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**Chapter 1 – Rational Numbers**

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**Download chapter wise detailed NCERT Solutions for Class 8 Maths / CBSE Class 8 Maths NCERT Solutions**

**Chapter 1 – Rational Numbers**

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**Download chapter wise detailed NCERT Solutions for Class 7 Maths / CBSE Class 7 Maths NCERT Solutions**

**Chapter 1 – Integers****Chapter 2 – Fractions and Decimals****Chapter 3 – Data Handling****Chapter 4 – Simple Equations****Chapter 5 – Lines and Angles****Chapter 6 – Triangle and its properties****Chapter 7 – Congruence of Triangles****Chapter 8 – Comparing Quantities****Chapter 9 – Rational Numbers****Chapter 10 – Practical Geometry****Chapter 11 – Perimeter and Area**

**Download chapter wise detailed NCERT Solutions for Class 6 Maths / CBSE Class 6 Maths NCERT Solutions**

**Chapter 1 – Knowing our Numbers****Chapter 2 – Whole Numbers****Chapter 3 – Playing with Numbers****Chapter 4 – Basic Geometrical Ideas****Chapter 5 – Understanding Elementary Shapes****Chapter 6 – Integers****Chapter 8 – Decimals****Chapter 9 – Data Handling****Chapter 10 – Mensuration****Chapter 11 – Algebra****Chapter 12 – Ratio and Proportion**

**Download chapter wise detailed NCERT Solutions for Class 9 Maths / CBSE Class 9 Maths NCERT Solutions**

**Chapter 1 – Number System**

**Exercise 1.1 – Number System****Exercise 1.2 – Number System****Exercise 1.3 – Number System****Exercise 1.4 – Number System****Exercise 1.5 – Number System****Exercise 1.6 – Number System**

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**Download chapter wise detailed NCERT Solutions for Class 10 Maths / CBSE Class 10 Maths NCERT Solutions**

**Chapter 1 – Real Numbers**

**Exercise 1.1 – Real Numbers****Exercise 1.2 – Real Numbers****Exercise 1.3 – Real Numbers****Exercise 1.4 – Real Numbers**

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**Download chapter wise detailed Maths NCERT Solutions for Class 6, Class 7, Class 8, Class 9 and Class 10 **

**Chapter 1 – Real Numbers**

**Exercise 1.1 – Real Numbers****Exercise 1.2 – Real Numbers****Exercise 1.3 – Real Numbers****Exercise 1.4 – Real Numbers**

**Chapter 2 – Polynomials**

**Chapter 3 – Pair of Linear Equations in two Variables**

**Exercise 3.3 – Pair of Linear Equations in two Variables****Exercise 3.4 – Pair of Linear Equations in two Variables****Exercise 3.5 – Pair of Linear Equations in two Variables****Exercise 3.6 – Pair of Linear Equations in two Variables**

**Chapter 4 – Quadratic Equations**

**Exercise 4.1 – Quadratic Equations****Exercise 4.2- Quadratic Equations****Exercise 4.3- Quadratic Equations****Exercise 4.4- Quadratic Equations**

**Chapter 5 – Arithmetic Progressions**

**Exercise 5.1 – Arithmetic Progressions****Exercise 5.2 – Arithmetic Progressions****Exercise 5.3 – Arithmetic Progressions**

**Chapter 6 – Triangles**

**Chapter 7 – Coordinate Geometry **

**Chapter 8 – Introduction to Trignometry**

**Exercise 8.1 – Introduction to Trignometry****Exercise 8.2 – Introduction to Trignometry****Exercise 8.3 – Introduction to Trignometry****Exercise 8.4 – Introduction to Trignometry**

**Chapter 9 – Some Applications of Trignometry**

**Chapter 10 – Circles**

**Chapter 11 – Constructions**

**Chapter 12 – Area related to Circles**

**Exercise 12.1 – Area related to Circles****Exercise 12.2 – Area related to Circles****Exercise 12.3 – Area related to Circles**

**Chapter 1 – Number System**

**Exercise 1.1 – Number System****Exercise 1.2 – Number System****Exercise 1.3 – Number System****Exercise 1.4 – Number System****Exercise 1.5 – Number System****Exercise 1.6 – Number System**

**Chapter 2 – Polynomials**

**Exercise 2.1 – Polynomials****Exercise 2.2 – Polynomials****Exercise 2.3 – Polynomials****Exercise 2.4 – Polynomials****Exercise 2.5 – Polynomials**

**Chapter 3 – Coordinate Geometry**

**Chapter 4 – Linear Equations in two Variables**

**Exercise 4.1 – Linear Equations in two Variables****Exercise 4.2 – Linear Equations in two Variables****Exercise 4.3 – Linear Equations in two Variables****Exercise 4.4 – Linear Equations in two Variables**

