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Integers: Numbers, Definition, Examples, Rules, Formula, Meaning

Integers Meaning

Integers: In mathematics, Integers are collections of numbers that include Positive, and negative numbers along with zero. Integers are the primary unit of mathematics. Examples of integers are -60,-55,-5, -20,0,4. 8,19,15 etc.In this article, we are going to learn about Integers, their definition, Examples, their types, and Properties, and solve some problems based on Integers.

Integer Definitions for Class 7

The meaning of Integer is Intact or whole. Groupings of positive and negative numbers along with zero are known as integers. Integers do not contain fractions and decimals just like whole numbers.
Integers can be positive, negative, or zero,  Integers can be used for all arithmetic operations, including addition, subtraction, multiplication, and division.
In mathematics, Integers are represented with ” Z”
Examples of Integers are -39,-20,-12- 5, 1 , 8, 10,20 -12, and so on.
Also, it can be written as,
Z ={ ……-4,-3,-2,-1,0,1,2,3,4,……….}

Integer Meaning in Hindi

Integer नाम का मतलब बरकरार या संपूर्ण होता है। शून्य के साथ धनात्मक और ऋणात्मक संख्याओं के समूहन को पूर्णांक कहते हैं। पूर्णांकों में पूर्ण संख्याओं की तरह ही भिन्न और दशमलव नहीं होते हैं।
पूर्णांक धनात्मक, ऋणात्मक या शून्य हो सकते हैं, पूर्णांकों का उपयोग सभी अंकगणितीय संक्रियाओं के लिए किया जा सकता है, जिसमें जोड़, घटाव, गुणा और भाग शामिल हैं।
गणित में, पूर्णांकों को “Z” से दर्शाया जाता है
पूर्णांकों के उदाहरण हैं -39,-20,-12- 5, 1 , 8, 10,20 -12, इत्यादि।
साथ ही, इसे इस प्रकार भी लिखा जा सकता है,
Z ={……-4,-3,-2,-1,0,1,2,3,4,……….}

Integers Number: Types

In maths, three types of Integers are there. Integers can be positive, negative, or zero, Those are- Negative, Positive integers, and Zero. Let’s briefly know about them.

• Negative integers

Integers with a negative value are those that are less than zero. They exclude decimals and fractions.
Examples of negative integers are -40,-7,-8,5,-4, -3-2,-1
Z- {∞,…….-10,-9,-8,-7,-6,-5,-4,-3,-2,-1}

•Positive  Integers

Positive Numbers or also known as Natural Numbers are a set of numbers from 1 to infinity. Natural numbers do not include 0, negative integers (-1, -35, etc.), fractions, or decimals. The positive numbers from 1 to 10 are 1, 2, 3, 5, 6, 7, 8, 9, and 10.

Z+ = {1,2,3,4,5,6,7,8,9,10…∞}

• Zero – The value of Zero is Null. But these are the most vital numbers in all Integers. Zero is neither a part of Positive integers nor a part of Negative integers.
Although Zero belongs to the whole Number set .in mathematics whole number ser is represented by the symbol W, which is written as { 0, 1, 2, 3, …….}
W = {0,1,2,3,4,5,6,7,8,9,10…∞}

Integers on Number line

A straight line of numbers is shown visually as a number line. As earlier we know, there are three types of integers -Zero, positive integers, and Negative integers. They can be simply represented on a number line.

• The center of an integer on a number line is zero. It cannot be positive or negative.
• On the number line, Positive integers are on the right side of zero
• Negative integers that are placed on the left side of the zero on the number line.

Integers: Numbers, Definition, Examples, Rules, Formula_40.1
Integers on Number line

 

Integer Rules

For integers, the following rules are specified:

Rules Examples
The sum of two positive integers is always an integer. (+ 5)+ (+40 )= + 45
The sum of two negative integers, also be always an Integer. (-8) +(-6) = -15
An integer and its inverse have total of zero. 3+ (-3) =0
9 + (-9) = 0
An integer’s product with its reciprocal is always one [ 9 ×1/9 ] = 1
If there is no sign in front of an integer then it is considered positive. 20 denotes +20.
An integer has a positive absolute value |5| =|+5|  =  5.

Arithmetic Operations on Integers

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 the operations with examples.

1. Addition of Integers

The addition of two integers always results in an integer. But there is some rule while performing the Addition.
• while multiplying two absolute values of the two integers having the same sign, then write the sum of the integers using the sign indicated by the numbers.

