Integers Class 7 Notes: Chapter 1

Introduction to Integers

Introduction to Numbers

Natural Numbers : The collection of all the counting numbers is called set of natural numbers. It is denoted by N = {1,2,3,4….}

Whole Numbers: The collection of natural numbers along with zero is called a set of whole numbers. It is denoted by W = { 0, 1, 2, 3, 4, 5, … }

For more information on Number Systems, watch the below video.

Properties of Addition and Subtraction of Integers

Closure under Addition and subtraction
For every integer $a$ and $b$,  $a+b$ and $a–b$ are integers.

Commutativity Property for addition
for every integer $a$ and $b$,  $a+b=b+a$

Associativity Property for addition
for every integer $a$,$b$ and $c$, $(a+b)+c=a+(b+c)$
To know more about Addition and Subtraction of Integers, visit here.

Additive Identity & Additive Inverse

Additive Identity
For every integer $a$, $a+0=0+a=a$ here 0 is Additive Identity, since adding 0 to a number leaves it unchanged.
Example : For an integer 2, 2+0 = 0+2 = 2.

Additive inverse
For every integer $a$, $a+(-a)=0$ Here, $-a$ is additive inverse of $a$ and $a$ is the additive inverse of-a.
Example : For an integer 2, (– 2) is additive inverse  and for (– 2), additive inverse is 2. [Since + 2 – 2 = 0]

Properties of Multiplication of Integers

Properties of Multiplication of Integers

Closure under Multiplication
For every integer $a$ and $b$, $a\times b=Integer$

Commutative Property of Multiplication
For every integer $a$ and $b$, $a\times b=b\times a$

Multiplication by Zero
For every integer $a$, $a\times 0=0\times a=0$

Multiplicative Identity
For every integer $a$, $a\times 1=1\times a=a$. Here 1 is the multiplicative identity for integers.

Associative property of Multiplication
For every integer $a$, $b$  and $c$,  $(a\times b)\times c=a\times (b\times c)$

Distributive Property of Integers
Under addition and multiplication,  integers show the distributive property.
i.e., For every integer $a$, $b$  and $c$,  $a\times (b+c)=a\times b+a\times c$

These properties make calculations easier.

To know more about “Properties of Multiplication of Integers”, visit here.

Division of Integers

Division of Integers

When a positive integer is divided by a positive integer, the quotient obtained is a positive integer.
Example: (+6) ÷ (+3) $=+2$

When a negative integer is divided by a negative integer, the quotient obtained is a positive integer.
Example: (-6) ÷ (-3) $=+2$

When a positive integer is divided by a negative integer or negative integer is divided by a positive integer, the quotient obtained is a negative integer.
Example: (-6) ÷ (+3) $=-2$ and Example: (+6) ÷ (-3) $=-2$
To know more about Multiplication and Division of Integers, visit here.

The Number Line

Number Line

Representation of integers on a number line

On a number line when we

(i) add a positive integer for a given integer, we move to the right.
Example : When we add +2 to +3, move 2 places from +3 towards right to get +5

(ii) add a negative integer for a given integer, we move to the left.
Example : When we add -2 to +3, move 2 places from +3 towards left to get +1

(iii) subtract a positive integer from a given integer, we move to the left.
Example: When we subtract +2 from -3, move 2 places from -3 towards left to get -5

(iv) subtract a negative integer from a given integer, we move to the right
Example: When we subtract -2 from -3, move 2 places from -3 towards right to get 1

To know more about Number Lines, visit here.

Addition and Subtraction of Integers

The absolute value of +7 (a positive integer) is 7
The absolute value of -7 (negative integer) is 7 (its corresponding positive integer)

Addition of two positive integers gives a positive integer.
Example : ($+\mathrm{3)+(}+\mathrm{4)}=+7$

Addition of two negative integers gives a negative integer.
Example : $(-3)+(-4)=-3-4=-7$

When one positive and one negative integers are added, we take their difference and place the sign of the bigger integer.
Example : $(-7)+\left(2\right)=-5$

For subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer.
Example : $56–(–73)=56+73=129$

Introduction to Zero

Integers

Integers are the collection of numbers which is formed by whole numbers and their negatives. 
The set of Integers is denoted by Z or I. I =  { …, -4, -3, -2, -1, 0, 1, 2, 3, 4,… }

Properties of Division of Integers

Properties of Division of Integers

For every integer $a,$
(a) a$÷$0 is not defined
(b) a$÷$1 $=a$

Note:  Integers are not closed under division
Example: (– 9) $÷$ (– 3) = 2. Result is an integer.
and $(-3)÷(-9)=\; 1/3$. Result is not an integer.

Multiplication of Integers

Multiplication of Integers

Product of two positive integers is a positive integer.
Example : $(+2)\times (+3)=+6$

Product of two negative integers is a positive integer.
Example :$(-2)\times (-3)=+6$

Product of a positive and a negative integer is a negative integer.
Example :$(+2)\times (-3)=-6$ and $(-2)\times (+3)=-6$

Product of even number of negative integers is positive and product of odd number of negative integers is negative.
These properties make calculations easier.