**Chapter 5 – Introduction to Euclid’s Geometry**

**Chapter 6 – Lines And Angles**

**Chapter 8 – Quadrilaterals**

**Chapter 9 – Areas of Parallelograms And Triangles**

**Exercise 9.1 – Areas of Parallelograms And Triangles****Exercise 9.2 – Areas of Parallelograms And Triangles****Exercise 9.3 – Areas of Parallelograms And Triangles**

**Chapter 10 – Circles**

**Exercise 10.1 – Circles****Exercise 10.2 – Circles****Exercise 10.3 – Circles****Exercise 10.4 – Circles****Exercise 10.5 – Circles**

**Chapter 12 – Heron’s Formula**

**Chapter 15 – Probability**

**Chapter 1 – Rational Numbers**

**Chapter 2 – Linear Equation in one Variable**

**Exercise 2.1 – Linear Equation in one Variable****Exercise 2.2 – Linear Equation in one Variable****Exercise 2.3 – Linear Equation in one Variable****Exercise 2.4 – Linear Equation in one Variable****Exercise 2.5 – Linear Equation in one Variable****Exercise 2.6 – Linear Equation in one Variable**

**Chapter 3 – Understanding Quadrilaterals**

**Exercise 3.1 – Understanding Quadrilaterals****Exercise 3.2 – Understanding Quadrilaterals****Exercise 3.3 – Understanding Quadrilaterals****Exercise 3.4 – Understanding Quadrilaterals**

**Chapter 4 – Practical Geometry**

**Exercise 4.1 – Practical Geometry****Exercise 4.2 – Practical Geometry****Exercise 4.3 – Practical Geometry****Exercise 4.4 – Practical Geometry****Exercise 4.5 – Practical Geometry**

**Chapter 5 – Data Handling**

**Chapter 6 – Square and Square roots**

**Exercise 6.1 – Square and Square roots****Exercise 6.2 – Square and Square roots****Exercise 6.3 – Square and Square roots****Exercise 6.4 – Square and Square roots**

**Chapter 7 – Cube and Cube roots**

**Chapter 8 – Comparing Quantities**

**Exercise 8.1 – Comparing Quantities****Exercise 8.2 – Comparing Quantities****Exercise 8.3 – Comparing Quantities**

**Chapter 9 – Algebraic Expressions and Identities**

**Exercise 9.1 – Algebraic Expressions and Identities****Exercise 9.2 – Algebraic Expressions and Identities****Exercise 9.3 – Algebraic Expressions and Identities**

**Chapter 11 – Mensuration**

**Exercise 11.1 – Mensuration****Exercise 11.2 – Mensuration****Exercise 11.3 – Mensuration****Exercise 11.4 – Mensuration**

**Chapter 12 Exercise 12.1 – Exponents and Powers**

**Chapter 13 – Direct and Inverse Proportions**

**Chapter 14 – Factorisation**

**Chapter 15 – Introduction to Graphs**

**Exercise 15.1 – Introduction to Graphs****Exercise 15.2 – Introduction to Graphs****Exercise 15.3 – Introduction to Graphs**

**Chapter 1 – Integers****Chapter 2 – Fractions and Decimals****Chapter 3 – Data Handling****Chapter 4 – Simple Equations****Chapter 5 – Lines and Angles****Chapter 6 – Triangle and its properties****Chapter 7 – Congruence of Triangles****Chapter 8 – Comparing Quantities****Chapter 9 – Rational Numbers****Chapter 10 – Practical Geometry****Chapter 11 – Perimeter and Area****Chapter 12 – Algebraic Expressions**

**Chapter 1 – Knowing our Numbers****Chapter 2 – Whole Numbers****Chapter 3 – Playing with Numbers****Chapter 4 – Basic Geometrical Ideas****Chapter 5 – Understanding Elementary Shapes****Chapter 6 – Integers****Chapter 8 – Decimals****Chapter 9 – Data Handling****Chapter 10 – Mensuration****Chapter 11 – Algebra****Chapter 12 – Ratio and Proportion**

Maths Class 6 Maths Class 7 Maths Class 8

Coordinate geometry is one of the most important and exciting ideas of mathematics. In particular it is central to the mathematics students meet at school. It provides a connection between algebra and geometry through graphs of lines and curves.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.

Here +x indicates about positive x-axis and –x indicates negative x-axis.

Similarly +y indicates positive y-axis and –y indicates negative y-axis.

