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Numbers

Integers

Written by Prerit Jain

Updated on: 15 Aug 2023

Integers

Integers

Introduction

In mathematics, an integer is  a set of  negative numbers, positive numbers, and zero. The symbol used to denote integers is “Z.” Integers are an essential concept in mathematics and are used in a wide range of applications in everyday life. Understanding their properties and rules is essential for solving mathematical problems and making informed decisions. This article aims to provide a comprehensive overview of integers, including their definition, types, operations, properties, rules, and applications.

Definition of Integers:

Integers are defined as a set of  positive numbers, negative numbers, and zero. The symbol used to denote integers is “Z.” integers set  can be written as:

    \[Z = \{ ....., - 3, - 2, - 1,0,1,2,3,.....\} \]

    \[\]

Types of Integers:

Integers can be classified into three types:

A. Zero (0): Zero is an integer that represents absence of quantity.

B. Positive Integers (Natural Numbers): Integers greater than zero are  Positive integers 

C. Negative Integers (Additive inverse of Natural Numbers):Integers less than zero and represent additive inverse of natural numbers are negative integers.

Integers on a Number Line:

Integers can be plotted on a horizontal number line that represents all possible values of integers. It has zero at center and positive integers to right side of zero and negative integers to left side of zero.

Understanding position of integers in relation to one another on number line is essential for performing integer operations.

Integer Operations:

The basic operations of arithmetic includes four, namely addition, subtraction,

division ,multiplication can be done on integers.

A. Addition of Integers:

Addition of integers involves adding two or more integers to find their sum. sum of two integers with same sign is always an integer with that sign itself, while sum of two integers with opposite signs is always an integer with sign of larger absolute value. For example, sum of -5 and -3 is -8, while sum of -5 and 3 is -2.

B. Rules of Integers in Addition:

i. We get a positive integer if we add two positive integers

ii. We get a negative  integer if we add two negative integers

iii. If  positive and a negative integer are added ,it  is equal to their difference, with sign of larger absolute value

C. Subtraction of Integers:

Subtraction of integers involves subtracting one integer from another to find their difference. difference of two integers with same sign is always an integer with sign of larger absolute value, while two integers differences with different signs is always an integer with sign of larger  value. Here difference between -5 and -3 is -2, while difference between -5 and 3 is -8.

D. Rules of Integers in Subtraction:

i. We get a  positive integer if we subtract two positive integers.

ii. We get a negative integer if we subtract two negative integers

iii. If we do subtraction  between a positive and a negative integer is equal to their sum, with sign of larger absolute value.

E. Multiplication of Integers:

Multiplication of integers involves multiplying two or more integers to find their product. Two integers product with same sign is always a positive integer, while two integers  product with opposite signs is always a negative integer. For example, product of -5 and -3 is 15, while product of -5 and 3 is -15

F. Rules of Integers in Multiplication:

i. If we do multiplication  of two positive integers,we get    a positive integer..

ii. If we do multiplication  of two negative integers ,we get  a positive integer.

iii. If we do multiplication of a positive and a negative integer ,we get   a negative integer

G. Division of Integers:

Division of integers involves dividing one integer by another to find their quotient. Division of integers is not closed under division, as quotient of two integers may not be an integer. When dividing integers, sign is important for integers to find sign of quotient. For example, quotient of -6 and -3 is 2, while quotient of -6 and 3 is -2.

H. Rules of Integers in Division:

i. If we do division of  of two positive integers ,we get quotient of   positive integer.

ii. If we do division of two negative integers,the quotient  is  a positive integer.

iiiIf we do division of a positive and a negative integer,the quotient  is  a negative integer

.

Properties of Integers:

Integers have several properties that are important to understand when performing integer operations. These properties include:

A. Closure Property:

The closure property of integers states that  difference, sum,quotient, and product of any two integers is always an integer.

B. Associative Property:

The associative property of integers states that grouping of two or more integers in an operation, there will be no of change result. 

    \[(2 + 3) + 4 = 2 + (3 + 4) = 9\]

is an example

C. Commutative Property:

The commutative property of integers states that when the order of two or more integers in an operation , there will be no change of result.  

    \[2 + 3 = 3 + 2 = 5\]

is an example

D. Distributive Property:

The distributive property of integers states that product of an integer and sum or difference of two or more integers is  sum or difference of products of integer and each individual integer. For example,

    \[2 \times (3 + 4) = (2 \times 3) + (2 \times 4) = 14\]

E. Additive Inverse Property:

The additive inverse property of integers states that every integer has an additive inverse, which is integer that, when added to original integer, results in zero. Additive inverse of 5 is -5, and additive inverse of -5 is 5.

F. Multiplicative Inverse Property:

The multiplicative inverse property of integers states that every nonzero integer has a multiplicative inverse, which is integer that, when multiplied by original integer, results in 1.  Multiplicative inverse of 5 is

    \[\frac{1}{5}\]

, and multiplicative inverse of -5 is

    \[ - \frac{1}{5}\]

.

