Properties of Multiplication of Integers: We know that numbers \(\left\{ { – 6,~ – 5,~ – 4,~ – 3,~ – 2,~ – 1,~0,~1,~2,~3,~4,~5, \ldots } \right\}\) are integers. These integers can be classified into different sets. The numbers with a minus sign along with them like \(-1,-5\) and \(6\) are called negative integers, whereas numbers like \(1,2,3,4\) and \(5\) are called positive integers. Zero is an integer that is neither positive nor negative.

We can perform all the mathematical operations like addition, subtraction, multiplication, and division on integers. But to know them better, we need to study the properties of integers. In this article, we will learn about the properties of the multiplication of integers in detail.

What are Integers?

Integers can be defined as the set of positive numbers, negative numbers, and zero. The set of integers is denoted by

\(Z = \left\{{ \ldots .. – 3,~ – 2,~ – 1,~0,~1,~2,~3 \ldots .} \right\}\)

Integers do not include fractions or decimals. Integers are an extension of whole numbers and natural numbers. Natural numbers, along with zero and negative of natural numbers, make integers. Whole numbers, along with negative of natural numbers, make integers.

That is,

Whole Numbers \(+\) Negative of Natural numbers \(=\) Integers

Natural Numbers \(+\) Zero \(+\) Negative of Natural Numbers \(=\) Integers

Multiplication of Integers

The multiplication of integers can be regarded as repeated addition.

For example, if we consider the integers \(4\) and \(5\)
\(4 \times 5=5+5+5+5=20\)

Extending this idea to the product of a negative number and a positive number, we have
\((5)\times(-3)=(-3)+(-3)+(-3)+(-3)+(-3)=-15\)

For a product of two negative integers, the multiplication will be
\((-3) \times(-4)=(-4)+(-4)+(-4)=12\)

Rules for Multiplication

For any two positive integers \(a\) and \(b\),

\(a \times b=+(a \times b)\)
\((-a) \times(-b)=+(a \times b)\)
\(a \times(-b)=-(a \times b)\)
\((-a) \times b=-(a \times b)\)

That is, 

1. If the integers have the same sign, their product is positive.
2. If the integers have opposite signs, their product is negative.

Properties of Multiplication of Integers

Let us study the properties of multiplication of integers.

1. Closure Property: To explore the closure property of multiplication, let us first observe the given table.

From the table, we observe that,

\((-2) \times(-2)=4\), and \(4\) is an integer.
\((-1) \times(-2)=2\), and \(2\) is an integer.
\((0) \times(-2)=0\), and \(0\) is an integer.
\((1)\times(-2)=-2\), and \(-2\) is an integer.

Thus, in general, if \(a\) and \(b\) are any two integers, then \((a \times b)\) is also an integer.
Thus, we can say that integers are closed under multiplication. This is called the closure property of the multiplication of integers.

2. Commutative Property: The commutative property of multiplication states that the final result remains the same when multiplying numbers, even if the order of numbers is changed. Changing the order of multiplication doesn’t change the product.

Let us take \(2\) integers and multiply them in \(2\) different ways.
\((-5) \times(4)=20\), and \(4 \times(-5)=20\)

We observe that their product is the same in both cases,
i.e., \((-5) \times(4)=4 \times(-5)\)

Let us take another example.
\((-10) \times(-5)=50\) and \((5) \times(-10)=50\)

Here also, \((-10) \times(-5)=(5) \times(-10)\)
We can say that the product of two integers remains the same on interchanging their order. Thus, we conclude that multiplication is commutative for integers.

In general, we can say that if \(a\) and \(b\) are any two integers, then \(a \times b=b \times a\).

3. Associative Property: Associative property of multiplication states that if we want to multiply any \(3\) numbers together, the answer will always be the same irrespective of the order in which we multiply the numbers.

Let us consider any \(3\) integers, say, \(4, 10\) and \(-5\). To find their product, we can either multiply \(4\) and \(12\) first and then multiply their product by \(-5\), i.e.,
\((4 \times 10) \times(-5)=40 \times(-5)=-200\)

Or, we can multiply \((-5)\) and \(10\) first and then multiply the product with \(4\).
i.e., \((4) \times(10 \times-5)=4 \times(-50)=-200\).