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

For Example,
(+4) + (+7) = +11
(-6) + (-4) = -10
(-12)+ 3 = – 9

2. Subtraction of integers

We follow the below guidelines while subtracting two integers:

•Change the subtrahend’s sign to make the operation become an additional problem.
•Then apply the same integer addition rules to solve the problem.
For example,
Substrate, 20 – 7
= 20 – (+7)
= + 13
Substrate , – 35 – 7
= -35 – (+7)
= – 42

4. Multiplication of integers

The following guidelines are used when multiplying two integers:

Operation Results Examples
(+) × (-) 5× (-3) = -15
(-) × (+) (-5) × 3 = – 15
(-) × (-) + (-5) × (-3) = 15

4. Division of Integers

While the division of two integers follows the guidelines,

Operation Results Examples
(+) ÷ (+) + 21 ÷ 3 = 7
(+) ÷ (-) 21 ÷ (-3) = -7
(-) ÷ (+)  – (-21) ÷ 3 = – 7
(-) ÷ (-) + (-21) ÷ (-3) = 7

Integers: Properties

Major characteristics of integers include:

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

Additive Inverse Property

A number’s opposite number is the additive inverse of that number. If 9 is an integer, its additive inverse is -9.
When an integer is multiplied by its additive inverse, the result is zero.
The simple rule is to flip from the positive to negative numbers or vice versa.
As The 9’s Additive integer is -9 then the sum of them is equal to 0.
[9+(-9]= 0

Multiplicative Inverse Property

The word Inverse means the opposite. Here Multiplicative Inverse means the reciprocal of an integer. Eg – Integer 4, its reciprocal is 1/4.
The Multiplicative Inverse Property of Integers suggests that if we multiply an integer with the reciprocal of that number the result is always  One.
Integer 5, reciprocal 1/5
[ 5 ×1/5 ] = 1

Closure Property

The Closure property of integers indicates that the set is closed for the given math operation. When two integers are added or multiplied, only an integer is produced.

For any two integers a and b:

a + b = integer
a – b = integer
a × b = integer
a/b = Integer

Associative Property

Due to the associative Property of integers, when three or more numbers are added or multiplied, the resultant sum or product always comes out the same, regardless of how the numbers are arranged. Brackets are used to group the items in this sentence.
a × (b × c) = (a × b) × c

Commutative Property

According to the commutative property, If we move or swap the integers present in the operation then the solution will remain the same. No changes are there. But Subtraction and division are not affected by the property, which only applies to addition and multiplication.
Let’s suppose a and b are two integers.
a × b = b×a
a + b = b +a
Above both cases, the sum or the product will see although changes are made.

Distributive Property

When we distribute something, it means dividing that thing into equal parts.  The distributive property of integers says that
Let three integers A, B and C then,
A ( B+ C) = A× B + A× C

Example:  8 x ( 4 +  6) = 8x 4 + 8x 6

LHS = 8 x (4+6 ) = 8 x 10= 80

RHS = 8 x 4 + 8 x 6 = 32+ 48 = 80

Hence, LHS = RHS (proved)

Integers: solved Examples

Example 1. Prove, 10 ( 4 + 5) = 90
Solution- L.H.S- 10 ( 4 + 5)

= 10 x 9 =90= R.H.S ( Proved)

Example 2. Solve the following integer product:

  1. (+6) × (+12)
  2.  (30) × (6)
  3. (- 9) × (6)
  4. 5 × (-4)

Solution:

  1. (+6) × (+10) = +50
  2. (30) × (6) = 1800
  3. (- 9) × (6) = -54
  4. 16 × (-5) = -80

Example 3. Solve the following Problems

  • (-20) ÷ (4) = ?
  • (-8) × (-4) = ?
  • (169) ÷ (-13) = ?

Solution:

  • (-20) ÷ (4) = -5
  • (-8) × (-4)= 31
  • (169) ÷ (- 13) = -13

Related Post:

Integer Meaning in Maths: FAQs

Q. What do you mean by integers?

The meaning of Integer is Intact or whole. Groupings of positive and negative numbers along with zero are known as integers. Integers do not contain fractions and decimals just like whole numbers.Examples of Integers are -39,-20,-12- 5, 1 , 8, 10,20 etc

Q. What are the types of  Integers?
In maths, three types of Integers are there. Integers can be positive, negative, or zero, Those are- Negative, Positive integers, and Zero.

Q. Should I say decimals are called integers?
No. Decimals are not integers. An integer is a collection of positive integers, negative integers,s, and zero except for Decimals and fractions.

Q. Give some examples of Positive integers.
Some examples if positive intergers are – 50,77,157,8, 24,2999,456 etc.

Q. Give some examples of a negative integer.
Some examples of negative integers are -60,-55,-5, -20,-245 etc

Q. Explain the Multiplicative Inverse Property of an integer.
When an integer is multiplied by its additive inverse, the result is zero.
The straightforward rule is to reverse the positive and negative numbers.
As The 9’s Additive integer is -9 then the sum of them is equal to 0.
[9+(-9]= 0
Q. What are Consecutive integers?
Consecutive integers are those numbers that come after one another in a particular order.   For example-  the integers 2, 3, 4, and 5 are all Consecutive.

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