What is an Integer

In Mathematics we deal with different types of numbers. Each number is different from another, but they may share some common characteristics. To understand these numbers they are categorized in the different groups according to their characteristics. 

Some of the groups in the system are:

• Natural numbers

• Whole numbers

• Integers

• Rationals

• Real numbers

In this article let us study what is an integer? And all about integers.

Integers are numbers, which includes positive numbers, negative numbers, and also zero but no fractions are allowed.

So, integers can be positive {1, 2, 3, 4, 5, 6 ... }, negative {−1, −2,−3, −4, -5, -6. .... }, or zero {0}

We can represent this all integer numbers together as:

Integers( Z ) = { ...6, -5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6... }.. The dots represent the infinite numbers.

All About Integers

An integer is any number that can be either 0, or positive number, or negative number. An integer can never be a fraction, a decimal, or a percent. Some examples of integers include 2, 3, -4, 8, 99, -81, -56, etc.

The integers can be represented as:

Z = {……-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ……….}

On the number, line integers are represented as follows

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Here 0 is at the center of the number line and is called the origin.

All integers to the left of the origin (0) are negative integers prefixed with a minus(-) sign and all numbers to the right are positive integers prefixed with positive(+) sign, they can also be written without + sign.

Types of Integers

All integer numbers are basically of three types:

• Zero

• Positive Numbers and

• Negative Numbers

Positive Numbers

Positive numbers are those numbers that are prefixed with a plus sign (+). Most of the time positive numbers are represented simply as numbers without the plus sign (+). Every positive number is greater than zero, negative numbers, and also to the number to its left. On a number line, positive numbers are represented to the right of origin( zero).

Example: 1, 5, 500, 66 ,89, etc.

Negative Numbers

Negative numbers are those numbers that are prefixed with a minus sign (-). It is mandatory to mention the sign of negative numbers. Negative numbers are represented to the left of the origin(zero) on a number line.

Example: -8 , -10, -1000, -1919, etc.

Zero

Zero is a neutral integer because it can neither be a positive nor a negative integer, i.e. zero has no +ve sign or -ve sign.

Arithmetic Rules for Integers

• Addition Rule

To add two integers with the same sign:

If the sign of both the integers is the same, then add the absolute value and the result is assigned the same sign as both the values.

Rules:

• (+) + (+) = +

• (-) + (-) = –

For Example:

• 4 + 9 = 13

• (-4) + (-9) = -(4+9) = -13

To add two integers with different signs:

If one of the numbers has a different sign, then it will lead to subtraction, and output will have a sign of the larger number.

Find the difference of the absolute values and give the difference the same sign as the number with the largest absolute value.

Integers on a Number Line

In the study of mathematics, a number line is a straight line on which we can add or subtract real numbers. Number lines can be used to compare numbers, which are placed at equal intervals. A number line is also called an infinite line as it extends on both sides horizontally. Just like other numbers, the set of integers can also be represented on a number line.

Graphing Integers on a Number Line

• The number is written on the right hand side of the line are always greater than the numbers that are written on the left side of the line

• Positive numbers or integers are placed on the right side of the number line that is the right side of zero, these numbers are greater than zero.

• Negative numbers or integers are placed on the left side of the number line that is the left side of zero, these numbers are less than zero.

• Zero is always marked at the centre of the number line as it is not positive or negative.

Integer Operations

The main arithmetic operations related to integers are as follows- 

• Addition of integers

• Subtraction of Integer

• Multiplication of Integers

• Division of Integers

These operations can only be performed by following certain rules, they are as follows –

A number that is written individually that is there is no sign in front of the number, that number is a positive number

For example, 5 = +5.

While a negative sign indicates that the number is a negative number.

Addition of Integers

While adding two integers, we come across the following cases:When both integers are having the same sign, we can calculate their absolute values and allot them the same sign.

If an integer is positive and the other is negative, then their result will have the sign of the large number

Let us understand it with following steps

For example: (-7) + (4)

1. Find the absolute values of -7  and 4.

2. Find the difference between 7 and 4 i.e is 7 – 4 = 3.

3. Find the sign of the largest absolute value. Here the largest number is 7. It has a negative sign.

4. So the result obtained is -7 + 4 = -3

• Subtraction Rule: 

Subtracting means adding the opposite of a given number or changing the signs of the number. The sign of the first number remains the same, change subtraction to addition and change the sign of the second number. Once you have applied this rule, follow the rules for subtracting integers

Subtraction of Integers

Subtraction of integers is as follows- 

Convert the operation into an additional problem by changing the sign of the subtrahend.