You have observed that the axes have divided the plane into four parts. These four parts are called as quadrants which are numbered as I, II, III and IV anticlockwise from +x. So the plane consists of the axes and quadrants. We call the plane, the Cartesian plane or the coordinate plane or xy plane and the axis are called as coordinate axis.

**Example 1**

In which quadrant does the point P(-2, 3) lie?

**Solution**

Coordinate axes divide the plane of the graph paper into four quadrants.

In Quadrant I, sign of coordinates are (+, +)

In Quadrant II, sign of coordinates are (-, +)

In Quadrant III, sign of coordinates are (-, -)

In Quadrant IV, sign of coordinates are (+, -)

Since, In Point P (-2, 3) x-coordinate is negative while y-coordinate is positive.

Hence, Point P (-2, 3) lie in Quadrant II.

**Example 2**

In which quadrant does the point Q (3, -4) lie?

**Solution**

Coordinate axes divide the plane of the graph paper into four quadrants.

In Quadrant I, sign of coordinates are (+, +)

In Quadrant II sign of coordinates are (-, +)

In Quadrant III sign of coordinates are (-, -)

In Quadrant IV sign of coordinates are (+, -)

Since, In Point Q (3, -4) x-coordinate is positive while y-coordinate is negative.

Hence, Point Q (3, -4) lie in Quadrant IV.

**Example 3**

In which quadrant does the point O (-3, -2) lie?

**Solution**

Coordinate axes divide the plane of the graph paper into four quadrants.

In Quadrant I, sign of coordinates are (+, +)

In Quadrant II sign of coordinates are (-, +)

In Quadrant III sign of coordinates are (-, -)

In Quadrant IV sign of coordinates are (+, -)

Since, In Point O (-3, -2) both x and y coordinate are negative.

Hence, Point O (-3, -2) lie in Quadrant III.

**Example 4**

In which quadrant does the point P(1, 4) lie?

**Solution**

Coordinate axes divide the plane of the graph paper into four quadrants.

In Quadrant I, sign of coordinates are (+, +)

In Quadrant II, sign of coordinates are (-, +)

In Quadrant III, sign of coordinates are (-, -)

In Quadrant IV, sign of coordinates are (+, -)

Since, In Point P (1, 4) both x and y coordinate are positive

Hence, Point P (1, 4) lie in Quadrant I.

We write the coordinates of a point using the following instruction.

1) The x-coordinates of a point is its perpendicular distance from y-axis measured along x-axis (Positive along the positive x-axis and Negative along the negative x-axis). For the point P it is 3. The **X-****coordinate is also called as the abscissa**.

2) The y coordinates of a point is its perpendicular distance from x-axis measured along the y axis (Positive along the positive y-axis and negative along the negative y-axis). For the point P that will be 4. **Y-coordinate is also known as the ordinate**.

3) In stating the coordinates of a point in the coordinate plane, the x-coordinate comes first and then the y-coordinate. We place the coordinates in the bracket i.e, **(3,4)**

**Example 1**

Write the coordinates of point P.

**Solution**

The point P is at 1 unit distance from y-axis and 3 units distance from x-axis. Therefore point P is (1,3)

**Example 2**

Write the coordinates of point B.

**Solution**

Point is at 2 unit distance from y-axis along positive x-axis so the x coordinate is 2 and from x-axis along negative y-axis the distance is 2 so the y-coordinate is -2. So the point B will be (2, -2).

**Example 3**

Write the coordinates of point Q.

**Solution**

The point is on y-axis so the distance from y-axis along the x-axis will be zero because it is on the y-axis. So the x coordinate will be zero. Similarly, the distance from x-axis along positive y-axis is 2 so the y-coordinate will be 2. So point Q is (0,2)

Let the coordinates of a point M be (-3, 2). We want to plot this point in the coordinate plane. We draw the coordinate axis and choose our units such that one centimeter represents one unit on both the axes. The coordinates of the point (-3, 2)tell us that the distance of this point from y axis along negative x-axis is 3 units and the distance of the point from the x-axis along the positive y-axis is 2 units. Starting from the origin we count 3 units on the negative x-axis and mark the corresponding point. Now starting from this point we move towards positive y axis and count 2 unit distance and mark the point. This point will be (-3, 2).

**Convert Mixed Fraction to Decimal – Example 1**

Express into Decimal.

**Solution**

Whenever, we have a mixed fraction, it would consist of two parts.

Whole Number Part = 5

Fraction Part =

The Whole Number Part would remain the same.

The Fraction Part should be converted into decimal

When we convert into decimal we would get 0.6

The Decimal value of the given fraction = Whole Number Part . Decimal value of Fraction Part

The Decimal value of the given fraction = 5.6

**Convert Mixed Fraction to Decimal – Example 2**

Express into Decimal.

