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Integers

Operations on Integer Numbers

Integer numbers are whole numbers. These can be negative, positive or a zero. When we perform mathematical operations of integers, we follow a different set of rules that are specific to the character of the number. Not every rule of mathematical operation applies to the integers. Let’s see how handling integers are different from the normal mathematical operations.

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Integer Numbers and Mathematical Operations

Mathematical operations include addition, subtraction, multiplication, and division of any number. When we perform these operations with integer numbers we always keep in mind the sign before every number.

As we already know that an integer includes a number with a positive or negative sign, therefore, these have to be dealt with different perceptions. Before delving into further operations, we first need to know the properties related to these mathematical operations.

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Addition and Subtraction of Integer numbers

integer numbers

Source: emaze

Property of Closure in Addition and Subtraction

When we add two whole numbers, the answer is a whole number. This is called the closure property of additions. Now, do integers also follow this property? Let’s see:

10 + (-12) = -2

Is -2 an integer number. Since it has a negative sign with it, therefore we call it an integer number. From the above example, we can say that the sum of two integers is also an integer. So when an integer a is added with another integer b the answer is always an integer.

Let’s take some more examples to support our observation:

  • 19+10= +29
  • 15 + (-20)= -5

In both the examples, the solution is an integer. Therefore every integer when summed up with an integer gives an integer. What happens when an integer is subtracted from an integer. Let’s see the following examples:

  • 10- 3 = +7
  • 15 – (-7) = -22

In both the examples, the difference between two integers gives an integer in solution. This means that integers are also closed under subtraction. Hence, in the case of subtraction of two integers a and b the answer to a-b is also an integer.

Commutative Property

Commutative property states the following:

a+b =b+a

For whole numbers, we already know that the sum of two whole numbers is always the same. But is the case true for integer numbers as well? The sum of two integer numbers is also, always the same. This means that integer numbers also follow the commutative property like whole numbers. Let’s see the following examples:

  • 15 + 20 =35; 20 +15=35
  • -10 + (-5) = -15; -5 + (-10) = -15

The above examples prove that the addition of integers is commutative. Is the case true with subtractions? Are subtractions also commutative? The following examples will let us know this:

  • 5-(-3) = +8
  • -3-5 = -8

Are the integer numbers same? The answer is a no! This brings us to a conclusion that subtractions of integers are not commutative. Therefore, a-b ≠ b-a

Associative Property

According to associative property of whole numbers a,b,c,

[a+b]+c =a+[b+c]

Does this property also apply to integer numbers? Let’s see:

  • [(-4)+(-6)]+(-2) = -12
  • (-4) + [(-6)+(-2)] = -12

The answer in both the cases is the same. So the sum of integer numbers abides the associative property of addition.

Additive Identity

The additive identity of any number is checked with the help of zero. For all the whole numbers, zero proves to be their additive identity. This means that when a whole number is added with a zero, the answer is the whole number itself. Is the case same with integer numbers as wel?. The answer here is a yes. The following examples prove our observation:

  • +10 +0 = +10
  • -9+0=-9

Therefore for any integer, x,

x+0 = x = 0+x

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Operations on Integers

Multiplication of Integers

After addition and subtraction, we need to understand the multiplication of two integer numbers. Let’s look at some of its properties.

Closure Property

Is the product of two integer numbers also an integer number? Well, yes the product of every integer is an integer.

Commutative Property

The product of two same integers is always the same. This means that

a × b = b× a

Multiplication with Zero

Every integer, when multiplied with a zero, gives zero as the answer.

a×0 = 0

Multiplicative Identity

Every integer number, when multiplied with 1, gives an integer number in an answer.

  • a×1 = a
  • (-a) ×1 = -a

Associative Property

Integer numbers also abide by the associative property of multiplication. This implies that like whole numbers the grouping of integers does not affect the product of integers. Thus,

(a×b)×c = (a) ×(b×c)

Distributive Property

Here we shall check the distributivity of integers over addition is true or not.

  • (-4)  × (2 + 6) = -32
  • (-4) × (2) + (-4) ×(+6) =(-8) +(-24) = -32

The example above shows that multiplication of integers also shows distributivity of multiplication over addition. Thus,

a × (b+c) = (a×b) + (a×c)

Does the distributive property also comply with subtractions in multiplication? The answer is a yes! The example below will help you understand:

  • (-8) × (5 – 3) = -16
  • [(-8) ×(+ 5)] – [(-8) × (-3)] =( -40)- (+24) = -16

From the above example we can thus state that;

a × (b-c) = (a×b) – (a×c)

Multiplication of a Negative and Positive Integer

As already said earlier, while performing the mathematical operations of integer numbers, we must always keep in mind the respective sign before every number. When multiplying two integers with a negative and positive sign, the product shall always be negative.

