve the Multiplication of Integers problems.

Q.2. What are the four rules of Multiplication of Integers?
Ans: There are four rules of Multiplication of Integers
1. The product of a Positive Integer and a Negative Integer is negative.
Examples: \(7 \times ( – 6) = ( – 42)\); \(( – 8) \times 5 = ( – 40)\)
2. The product of a Negative Integer and a Positive Integer is negative.
Examples: \(( – 3) \times 5 = ( – 15)\); \(( – 2) \times 3 = ( – 6)\)
3. The product of two Positive Integers is positive.
Examples: \(5 \times 20 = 100\); \(10 \times 12 = 120\)
4. The product of two Negative Integers is always positive.
Examples: \(( – 7) \times ( – 5) = + 35\); \(( – 5) \times ( – 8) = + 40\)

Q.3: Explain the multiplication of integers with example?
Ans: The Multiplication of Integers is the process of repetitive addition, including positive or negative integers.
Multiplication (denoted by the symbol ‘\( \times\)’) is a method of finding the product of two or more integers.
Examples:
1. \( 5 \times 8\)
2. \( 8 \times 3 \times ( – 2) = (- 48)\)
If \(m\) is multiplied by \(n\), either ‘\(m\)’ is added to itself ‘\(n\)’ number of times or vice versa.
For example, \(5 \times 4\) means \(4\) times of \(5\), such as:
\(5 + 5 + 5 + 5 = 20\)

Q.4: What are the types of Properties of Multiplication?
Ans: Properties of multiplication are,
1. Closure property of Multiplication
2. Commutative Property of Multiplication
3. Associative Property of Multiplication
4. Distributive Property of Multiplication
5. Identity Property of Multiplication
6. Zero Property of Multiplication

Q.5: How do you multiply Negative Integers?
Ans: We know that the product of a positive integer and a Negative Integer is negative.
Examples: \(5 \times ( – 6) = ( – 30)\); \(( – 6) \times 4 = ( – 24)\)
And, the product of two Negative Integers is positive.
Examples: \(( – 8) \times ( – 5) = + 40\); \(( – 6) \times ( – 7) = + 42\)

Q.6. What are examples of Properties of Multiplication?
Ans:
1. Example of commutative property:
\(4 \times 5 = 5 \times 4\)
\( \Rightarrow 20 = 20\)
2. Example of associative property:
\((3 \times 4) \times 5 = 3 \times (4 \times 5)\)
\( \Rightarrow 12 \times 5 = 3 \times 20\)
\( \Rightarrow 60 = 60\)
3. Example of distributive property:
\(5 \times (2 + 3) = (5 \times 2) + (5 \times 3)\)
\( \Rightarrow 5 \times (5) = (5 \times 2) + (5 \times 3)\)
\( \Rightarrow 25 = 10 + 15\)
\( \Rightarrow 25 = 25\)
4. Example of identity property:
\(11 \times 1 = 11\)
5. Example of zero property:
\(24 \times 0 = 0 \times 24 \Rightarrow 0\)

Q.7: What is the Associative Property of Multiplication?
Ans: The result of the product of three or more integers is irrespective of the grouping of these integers.
In general, if \(p,q\) and \(r\) are three integers then,
\(p \times (q \times r) = (p \times q) \times r\)
Examples:
1. \(3 \times (4 \times 5) = (3 \times 4) \times 5 \Rightarrow 60 = 60\)
2. \( – 2 \times \left( { – 4 \times – 3} \right) = \left( { – 2 \times – 4} \right) \times – 3 \Rightarrow – 24 = – 24\)

Q.8: What is the Commutative Property of Multiplication?
Ans: Commutative Property of Multiplication states that if we change the orders of the integers, the result will remain the same.
The commutative property of multiplication states, if \(p\) and \(q\) are any two integers, then,
\(p \times q = q \times p\)
Examples:
1. \(6 \times 4 = 4 \times 6 \Rightarrow 24 = 24\)
2. \(7 \times \left( { – 5} \right) = \left( { – 5} \right) \times 7 \Rightarrow – 35 = – 35\)

Students can take Multiplication of Integers notes from the above questions to revise the concept quickly.

We hope this detailed article on the Multiplication of integers is helpful to you. If you have any queries, ping us through the comment box below. We will get back to you as soon as possible.

Stay tuned to Embibe for the latest news and updates!