G. Identity Property:

The identity property of integers states that sum of any integer and zero is equal to original integer, and product of any integer and one is equal to original integer. For example,

    \[5 + 0 = 5\]

, and

    \[5 \times 1 = 5\]

Rules of Integers:

There are several rules of integers that are important to understand when performing integer operations. These rules include:

A. we get a integer if we add two positive integer

B. . we get a integer if we add two negative integer

C. . we get a integer if we multiply two positive integer

D. . we get a integer if we multiply two negative integer

E.we get zero if we add an integer and its additive inverse 

F. we get 1 if we multiply an integer and its additive inverse 

Integers in Everyday Life:

Integers are used in a wide range of applications in everyday life. For example, integers are used in counting, measuring, and financial transactions. Understanding importance of integers in mathematical problem-solving and decision making is essential for making informed decisions. Integers are used in many fields such as economics, physics, computer science, and engineering, among others.

In economics, integers are used to represent quantities such as prices, profits, and taxes. For example, when calculating total cost of a product, price is multiplied by quantity, which is an integer.

In physics, integers are used to represent quantities such as distance, time, and velocity. For example, when calculating speed of an object,  distance  is divided by  time it took to travel that distance, which are both integers.

In computer science, integers are used to represent data such as memory addresses, file sizes, and time durations. For example, when allocating memory for a program, amount of memory needed is represented by an integer.

In engineering, integers are used to represent quantities such as force, mass, and distance. For example, when designing a bridge, weight of bridge must be considered, which is represented by an integer.

Real- life application of integers:

In context of temperature, a positive integer indicates a temperature above

    \[0^\circ \]

Celsius, while a negative integer indicates a temperature below

    \[0^\circ \]

Celsius. This is useful for understanding weather patterns and dressing appropriately for conditions.

In context of gain and loss, a positive integer represents a gain or increase in value, while a negative integer represents a loss or decrease in value. This is commonly used in finance and accounting to track revenue and expenses.

In context of vertical direction, a positive integer represents upward movement, while a negative integer represents downward movement. This is useful for measuring height or elevation.

In context of bank accounts, a positive integer represents a deposit or addition of funds, while a negative integer represents a withdrawal or subtraction of funds. This is important for managing finances and keeping track of account balances.

Finally, in context of acceleration, a positive integer means increase in velocity, while a negative integer means decrease in velocity. This is useful in physics and engineering for understanding motion of objects and predicting their behavior.

By understanding these different contexts and meanings of positive and negative integers in each, we can apply mathematical principles to solve problems and make informed decisions in a wide range of real-world situations

Conclusion:

Integers are a fundamental concept in mathematics that are used in a wide range of applications in everyday life. Understanding their properties and rules is essential for solving mathematical problems and making informed decisions. Integers are used in many fields such as economics, physics, computer science, and engineering, among others. Knowing how to perform integer operations and how to graph integers on a number line is crucial for understanding their position in relation to one another. rules and properties of integers, such as commutative property,  additive inverse property,closure property, multiplicative inverse property, associative property, distributive property, and identity property, are important to understand when performing integer operations. Knowing how to apply integers in real-world scenarios, such as counting, measuring, and financial transactions, is also crucial for understanding their importance in mathematical problem-solving and decision making.

Solved examples:

1. What is sum of -5 and 8?.

Ans: To find sum of -5 and 8, we simply add them together.

 Therefore,

    \[ - 5 + 8 = 3\]

 

The sum of -5 and 8 is 3.

2. What is product of -4 and -6?

Ans: To find product of -4 and -6, we simply multiply them together.  

Therefore,

    \[( - 4) \times ( - 6) = 24\]

The product of -4 and -6 is 24

3. What is difference between -10 and -3?

Ans: : To find difference between -10 and -3, we simply subtract them. 

Therefore,

    \[ - 10 - ( - 3) =  - 10 + 3 =  - 7\]

The difference between -10 and -3 is -7.

4. Find value of

    \[ - 6 - 3 + 8\]

Ans: To find value

    \[ - 6 - 3 + 8\]

,here -6 and -3 are both negative, their sum will be a negative number and then add 8 

    \[ - 6 - 3 + 8 = ( - 6) + ( - 3) + 8\]

    \[ =  - 9 + 8 =  - 1\]

Therefore, value of

    \[ - 6 - 3 + 8\]

is

    \[ - 1\]

5. Find quotient of -16 and 4.

Ans: quotient of

    \[( - 16) \div 4\]

  can be calculated as follows:

    \[( - 16) \div 4 =  - 4\]

 

Therefore, quotient of

    \[( - 16) \div 4\]

  is -4. 

FAQs

1: Are fractions considered integers?

Ans: No, fractions are not considered as integers. Integers are set of numbers including positive numbers, negative numbers, and zero.

2: What is the opposite of an integer?

Ans: The opposite of an integer is its additive inverse of an integer. For example, the opposite of 5 is -5, and the opposite of -3 is 3.

3: What is the difference between a negative integer and a positive integer?

Ans: A positive integer means an integer greater than zero, while a negative integer means an integer less than zero.

4: Can integers be decimals or fractions?

Ans: No, integers cannot be decimals or fractions. Integers are set of numbers including positive numbers, negative numbers, and zero.

5: What are some real-life examples of integers?

Ans: Integers are used in a wide range of real-life scenarios, such as counting objects, measuring distances, and financial transactions.

Reference:

S.Chand’s smart maths book

Written by

Prerit Jain

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