Integer Definition

The word "integer" comes from the Latin, and it is used for expressing "whole" or "intact". Because of this, fractions and decimals are not considered integers. In math, Integers play a crucial role in various ways. Let us read this article to learn more about integers, their definitions, and their characteristics.

What is an Integer?

A numerical value or a number is said to be an integer if it is a whole number, be it positive or negative. Therefore, the three types of integers are positive, negative, and zero. Accordingly, a set of integers is formed if we combine multiple positive whole numbers and/ or negative numbers without fractions and decimals.

The group of counting numbers that can be expressed without the need for a fractional component also includes a zero. So, as was already established, an integer can be either positive, negative, or zero.

Integers with a base of 1 and an infinite range make up all natural numbers. Beginning at 0 and going all the way to infinity on both sides (positive and negative), all whole numbers are integers.

Integer can be represented as below:

…… -3, -2, -1, 0, 1, 2, 3……

Integers, which can comprise positive and negative numbers, including zero, represent the numbers without a decimal or fractional component. The following are a few random examples of integers: -4, 0, 1, 3, 7, 99, and 4960.

Classification of Integers

Integers are mainly categorized into the following three categories:

• Positive Integers: A number becomes positive if it is higher than zero. Any specific positive number without fractions or decimals is called a positive integer.

Example: Z = (1, 2, 3), etc.

• Negative Integers: A number is considered negative if it is less than zero. Any specific negative number without fractions or decimals is called a negative integer.

Example: Z = (-1, -2, -3), etc.

• Zero Integer: Zero is characterized as being neither a negative nor a positive number. It is a complete number and called a zero integer.

Example: Z = (0)

In all the above examples, Z is an integer set that consists of the respective integers.

Integers on Number Lines

A straight line of numbers is shown visually as a number line. This line is used to compare integers spaced equally apart on an infinite line extending horizontally on both sides. The collection of integers can be shown on a number line, much like other numbers.

Graphing Integers on a Number Line

Number lines can be used to depict both positive and negative numbers visually. A number line's integers assist in carrying out mathematical operations. While arranging numbers on a number line, it is important to keep the following in mind:

• Always, the right side's number is higher than the left side's number.
• Being bigger than 0, positive numbers are positioned to the right of zero.
• Due to their smaller value than 0, negative numerals are positioned to the left of zero.
• It is customary to place zero in the middle, which is neither positive nor negative.

In the following image, the integers -6, -2, and 3 are graphed on a number line with a pink colored circle (dot):

Integer Operations

Integers can be used in the following four fundamental arithmetic operations:

1. Addition of Integers
2. Subtraction of Integers
3. Multiplication of Integers
4. Division of Integers

Let us understand each in detail:

Addition of Integers

Calculating the sum between two or more integers is the process of adding them, and depending on whether they are positive or negative, the value may increase or decrease.

The following section lists the many guidelines and potential scenarios for adding integers.

Guidelines for Adding Integers

We employ the following principles when adding two integers:

• The absolute values of the numbers are added, and the result is assigned the same sign as the given integers when both of the integers have the same sign.
• Whenever a positive integer and a negative integer are added, give the sign of the greater of these two numbers to the outcome after determining the difference between the absolute values of the two integers.

Example

5 + 8 = 13

-5 + (-8) = -13

The output will contain the signature of the greater integer if any of the integers has a different orientation, resulting in subtraction. Let us clarify with the use of examples:

(-10) + (2) = -10 + 2 = -8

(-2) + (10) = -2 + 10 = 8

That is how the addition of integers works.

Subtraction of Integers

Determining the differences between two or more integers and allowing for the possibility that the result may be positive or negative, subtracting integers involves finding the differential value between the integers.

The next section will review the various guidelines and potential scenarios for subtracting integers.

Rules of Integer in Subtraction

The following guidelines are used when subtracting two integers:

• Change the sign of the 2^nd number which is being subtracted and follow the rules or operation same as the addition. Utilizing the same integer addition rules resolve the issue discussed in the previous phase.

Example

7 - 10 = -3

Now, let us discuss the operation process for this calculation. First, change the sign of the subtractor to produce the following result (by converting the provided statement to look like an addition problem):

7 + (-10)

The guidelines for this operation will now be the same as those for adding two numbers. In this instance, the absolute values of 7 and (-10) are 7 and 10, respectively.

They differ by 10 - 7 (bigger number - smaller number), which equals 3. Since 'ten' is the greater number in this instance, and so the output (result) bears its original sign, i.e., "-", instead of the sign of the seven. As a result, the outcome is denoted by the "-".

