We know that an integer is a bigger collection of numbers which includes the whole number and negative numbers as well. Have you ever noticed the difference between the whole numbers and the integers? Here we will discuss the integers, their properties, and operations. 

Integers

We already know how to represent the integers on the number line. We also know the method of addition and subtraction of integers. Talking about the number line, we know how to

• To add a positive integer on a number line, we move towards the right.

• Add a negative integer, then we move towards the left.

• Subtract a positive integer, then we move towards the left.

• Subtract a negative integer, then we move towards the right.

Integers 7th class notes-

We have already discussed the integers. Let us talk about the different properties of integers in detail.

PROPERTIES OF ADDITION AND SUBTRACTION OF INTEGERS-

Closure under Addition-

We already know that the sum of the whole number is the whole number. For example- adding 28+ 32=60, which is again a whole number. We know that this type of property is referred to as the closure property for the addition of the whole number. Let us understand for example if this type of property is also true for the integers. Add (-21) + 13 = -8. We observe that the result is an integer. So can we say that is the sum of two integers always an integer? Or is there any sum of pair whose integer is not an integer? Since the addition of integers gives integers, we can say that integers are closed under addition. In general, we say that for any two integers say a and b, a + b is an integer. 

Closure under Subtraction-

What will happen when we subtract an integer from another integer? Can we say that their difference will also be an integer? Let us understand this with an example. 

Subtract 4 – 6 =- 2 , -2 is an integer.

Subtract (-42) – 64 = -106 , -106 is an integer.

What did you observe? Is there any pair of integers whose difference is not an integer? Can we say integers are closed under subtraction? So, we can see that integers are closed under subtraction. 

Thus, if a and b are two integers then a –b is also an integer. 

Commutative Property-

We know that 6 + 4= 4 + 6 = 10, which means the whole number can be added in any order. In other words, addition is commutative for the whole number. In general, for any two integers a and b, we can say that a + b= b + a 

We know that subtraction is not commutative for whole numbers. Is it commutative for integers? Let’s take an example of some integers.  

Is 4-(-2) the same as (-2)-4? Its big no, because 4 – (-2) =4+2 = 6, and (-2) – 4= -6 

The answers to both of them are different integers. Hence, we conclude by saying that subtraction is not commutative for integers. 

Associative Property-

Consider the integers -4, -2 and -3 

Do the sum of them like-

(-4) + [(-2) +(-3)] and [(- 4) + (-2 )] + (-3)

In the first part, the sum of (-2) and (-3) are grouped together and in the second (-4) and (-2) are grouped together. We will solve and see if the results are the same or different. 

So, in both of the cases, we get -9. 

You will not find examples where the sums are different. Hence, Addition is associative for integers. In general, for any integers a, b, and c we can say that 

                                                                 a+b+c=a+b+c

Multiplication of Integers-

We are already aware that repeated addition is called multiplication. 

For example- 4+4+4=3 ×4=12

So, can you write the addition of integers in this way?

While multiplying a positive integer and a negative integer, we multiply them as a whole number and put the minus sign before the product. So, the overall result is a negative integer. 

In general, for any two positive integers a and b, we write

            a ×-b=-a×b=-(a×b)

Multiplication of two negative integers

When the product of two negative integers is done, then the result is a positive integer. We multiply the two negative integers as whole numbers and put the positive sign before the product. Writing the equation in general, for any two positive integers a and b, 

                                     (-a) ×(-b)=a ×b

Integers related solved examples

1. Find the product of -21 ×-4-3 ×15

The product will be -21 ×(-4)] ×[-3 ×15 ]

                                   =84 ×(-45)]

                                   = -3780

2. Calculate 125 ÷-5

Dividing 125 by 5 first and using the negative sign with the result. 

         125 ÷-5= -25 

Conclusion–

We have discussed in this article, the integers, what are properties of integers and tried to explain the whole concept with examples. We have even solved a few examples step-by-step to explain how to carry out the mathematical problems. Above the discussed the addition, subtraction, and multiplication of integers.