In both cases, we get the same product.
\((4 \times 10) \times(-5)=(4) \times(10 \times-5)\)

Thus, the product of any \(3\) or more integers remains the same irrespective of the order in which the multiplication is carried out. So, multiplication is associative for integers.

In general, if \(a, b\) and \(c\) are any \(3\) integers, then \((a \times b) \times c=a \times(b \times c)\).

4. Distributive Property Over Addition and Subtraction: The distributive property of multiplication states that multiplication can be distributed over addition and subtraction.

Consider any \(3\) integers, say, \(20, -6\) and \(13\). Multiply one of them by the sum of the other two numbers in different ways.

\(20×(-6+13)\) \(=20×(7)\) \(=140\)

\(20×(-6+13)\) \(=20×(-6)+20×13\) \(=-120+260=140\)

In both cases, we get the same result. Therefore, we can say that \(20 \times(-6+13)=20 \times(-6)+20 \times 13\).

Let us take another group of \(3\) integers, \((-4), 10\) and \((-5)\), and repeat the process.

\((-4)×(10+(-5))\) \(=(-4)×(10-5)\) \(=(-4)×5\) \(=-20\)

\((-4)×(10+(-5))\) \(=(-4)×(10)+(-4)\times(-5)\) \(=-40+20\) \(=-20\)

Here also, we get the same result in both cases. Therefore, \((-4) \times(10+(-5))=(-4) \times(10)+(-4) \times(-5)\)

In general, if \(a, b\) and \(c\) are any \(3\) integers, then \(a \times(b+c)=(a \times b)+(a \times c)\) This is called the distributive property of integers over addition.

Similarly, we can show for the subtraction as well. Let us take another group of \(3\) integers, \((-5), 10\) and \((5)\).

\((-5)×(10-5))\) \(=(-5)×(10-5)\) \(=(-5)×5\) \(=-25\)

\((-5)×(10-5)\) \(=(-5)×(10)-(-5)\times(5)\) \(=-50+25=-25\)

Thus, multiplication is distributive over addition and subtraction.

5. Multiplicative Identity: The multiplicative identity property of multiplication states that if we multiply any number by \(1\), the answer will always be the same number.

Let us observe the product of \(1\) and any other integer.
\(1 \times 8=8\) or \(8 \times 1=8\)
\(1 \times(-6)=-6\) or \((-6) \times 1=-6\)
\(1 \times 40=40\) or \(40 \times 1=40\)

In general, we can say that for any integer \(a\), we have
\(1 \times a=a \times 1=a\)
i.e., the product of any integer and \(1\) is always the integer itself.

Thus, \(1\) is called the multiplicative identity for integers.

6. Multiplicative Property of Zero: The multiplicative identity of zero states that, on multiplying any integer by \(0\), the result is always \(0\).

We observe that:
\(8 \times 0=0\), and \(0 \times 8=0\) i.e., \(8 \times 0=0 \times 8=0\)
\(-6 \times 0=0\) or \(0 \times(-6)=0\), i.e., \(-6 \times 0=0 \times(-6)=0\)

In general, for any integer \(a, a \times 0=0 \times a=0\). So, the product of \(0\) and any integer is always \(0\).

Solved Examples – Properties of Multiplication of Integers

Q.1. Find the value of \(700 \times(-5)+(

-700) \times 95\)
Ans: Given, \(700 \times(-5)+(-700) \times 95\)
Using the distributive property of multiplication over addition, we get,
\(700 \times(-5)+(-700) \times 95=700 \times[(-5)+(-95)]\)
\(=700 \times[-100]=-70000\)

Q.2. Verify and name the property used in the following.
a) \((-18) \times[(-5)+(-3)]=(-18) \times(-5)+(-18) \times(-3)\)
b) \((-72) \times(-45)=(-45) \times(-75)\)
Ans: a) \((-18) \times[(-5)+(-3)]=(-18) \times(-5)+(-18) \times(-3)\)
LHS \(=(-18) \times[(-5)+(-3)]\)
\(=(-18) \times(-8)=144\)
RHS \(=(-18) \times(-5)+(-18) \times(-3)\)
\(=90+54\)
\(=144\)
Therefore, LHS \(=\) RHS
We can see that in RHS \((-18)\) is distributed to \((-5)\) and \((-3)\).
Hence, we have used the distributive property of multiplication over addition.
b) \((-72) \times(-45)=(-45) \times(-72)\)
\(\mathrm{LHS}=(-72) \times(-45)\)
\(=+(72 \times 45)\)
\(=3240\)
RHS \(=(-45) \times(-72)\)
\(=+(45 \times 72)\)
\(=3240\)
Therefore, LHS \(=\) RHS
We can see that in LHS and RHS, the order of \((-45)\) and \((-72)\) is interchanged, but still, the answer came out the same.
Hence, we have used the commutative property of multiplication.

