Integers Introduction: To score well in the exam, students must check out the Integers introduction and understand them thoroughly. The collection of negative numbers and whole numbers is known as an integer in mathematics. Integers, like whole numbers, do not contain the fractional portion. Integers can therefore be defined as numbers that can be positive, negative, or zero but not as fractions. We can carry out all arithmetic operations on integers, including addition, subtraction, multiplication, and division. Integer examples include 1, 2, 5, 8, -9, -12, etc. “Z” stands for an integer.

Let’s now go over the definition of integers, their symbol, types, operations, laws, and properties, and how to display them on a number line with numerous worked-out examples.

Integers Introduction: What are Integers?

The word integer came from the Latin word “Integer.” It means intact or whole. Integers are a unique set of zero, negative, and positive numbers.

Examples of Integers: – 1, 6, -12, 15.

Symbol

The symbol ‘Z’ represents integers. 

Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}

Types of Integers

Integers are of three types:

• Zero (0)
• Negative Integers (Additive inverse of Natural Numbers)
• Positive Integers (Natural numbers)

Define Zero

Zero is neither a negative nor a positive integer. Zero is a neutral number with no sign. Plus or Minus sign is not added with zero.

What are Positive Integers?

The positive integers are the natural numbers. The positive integers are also known as counting numbers. Z+ denotes these integers. On a number line, the positive integers can be found the right side of 0.

Z+ shows 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30..

What are Negative Integers?

The symbol for negative integers, which are the opposite of natural numbers, is Z-. On a number line, the negative integers are located to the left of zero.

Z– shows -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, -19, -20, -21, -22, -23, -24, -25, -26, -27, -28, -29, -30,…..

Integers Introduction: How Should Integers Be Displayed on a Number Line?

We can represent the three types of integers on a number line based on positive integers, negative integers, and zero because we have already discussed them.

The centre of an integer on a number line is zero. Positive integers can be found on the right side of zero, whereas those that are negative are on the left.

Integers Introduction: Rules of Integers

Along with the integers introduction, students must check out the rules defined for integers. They are as follows:

• Two negative integers are added together to form an integer.
• Two positive numbers are added together to create an integer.
• An integer’s inverse and sum are both equal to zero.
• A positive integer is the result of two positive integers.
• A positive integer is the result of two negative integers.
• An integer’s product with its reciprocal is one.

Arithmetic Operations on Integers

We can perform the basic mathematical operations on integers. They are as follows:

• Addition of integers
• Division of integers
• Multiplication of integers
• Subtraction of integers

Addition of Integers

If the two integers are added, and if those two integers have the same sign, add the absolute values, and add the same sign in the sum that is provided with the numbers.

Check the example below.

(+4) + (+7) = +11

(-6) + (-4) = -10

The absolute values of two numbers with different signs added should be subtracted, and the difference should be expressed using the sign of the number with the highest absolute value.

Check the example below.

(+6) + (-4) = +2

(-4) + (+2) = -2

Division of Integers

Similar to multiplication, there is a rule for dividing integers.

• The outcome is positive if the signs of the two integers match.
• The outcome is negative if the integers have different signs.

Here is an example below.

• (+6) ÷ (+2) = +3
• (-16) ÷ (+4) = -4

Division of Signs	Resulting sign	Examples
– ÷ +	_	-15 ÷ 3 = -5
+ ÷ –	–	15 ÷ -3 = -5
+ ÷ +	+	15 ÷ 3 = 5
– ÷ –	+	-15 ÷ -3 = 5

Multiplication of Integers

The formula for multiplying two integers is straightforward.

• The outcome is positive if the signs of the two integers match.
• The outcome is negative if the integers have different signs.

Check out the example below.

• (+2) x (+3) = +6
• (+3) x (-4) = – 12

Thus, using the examples in the table below, we may summarise the multiplication of two integers:

Multiplication of Signs	Resulting Sign	Examples
– × +	_	-3 × 4 = -12
+ × –	–	3 × -4 = -12
+ × +	+	3 × 4 = 12
– × –	+	-3 × -4 = 12

Subtraction of Integers

Change the sign of the second number being subtracted and stick to the addition rules when subtracting two integers.

Check out the example below.

(-7) – (+4) = (-7) + (-4) = -11

(+8) – (+3) = (+8) + (-3) = +5

Properties of Integers

Check out the major Properties of Integers below.

1. Closure Property
2. Commutative Property
3. Associative Property
4. Distributive Property
5. Identity Property
6. Multiplicative Inverse Property
7. Additive Inverse Property

Closure Property

When two integers are multiplied or added together, according to the closure property of integers, only an integer is produced. Let us assume a and b are integers. 

