Multiplication and Division of Integers

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What are Integers?

While studying mathematics, we notice some arithmetic operations that include the processes of addition, subtraction, division and multiplication. Integers are the whole numbers that are non-fractional. In integers, we can find,

The numbers which we count (1,2,3….)

The number 0

The negative numbers ( -1, -2, -3,...)

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Multiplication and Division of Integers

We multiply and divide integers in the same way we multiply by counting numbers. The only thing that makes the multiplication and division different is the sign. We need to be very careful regarding the signs while doing multiplication and division. When we do the multiplication of two integers having the same signs, the sign of the product is always positive. If we multiply positive to positive, we get positive. If we multiply negative to negative, we get positive. Similarly, when we divide the two numbers of integers having the same set of signs, the solution comes in positive.

Now, if we divide positive by positive, we get the solution in positive.

And if we divide negative by negative, we get the solution in positive.

Now, when we do multiplication of two integers having a different set of signs. For say, if we multiply positive to negative, we will get the product in the sign of negative.

Similarly, if we divide the positive integer by a negative integer, we will get the solution in the sign of negative.

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Some examples: 

  1. (-7) X (-5) = 35

  2. (-8) X 3 = 24

  3. 9 X 3 = 27

  4. -10 ÷ (-5) = 2

  5. -12 ÷ 6=2


Rules of Multiplication and Division of Integers

To simplify the calculation of multiplication and division, we need to follow some rules as below:


1. Closure Rule: 

In this closure rule, we multiply two integers suppose p x q, then the product of p x q is also an integer.

Here, p x q will be an integer, for each integer p and q.


2. Commutative Rule: 

In this commutative rule of multiplication of integers, we change the way of integers which doesn’t change the solution of the multiplication.

Here,

P X Q= Q X P, for each integer P and Q.


3. Associative Rule: 

In this rule of associative, the solution of three or more integers of grouping doesn’t change the solution of the product.

Here,

(m x n) x r =  m x (n x r )


4. Distributive Rule: 

In this rule of Distributive property of multiplication of integers which has p, q and r as three integers, then

p x ( q + r ) = (p x q ) + (q +r)

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Multiplication of Integers

Multiplication is the addition of numbers, but the rules for multiplication of integers are different from the addition of integers.

In the multiplication of integers, we get three cases:

  • Multiplication is occurring between two positive integers.

  • Multiplication is occurring between two negative integers.

  • Multiplication is occurring between one negative integer and one positive integer.

We should always remember the following points while multiplication.

  1. The product of two integers having positive integers value is always positive.

  2. The product of negative integers is always positive.

  3. The product will be negative if one integer is positive and another integer is negative.

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Division of Integers

The multiplication is the addition of numbers while the division of integers is the distribution of integers. Division of integers is the opposite way of doing multiplication. The rules for the division of integers and rules for multiplication of integers are quite similar. However, it’s not necessary to always find integers as your quotient value. Here too, we can see three cases of division of integers.

  1. The quotient value of two positive integers will always be a positive integer.

  2. The quotient value for two negative integers will always be a positive integer.

  3. The quotient value of one positive integer and one negative integer will always be a negative integer.

So, when we are dividing, we should always divide without the signs but after getting the solution of the integer, give the sign according to the sign given in the problem.

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Similarly, like the division of integers, we will also multiply the integers without using the sign. However, after getting the value of the product integer, we will give the sign of the product according to the sign given in the question.

FAQ (Frequently Asked Questions)

Q1. Why is it Important to Use Proper Sign Convention for Solving the Multiplication and Division of Integers?

Ans: It is important to use proper sign convention to find the proper solutions for multiplication and division of integers. While using the proper sign, we will get the correct answer to every problem we solve. A single wrong sign can deduct your marks by proving that solution as wrong.


Suppose a problem is given to you which has two integers in the problem, one positive and one negative, the answer should be in negative. Let us see take an example for a better understanding.


By solving 2 x (-4) = -8


However, we write the answer as 8, it will be incorrect because you have not provided the correct sign convention for this problem. It is always good to keep in your mind to use proper sign convention.

Q2. Why it is Required to Follow the Rules of Division and Multiplication of Integers?

Ans: Rules are introduced to keep the solving method easier which will reduce the amount of confusion we create while solving sums. Any alternative way or any properties or rules is beneficial to solve sums. The rules of multiplication and division of integers include some ways or formulas which are key to simplify the problems in much easier ways but there is always an exception.


So, no problem is specifying one rule. Every rule has its different ways and different rules to solve. Rules are provided to use to find correct answers with a hustle free method. By using rules in multiplication and division of integers, we get to know that if we multiply positive to positive, we find the solution is positive.

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