Q.3. Find the value of \(x\) in \(x \times[15+(-7)]=4 \times 15+4 \times-7\)
Ans: Using the distributive property, we can write
\(4 \times 15+4 \times-7\) as \(4 \times[15+(-7)]\)
Thus, \(x \times[15+(-7)]=x \times 15+x \times-7\)
or \(x \times[15+(-7)]=4 \times[15+(-7)]\)
By comparing LHS and RHS, we get,
\(x=4\)
Hence, the value of \(x\) is equal to \(4\).

Q.4. Use the property to evaluate \(18 \times 102\).
Ans: \(18 \times 102\) can be written as \(18 \times(100+2)\)
Using distributive property, we get,
\(18 \times(100+2)=(18 \times 100)+(18 \times 2)\)
\(=1800+36\)
\(=1836\)
Hence the value of \(18 \times 102\) is \(1836\).

Q.5. Find the product of \((-11) \times(103)+(-11) \times(-3)\) using suitable properties of multiplication.
Ans: Given, \((-11) \times(103)+(-11) \times(-3)\)
Using the distributive property of multiplication over addition, we get,
\(=(-11) \times(103+(-3))\)
\(=(-11) \times(103-3)\)
\(=(-11) \times(100)\)
\(=-1100\)
Hence, the value of \((-11) \times(103)+(-11) \times(-3)\) is \(-1100\).

Q.6. Evaluate the expression \(3333 \times 87+13 \times 3333\), using the distributive property.
Ans: For any three integers \(x, y\), and \(z, x \times y+x \times z=x \times(y+z)\)
Now, the given expression is, \(3333 \times 87+13 \times 3333\)
Thus, \(3333 \times 87+13 \times 3333\) can be written as \(3333 \times(87+13)\)
Therefore, \(3333 \times(87+13)=3333 \times 100\)
\(=333300\).
Hence, the required answer is \(333300\).

Summary

In this article, we have a brief review of the integers. We learnt about the multiplication of integers, and then we learnt in detail about the properties of the multiplication of integers. Further, we solved some examples to master learning the properties of the multiplication of integers.

Frequently Asked Questions (FAQ) – Properties of Multiplication of Integers

Q.1. What is the commutative property of multiplication?
Ans: The commutative property of multiplication states that the final result remains the same when multiplying numbers, even if the order of numbers are changed. Changing the order of multiplication doesn’t change the product.
We can say that the product of two integers remains the same on interchanging their order. Thus, we conclude that multiplication is commutative for integers.
In general, if \(a\) and \(b\) are any two integers, then \(a \times b=b \times a\)

Q.2. What is the associative property of multiplication?
Ans: The product of any \(3\) or more integers remains the same irrespective of the order in which the multiplication is carried out. This is known as the associative property of multiplication for integers.
In general, if \(a, b\) and \(c\) are any \(3\) integers, then \((a \times b) \times c=a \times(b \times c)\).

Q.3. What are the 6 properties of multiplication?
Ans: The \(6\) properties of multiplication of integers are;
1. Closure property
2. Commutative property
3. Associative property
4. Distributive property over addition and subtraction
5. Multiplicative identity
6. Multiplicative property of zero

Q.4. When do we use the distributive property?
Ans: Whenever there is a need to multiply a common number to a sum or difference of numbers, we use the distributive property.
That is,
\(x(y+z)=x \times y+y \times z\)
\(x(y-z)=x \times y-y \times z\)

Q.5. Define multiplicative identity and multiplicative property of 0 ?
Ans: Multiplicative Identity: The multiplicative identity property of multiplication states that if we multiply any number by \(1\), the answer will always be the same number.
In general, we can say that for any integer \(a\), we have
\(1 \times a=a \times 1=a\)
Multiplicative Property of zero: The multiplicative identity of zero states that, on multiplying any integer by \(0\), the result is always \(0\).
In general, for any integer \(a, a \times 0=0 \times a=0\).
So, the product of \(0\) and any integer is always \(0\).