Rules taught above about the addition of integers are the same rules that will be applied to the subtraction of integers.

Rules of Integers

Rules defined for integers are:

• Sum of two positives is an integer.

• Sum of two negatives is an integer.

• Product of two positive integers is an integer.

• Product of two negative integers is an integer.

• Addition operation between any integer and its negative value will give the result as zero

• Multiplication operation between any integer and its reciprocal will give the result as one.

Properties of Integers

The major Properties of Integers are:

• Closure Property

• Associative Property

• Commutative Property

• Distributive Property

• Additive Inverse Property

• Multiplicative Inverse Property

• Identity Property

Closure Property:

The closure property states that the set is closed for any particular mathematical operation. Z is closed under addition, subtraction, multiplication, and division of integers. For any two integers, a and b:

a + b ∈ Z

a - b ∈ Z

a × b ∈ Z

a/b ∈ Z

Associative Property:

According to the associative property, changing the grouping of two integers does not alter the result of the operation. The associative property applies to the addition and multiplication of two integers.

For any two integers, a and b:

a + (b + c) = (a + b) + c

a ×(b × c) = (a × b) × c

Commutative Property:

According to this property, swapping the positions of integers in an operation does not affect the result.

The addition and multiplication of integers follow the commutative property.

For any two integers, a and b:

a + b = b + a

a × b = b × a

Distributive Property:

According to this property, for an expression that is written in the form a (b + c), that means a × (b + c), a can be distributed among b and c as: (a × b + a × c) i.e.,

a × (b + c) = a × b + a × c

Additive Inverse Property:

According to this property, addition between any integer and its negative value will give the result zero.

For any integer, a:

a + (-a) = 0

Multiplicative Inverse Property:

The multiplicative inverse property states that the multiplication operation between any integer and its reciprocal will give the result as one.

For any integer,

a: a × 1/a = 1

Identity Property:

For addition and multiplication operations, the identity property is used in integers.

According to this property,

a × 0 = a

Similarly, multiplicative identity states that:

a × 1/a = 1\

Rules:

• (+) – (+) = (+) + (-)

• (-) – (-) = (-) + (+)

• (+) – (‐) = (+) + (+)

• (‐) – (+) = (‐) + (‐)

Always consider the greater number sign for the result.

For example:

Subtracting -5 from 10, we get:

10 – (– 5) 

Changing the signs or taking the opposite and adding

= 10 + 5 (5 is the opposite of -5) 

= 15.

• Multiplication Rule

To multiply or divide an integer, use the absolute value. If the signs of the integers are the same, multiply, or divide the answer is always positive. And if the two integers have different signs, the answer is negative.

Rules:

• (+) x (+) = + and (+) ÷ (+) = +

• (‐)  x  (‐) = + and (‐)  ÷  (‐) = +

If the signs are different:

• (+) x (‐) = – and (+) ÷ (‐) = ‐

• (‐) x (+) = – and (‐) ÷ (+) = ‐

For Examples: 

•  5 x 2 = 10 and 8 ÷ 2 = 4

• (-5) x (-2) = 10 and (-8) ÷ (-2) = 4

• (5) x (-2) = -10 and 8 ÷ (-2) = -4

• (-5) x (2) = -10 and (-8) ÷ (2) = -4 

Comparing the Integers

We can compare integers as we compare positive whole numbers. By using a number line we can easily compare the integers. The farther the number is to the right, the larger the value it has.

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• Compare -9 and -2

We will say that -9 is greater than -2, but no smaller the negative numbers greater the value. As you move towards the right the value increases.

Here, -2 is  to the right. Therefore, -2 is greater than -9 i.e -2 > -9.

• Compare -8 and 3

At first look, you will say -8 is greater than 3, but no negative values are always smaller than the positive ones.

Here we can see that 3 is larger than -8 i.e 3 > -8.

This was all about integers. 

Solved Examples

1. Compare -1 and 3

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Solution: -1 < 3 as -1 is to the left of the number 3

2. Compare -35 and 67

Solution: -35 < 67 as negative numbers are always smaller than positive numbers.

Quiz Time

Identify the integers

1. 67 

2. 2.98

3. 589000

4. 0.00008

Compare the integers

1. 45 and -89

2. 67 and -5678