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Integers

Integers is also considered as combination of positive integers and negative integers. We already know several properties of whole numbers and four fundamental integer operations namely addition, subtraction, multiplication and division.

But, we have not studied the differences such as 5 - 7, 6 - 10, 20 - 30 etc. in the whole number system.

This shows the need to extend our whole number system to represent such differences as well. This leads us to the study of integers. The study of integers enable us to study such numbers which gives us the new idea of representing numbers with direction. These numbers are known as directed numbers.

Integer numbers are starting from non-positive infinity to non-negative infinity. Mainly there are two kinds of integers are available. They are known as the positive integer type and the negative integer type. The numbers proceeds the 0 is known as the negative integers and the numbers following the zero is known as the positive integers.

Integer Definition

In mathematics, we define integer with one of the interesting topic in number representation. Integer has a set of numbers in which includes positive numbers, negative numbers and zero. An integer contains complete entity or unit. In integer, there is no fractional parts and also no decimal numbers.

Examples of integers:

489, - 546, 0, 84, etc.

Positive Integers

The whole number is called as positive integer and the positive integer is represented by the symbol ‘+’. The positive integers are involved in graph theory. The arithmetic operations are performed in positive integers.

Positive integers for addition and subtraction:
The positive integers are combined with negative integers also. So based on that some rules are followed for addition and subtraction operation.

( + ) + ( + ) = ( + )

( + ) + ( - ) or ( - ) + ( + ) = The resultant value is based on large number sign. Consider the numbers 4 and -2 and perform the addition operation as 4 + (-2) = 2.

Positive integers multiplication:

( + ) x ( + ) = ( + )

( + ) x ( - ) = ( - )

( - ) x ( + ) = ( - )

Positive integers division:
The sign of numerator value is deciding the resultant value sign.

( + ) / ( + ) = ( + )

( - ) / ( - ) = ( + )

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Negative Integers

The definition of negative integer, is one of the most important topic in mathematics. Negative integer is present before the zero value in the number line. The symbol used for representing the negative integer is known as the " - ". Most of the mathematical problems are also can be done using the negative integer. In this article, we are going to see about the negative integer with the example problems.

Explanation to negative integers definition :

Addition problem for the definition of negative integer.
Subtraction problem for the definition of negative integer.
Multiplication problem for the definition of negative integer.
Division problem for the definition of negative integer.

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Integer Rules

Integer rules are the procedures to be followed for doing problems in integers. For doing arithmetic operations on integers, rules for integers are very useful. Based on the sign of integers the rules differ. The integer rules are applicable for arithmetic operations with integers such as addition, subtraction, multiplication and division.

The integers rules follows for :

Adding integers
Subtracting integers
Multiplying integers
Dividing integers

Adding integers

To add a positive number and a negative number we can subtract the smaller number from the greater number without taking the sign into account and to the result, give the sign of the greater number.
A Positive number + a positive number = a positive number
A Negative number + a negative number = a negative number

Solved Examples

Question 1: Add the given two positive number 10, 23.
Solution:

Add 10 + 23 = 33

So, the answer is 33

Question 2: Add the given two negative number -10, -23.
Solution:

Given two numbers are negative numbers.

So the addition rules is applied to it. (-10) + (-23)

The answer is the -33.

Subtracting Integers

In whole numbers, we know that addition and subtraction are inverse operations.
The rules are similar to addition of integers but one different thing here we can do the subtraction.
A Positive number + a positive number = a positive number
A Negative number + a negative number = a negative number

Solved Examples

Question 1: Subtract -10 from -25.
Solution:

Given two numbers are negative numbers -25 and -10,

So using the subtraction rules we get, (-25) - (-10) This is written as -25 + 10 Subtracting the above two numbers.

The answer is -15 (subtracting the larger value to the smaller value and then put the larger value sign).

Question 2: Subtract -63 from -5
Solution:

Given two numbers are both negative So by the subtraction rules (-5) - (-63)

This is written as -5 + 63 Subtracting the above two numbers.

The answer is 58 (subtracting the larger value to the smaller value and then put the larger value sign).

Multiplying Integers

We know that multiplication is repeated process of addition. The rules are,

A Positive number × a positive number = a positive number
A Positive number × a negative number = a negative number
A Negative number × a positive number = a negative number
A Negative number ×a negative number = a positive number

Solved Examples

Question 1: Multiply 2 with -3.
Solution:

Here 2 is a positive number and 3 is a negative number

When we multiply a positive with negative the resultant will be negative 2 x (-3) = -6

Question 2: Multiply -5 and -4
Solution:

Here -5 and -4 are both negative number.

When we multiply a negative with negative the resultant will be positive (-5) x (-4) = 20

Dividing Integers

Division is a repeated process of subtraction. The rules are,

A Positive number / a positive number = a positive number
A Positive number / a negative number = a negative number
A Negative number / a positive number = a negative number
A Negative number / a negative number = a positive number

Below you could see examples for integer division

Solved Examples

Question 1: Divide 12 with 4
Solution:

Here 12 is a positive number 4 is a positive number.

When we divide positive with a positive resultant will be positive $\frac{12}{4}$ = 3

Question 2: Divide -50 by 5
Solution:

Here -50 is a negative number 5 is a positive number.

When we divide negative by positive the resultant will be negative $\frac{-50}{5}$ = -10

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Integer Word Problems

Integer word problems it is has the series (with the continuous of expression) we have to fit that expression in the mathematical expression so we called that kind of expression as the word problems.

Solved Examples

Question 1: Find the sum of the two consecutive integer number is 717, find the integer?
Solution:

The numbers are consecutive so we take one integer as x and the another integer as x+1(consecutive so we take x+1)In the question they are given the sum is 717
So here we add these two integer and then find the value of x
X+x+1=717.
Subtracting 1 on both sides we get
2x =717-1 =716
Then divide the 2 on both sides we get the answer of x (integer)
X = $\frac{716}{2}$ =358
So the one number is 358 and the number is 359(x+1) both the answer are the integer.

Question 2: The morning temperature of the New York was -17 ⁰F if the temperature will be dropped as 11⁰F, calculate the temperature of the New York now?
Solution:

Here the morning temperature as given as -17
And the drooped temperature will be given as 11
We have to calculate the temperature of the city now,
morning temperature - dropped temperature
-17-11 = -28 ⁰F

Integer Number Line

Number line is a line which helps to represent the numbers along with its sign. Below we can see integer number line explained with figure.

Example : Subtract an integer 3 from 6 using number line.

Solution: Mark the first integer 6 on number line.

Step 1: Make a count from 6 and move to left side integer until reaches the count of second integer 3.
Integer Number Line