##
Mathematics - Chapter 1 - Integers

(Key Points)

**⑴**Integers are a bigger collection of numbers which is formed by whole numbers and their negatives.

**⑵**You have studied in the earlier class, about the representation of integers on the number line and their addition and subtraction.

**⑶**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.

**⑷**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*.

**⑸**Product of even number of negative integers is positive, whereas the product of an odd number of negative integers is negative.

**⑹**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.

**⑺**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.

**⑻**The properties of commutativity, associativity under addition and multiplication, and the distributive property help us to make our calculations easier.

**⑼**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.

**⑽**For any integer a, we have the distributive property help us to make our calculations easier.

**⑾**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.

**⑿**For any integer a, we have

*a ÷ 0 = not defined*

*a ÷ 1 = a*

**☛See also:**

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)

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