As a result, 7 - 10 = -3

Multiplication of Integer

The rules listed in the following table are used for multiplying whole numbers (or integers). The following section lists the many methods for multiplying integers and the scenarios in which they might be used. Here, we can determine which sign we should use with the output value when multiplying two integers of the same or different signs:

Example

Now, observe the following multiplication where we have integers 4 and 2. Although results seem the same in numbers, they differ because of their resultant sign, plus or minus:

4 x 2 = 8

4 x (-2) = -8

(-4) x 2 = -8

(-4) x 2 = 8

Division of Integer

When an integer is divided, it is either evenly divided into groups or a predetermined number of groups. We follow the guidelines listed in the following table when dividing integers. The following section determines which sign we should use with the output value when dividing two integers of the same or different signs:

Example

Now, observe the following division where we have integers 10 and 2. Like the multiplication of integers, the results in integers division also seem the same in numbers, but they differ because of their resultant sign, plus or minus:

10 ÷ 2 = 5

10 ÷ 2 = -5

(10) ÷ 2 = -5

(-10) ÷ (-2) = 5

Properties of Integers

Listed below are the integers' primary characteristics:

Closure Property

According to the set's closure property, it can be mathematically closed for every given operation. Under integer addition, subtraction, multiplication, & division, Z is closed, meaning that the output (result) will also be an integer. When given any two integers a and b, the closure property states:

• a + b ∈ Z
• a - b ∈ Z
• a × b ∈ Z
• a/b ∈ Z

Where Z depicts a set of integers.

For instance,

2 + 8 = 10 (which is an integer)

2 x 8 = 16 (which is an integer)

Associative Property

The associative property states that altering how two integers are grouped will not change the operation's outcome. Two integers can be added together and multiplied using the associative property.

Suppose a and b are any two integers, so the associative property states:

• a + (b + c) = (a + b) + c
• a × (b × c) = (a × b) × c

For instance,

2 + (4 + 4) = (2 + 4) + 4 = 10

2 x (3 × 4) = (2 × 3) x 4 = 24

It is essential to note that the associative property is not applicable to the subtraction and division of integers.

Commutative Property

The commutative property states that an operation's outcome is unaffected by the order in which its operands are used. Integers are added and multiplied following the commutative property.

Suppose a and b are any two integers, so the commutative property states:

• a + b = b + a
• a × b = b × a

Similar to the associative property, the commutative property is also not applicable to the subtraction and division of integers.

For instance,

2 + 8 = 8 + 2 = 10

2 x 8 = 8 x 2 = 16

Distributive Property

According to the distributive property, any phrase of form a (b + c), which denotes the form "x (b + c)", allows the operand a to be divided among operands b and c as follows:

(a × b) + (a × c),

As a result,

• a × (b + c) = (a × b) + (a × c)

For instance,

Let us prove that: 4 x (4 + 1) = 4 x 4 + 4 x 1

LHS = 4 x (4 + 1) = 4 x 5 = 20

RHS = 4 x 4 + 4 x 1 = 16 + 4 = 20

Since LHS (which is 20) = RHS (which is also 20). Hence, the distributive property is proven to be true.

Additive Inverse Property

According to the additive inverse property, when adding a specific integer to its positive and negative values, zero (0) is the only result.

Whatever the integer a is, the additive inverse property states:

• a + (-a) = 0

In this way, -a is called the additive inverse of integer a.

Multiplicative Inverse Property

According to the multiplicative inverse property, the outcome is always one (1) when an integer is multiplied by its reciprocal.

Whatever the integer a is, the multiplicative inverse property states:

• Z = a x (1/a) = 1

In this way, 1/a is the multiplicative inverse of integer a.

Identity Property

When performing addition and multiplication operations, integers adhere to the identity property.

According to the additive identity property, an integer becomes the integer itself when a zero is added to it. As a result, this illustrates:

• a + 0 = a

Similarly, according to the multiplicative identity property, multiplying one by any integer yields the integer. As a result, this illustrates:

• a × 1 = a

Comparing the Integers

Integers can be compared in the same way as positive whole numbers. We can quickly compare the integers by utilizing a number line. The value of a number increases as it moves further to the right.

A number line is expressed like this:

……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……

For instance, let us now compare integers -8 and -2. People will argue that the integer -8 is bigger than -2, but this is not true. As already said, the value rises as you turn to the right. So, the value does not decrease with the smaller negative numbers but increases. If we check the integers on a number line, we can easily notice the integer -2 on the right side of -8. Because of this, -2 is more than -8, or -2 > -8.

Likewise, in another instance, let us consider integers -8 and 3. Although it may appear at first glance that integer -8 is greater than 3, it is not the case again. One universal rule is that negative values are never greater than positive ones. Also, one can see that the integer 3 stands at the right of the integer -8. This demonstrates that 3 is more than -8, or 3 > -8.