This means that in every case of multiplication between contradictorily signed integers the answer shall always have a negative sign. For example:

4 ×(-5) = -20

To simplify the product we write the above example as :

( -5)+( -5)+( -5)+( -5) = -20

Some more examples are:

  • -3×4 = -12
  • 12×-6=-72

Therefore  (-a) × b = a × (– b) = – (a × b)

The multiplication of integers is similar to multiplication of whole numbers. The only difference here is that after multiplying we put a minus sign (-) in the answer.

Multiplication of Two Negative Integers

The multiplication of two negative integer numbers is always a positive integer. This means the product of (-a) ×(-b) = +c. The negative integer numbers in multiplication are multiplied like ordinary whole numbers. Some examples will help you understand this in a better way:

  • (-3) × (-6) = 18
  • (-30)×(-15) = 450

This means that: (-a) ×(-b) = a×b. The minus (-) sign of the integer numbers while multiplication is ignored.

Product of Three or More Integers

In the above topics, we multiplied only two integers. Now we shall multiply three or more than three integers. Consider the following image:

integer numbers

In the above set of examples, what do you notice? When we multiply more than three integers, we multiply them in the set of two. The signs of the answer depend on the rules we already mentioned in the multiplication of positive and negative integer.

integer numbers

In both the example images we notice that when we multiply two negative integers the answer is a positive integer. When we multiply two integers with negative and positive signs respectively, the answer is a negative integer number.

In case you are confused about signs, you may follow another very simple way of applying a sign. Now, if the number of negatively signed numbers is odd, the answer shall be a negative integer. If the negative integers are in even number then the answer shall be a positive integer.

(-a) × (-b) × (-c) × (-d) = +z

In the example given above, the total number of integers are even in number, hence the answer is a positive integer.

(-a) × (-b) × (-c) × (-d) × (-e) = -z

See the example given above. The total number of integers are odd in number, hence answer is a negative integer.

Division of Integer numbers

We know that divisions are inverse of multiplications. How do we then perform divisions in integers? Division of integers is similar to the division of whole numbers. While dividing an integer with an integer we divide them like normal whole numbers and then put the sign respective to the integers involved.

This means that when we divide a positive integer with a positive integer the answer is a positive integer. Similarly, when the division involves one positive and one negative integer then, we divide them like normal whole numbers and put a negative sign in the quotient. For example:

  • 72÷(–8) = –9
  • (-80)÷(5) = -16

Now, let’s look at some of the properties for the division of integers.

Closure Property of Division

Is the division of two integer numbers also an integer number? Well, no. The division of two numbers does not give an integer in the answer.

  • (-8)÷(2) = -4
  • (-9)÷(2) = -9/2

So the division of integers does not give an integer in the answer.

Commutative Property of Division

This property does not apply to divisions between integers. This means that a÷b ≠b÷a

Division by Zero

Like whole numbers, an integer number also cannot be divided by (zero) 0, but when zero (0) is divided by an integer the answer is zero (0).

Division by One 

Every integer number, when divided by 1, gives the same integer number in the answer. But the case is not the same when an integer is divided by -1

  • a÷1 = a
  • (a)÷(-1) = -a

From the above illustrations, we have come to know that the mathematical operations of integer number involve cautious calculations as these follow a different set of rules.

Solved Examples for You

Question 1: In a test (+5) marks are given for every correct answer and (–2) marks are given for every incorrect answer.

  1. Shyam answered all the questions and scored 40 marks though he got 10 correct answers.
  2. Renu also answered all the questions and scored (–16) marks though he got only 4 correct answers.

How many incorrect answers had they attempted?

Answer : One correct answer gives = +5 marks. Shyam gave 10 correct answers, so, marks given for 10 correct answers = 5 × 10 = 50. Shyam’s score = 40.

Marks obtained for incorrect answers = 40 – 50 = – 10
Marks given for one incorrect answer = (–2)
Therefore, number of incorrect answers = (–10) ÷ (–2) =5
Shyam gave 5 incorrect answers.

Marks given for 4 correct answers = 5 × 4 = 20
Renu’s score = –16
Marks obtained for incorrect answers = –16 – 20 = – 36
Marks given for one incorrect answer = (–2)
Therefore number of incorrect answers = (–36) ÷ (–2) = 18
Renu gave 18 incorrect answers.

Question 2: Is a decimal an integer?

Answer: We can express any integers as a decimal however most of the numbers that can be expressed as a decimal are not integers. In addition, if all the digits after the decimal are zeros then the number is an integer, but if they are any other number except zero after the decimal point then it is not an integer.

Question 3: How to express answers in an integer?

Answer: Integers refers to all those numbers that do not contain any decimal point or are in fraction. Besides, fro converting any number to integers just count the number of significant figures in the number and write this down as a whole number.

Question 4: What is a valid integer value?

Answer: In a computer definition, integers are the whole number, which on sending the number 145.5732 to an integer function would return 145. Moreover, they can be signed (negative or positive) or unsigned (always positive).

Question 5: Is pi a real number?

Answer: It is an irrational number, which means that it is a real number that we cannot express using a simple fraction. When students first introduced to pi then they were told its value to be 22/7 or 3.14 or 3.14159.

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