## Mathematics - Chapter 1 - Integers (Key Points)

Integers are a bigger collection of numbers which is formed by whole numbers and their negatives.

You have studied in the earlier class,  about the representation of integers on the number line and their addition and subtraction.

We now study the properties satisfied by addition and subtraction.
(a) Integers are closed for addition and subtraction both. That is, a + b and a – b are again integers, where a and b are any integers.

(b) Addition is commutative for integers, i.e., a + b = b + a for all integers a and b.

(c) Addition is associative for integers, i.e., (a + b) + c = a + (b + c) for all integers a, b and c.

(d) Integer 0 is the identity under addition. That is, a + 0 = 0 + a = a  for every integer a.

We studied, how integers could be multiplied and found that product of a  positive and a negative integer is a negative integer, whereas the product of two negative integers is a positive integer. For example, – 2 × 7 = – 14 and – 3 × – 8 = 24.

Product of even number of negative integers is positive, whereas the product of an odd number of negative integers is negative.

Integers show some properties under multiplication.

(a) Integers are closed under multiplication.  That is,  a  ×  b  is an integer for any two integers a and b.

(b) Multiplication is commutative for integers. That is, a × b = b × a for any integers a and b.

(c) The  integer  1  is  the  identity  under  multiplication,  i.e.,  1  ×  a  =  a  ×  1  =  a  for  any
integer a.

(d) Multiplication is associative for integers, i.e., (a × b) × c = a × (b × c) for any three integers a, b and c.

Under addition and multiplication, integers show a  property called distributive property. That is, a × (b + c) = a × b + a × c for any three integers a, b and c.

The properties of commutativity, associativity under addition and multiplication, and the distributive property help us to make our calculations easier.

We also learnt how to divide integers. We found that,

(a) When a positive integer is divided by a negative integer, the quotient obtained is a negative integer and vice-versa.

(b) Division of a negative integer by another negative integer gives a positive integer as a quotient.

For any integer a, we have the distributive property help us to make our calculations easier.

We also learnt how to divide integers. We found that,

(a) When a positive integer is divided by a negative integer, the quotient obtained is a negative integer and vice-versa.

(b) Division of a negative integer by another negative integer gives a positive integer as a quotient.

For any integer a, we have
a ÷  0 = not defined
a ÷  1 = a

Ch 1 - Integers (NCERT Ex 1.2)
Ch 1 - Integers (NCERT Ex 1.3)
Ch 1 - Integers (Key Points)

Ch 2 - Decimals - Multiplication and Division (Worksheet)
Ch 2 - Fractions and Decimals - NCERT Exercise Answers 2.6

Ch 3 - Data Handling (NCERT Ex 3.4)

CH 4 - Simple Equations (NCERT Ex 4.4)
CH 4 - Simple Equations (Worksheet)

CH6 - Lines and Angles (Worksheet)

CH8 - Comparing Quantities (Ratio and NCERT Ex 8.1)
CH8 - Comparing Quantities (Percentage and NCERT Ex 8.2)
CH8 - Comparing Quantities(NCERT Ex 8.3)

Chapter 12 - Algebraic Expressions (Revision Sheet)
Chapter 12 - Algebraic Expressions - NCERT Solution 12.1
Chapter 12 - Algebraic Expressions (Worksheet)

Chapter 13: Exponents and Powers (MCQs)  