• a + b = integer
• a x b = integer

 Check out the examples below. 

• 2 x 5 = 10 (is an integer)
• 2 + 5 = 7 (is an integer)

Commutative Property

Let us assume a and b are two integers. According to the commutative property of integers

• a + b = b + a
• a x b = b x a

Examples are given below.

• 3 + 8 = 8 + 3 = 11
• 3 x 8 = 8 x 3 = 24

The commutative property, however, does not apply to the division and subtraction of integers.

Associative Property

If a, b and c are integers, as per the associative property

• a+(b+c) = (a+b)+c
• ax(bxc) = (axb)xc

Examples are available below.

• 2+(3+4) = (2+3)+4 = 9
• 2x(3×4) = (2×3)x4 = 24

As with commutativity, associativity only applies to integer addition and multiplication.

Distributive property

If a, b, and c are integers and have the distributive property of integers, then:

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

Check the example below.

How to prove 3 x (5 + 1) = 3 x 5 + 3 x 1?

RHS = 3 x 5 + 3 x 1 = 15 + 3 = 18

LHS = 3 x (5 + 1) = 3 x 6 = 18

Therefore, LHS = RHS

Hence, it is proved.

Additive Inverse Property

Let’s say a is an integer. According to the additive inverse property of integers, 

a + (-a) = 0

Therefore, -a is the additive inverse of integer a.

Multiplicative inverse Property

Let’s say a is an integer. According to the multiplicative inverse property of integers, 

a x (1/a) = 1

Therefore, integer a’s multiplicative inverse is 1/a.

Identity Property of Integers

Check out the elements of integers below.

• a+0 = a
• a x 1 = a

Example is -100,-12,-1, 0, 2, 1000, 989 etc…

We can also represent it as a set.

Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}

Applications of Integers

There are many practical uses for integers; they are more than just abstract numbers on a page. Positive and negative numbers have various effects in the real world. They typically represent two opposing circumstances.

For instance, positive numbers are used to represent temperatures above zero, whereas negative numbers are used to indicate temperatures below zero. They make it possible to compare and quantify two things, such as how big or little, how many or how few they are.

The ratings of movies or music, player scores in golf, football, and hockey competitions, the representation of credits and debits in banks, and other real-world scenarios all involve numbers.

Integers Introduction: Integers Examples

Check out some of the integer examples below.

Check out the 1st example.

Let us solve the following question.

• 5 + 3 = ?
• 5 + (-3) = ?
• (-5) + (-3) = ?
• (-5) x (-3) = ?

Here is the solution.

• 5 + 3 = 8
• 5 + (-3) = 5 – 3 = 2
• (-5) + (-3) = -5 – 3 = -8
• (-5) x (-3) = 15

Check out the 2nd example.

How to solve the following product of integers?

• (+5) × (+10)
• (12) × (5)
• (- 5) × (7)
• 5 × (-4)

Check the solution below.

• (+5) × (+10) = +50
• (12) × (5) = 60
• (- 5) × (7) = -35
• 5 × (-4) = -20

Check out the 3rd example.

How to solve the following division of integers?

• (-9) ÷ (-3)
• (-18) ÷ (3)
• (4000) ÷ (- 100)

Check the solution below.

• (-9) ÷ (-3) = 3
• (-18) ÷ (3) = -6
• (4000) ÷ (- 100) = -40

A Few Important Practice Questions on Integers Introduction

1. A positive integer is the sum of two positive integers. False or True?

2. What are the first five positive integers added together?

3. What is the first five positive odd-numbered integers’ product?

4. Draw a number line and plot the integers from -10 to +10.

Students who want to learn about the properties of integers and solve problems on this particular topic can download Embibe, the best learning App from Google Play Store. Many interactive videos are available there on this topic.

FAQs on Integers Introduction

Check out some of the frequently asked questions on integers.

Q1. What do integers mean?

A. The three elements that make up an integer are zero, the natural numbers, and their additive inverse. It can be shown on a number line but without the fractional portion, and Z stands for it.

Q2. Define an integer formula.

A. An integer is a collection of positive and negative numbers and zero, yet it lacks a mathematical formula.

Q3. Give us some integer examples.

A. Integers have instances like 3, 5, 0, 99, -45, etc.

Q4. Can Integers be negative?

A. Yes, negative integers are possible. Negative integers, such as -1,-2,-3,-4, and so on, are the additive inverse of natural numbers.

Q. Define the three categories of integers.

A. The three categories of integers are negative numbers, zero, and positive integers.

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