**Solution**

Whenever, we have a mixed fraction, it would consist of two parts.

Whole Number Part = 6

Fraction Part =

The Whole Number Part would remain the same.

The Fraction Part should be converted into decimal

When we convert into decimal we would get 0.8

The Decimal value of the given fraction = Whole Number Part . Decimal value of Fraction Part

The Decimal value of the given fraction = 6.8

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**To Convert Decimal to Fraction we have to follow the following steps.**

**Step 1:** Write the given Decimal without the Decimal point as the numerator of the fraction.

**Step 2:** In the Denominator, write 1 followed by as many zeroes, as there are Decimal places in the given Decimal.

**Step 3:** Convert the fraction into simplest form.

**Convert Decimal to Fraction – Example 1**

Convert 0.8 into a fraction in its Simplest form?

**Solution**

Step 1: Write the given Decimal without the Decimal point as the numerator of the fraction 8

Step 2: Write 1 followed by as many zeroes, as there are Decimal places in the given Decimal 10 (Since there was 1 decimal place in the given number, we have added 1 zero at the end of 1 )

=

Step 3: Convert the fraction into simplest form

HCF of 8 and 10 is 2

To simplify this we have to divide both the numerator and denominator of the fraction by their HCF i.e,

8 and 10 by 2

In other words = =

Hence, 0.8 =

**Convert Decimal to Fraction – Example 2**

Convert 0.55 into a fraction in its Simplest form?

**Solution**

Step 1: Write the given Decimal without the Decimal point as the numerator of the fraction 55

Step 2: Write 1 followed by as many zeroes, as there are Decimal places in the given Decimal 100 (Since there was 2 decimal place in the given number, we have added 2 zero at the end of 1 )

=

Step 3: Convert the fraction into simplest form

HCF of 55 and 100 is 5

To simplify this we have to divide both the numerator and denominator of the fraction by their HCF i.e,

55 and 100 by 5

In other words = =

Hence, 0.55 =

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Types of Fraction:

- Proper Fraction
- Improper Fraction
- Mixed Fraction

– Convert Improper Fraction to Mixed Fraction

– Convert Mixed Fraction to Improper Fraction

- Fraction is a part of whole number.
- In generalise form for any two natural numbers ‘a’ and ‘b’ fraction is:

Where,

a = Numerator

b = Denominator

**Example 1**

Write down the Numerator and Denominator of the Fraction:

**Explanation**

Numerator is the upper part (Number on Top) of the Fraction

So, the Numerator of is 4

Denominator is the lower number ( Number at Bottom) of the Fraction

So, the Denominator of is 3

**Example 2**

Write down the Numerator and Denominator of the Fraction:

**Explanation**

Numerator is the upper part (Number on Top) of the Fraction

So, the Numerator of is 5

Denominator is the lower number ( Number at Bottom) of the Fraction

So, the Denominator of is 6

Proper fraction is a fraction which represents the part of whole number. In proper fraction numerator is always less than its denominator.

In generalise form for any two integer ‘a’ and ‘b’

where (a < b)

i.e Numerator < Denominator

**Example 1**

Is is a proper fraction?

**Explanation**

Proper fraction is a fraction which represents the part of whole number. In proper fraction numerator is always less than its denominator.

In generalise form for any two integer ‘a’ and ‘b’

where (a < b)

i.e Numerator < Denominator

Numerator = 2

Denominator = 3

Here 2 < 3

i.e, Numerator < Denominator

Hence, is a proper fraction

**Example 2**

Is is a proper fraction?

**Explanation**

Proper fraction is a fraction which represents the part of whole number. In proper fraction numerator is always less than its denominator.

In generalise form for any two integer ‘a’ and ‘b’

where (a < b)

i.e Numerator < Denominator

Numerator = 7

Denominator = 5

Here 7 > 5

i.e, Numerator > Denominator

Hence, is not a proper fraction

Improper fractions are the combination of whole number and a proper fraction. Improper fraction is opposite of proper fraction, since in improper fraction numerator is greater than its denominator

In generalise form for any two integer ‘a’ and ‘b’

where (a > b)

i.e Numerator > Denominator

**Example 1**

Is is a improper fraction?

**Explanation**

Improper fractions are the combination of whole number and a proper fraction. Improper fraction is opposite of proper fraction, since in improper fraction numerator is greater than its denominator

In generalise form for any two integer ‘a’ and ‘b’

where (a > b)

i.e Numerator > Denominator

Numerator = 17

Denominator = 9

Here, 17 > 9

i.e, Numerator > Denominator

Hence, is an improper fraction.

**Example 2**

Is is a improper fraction?

**Explanation**

Improper fractions are the combination of whole number and a proper fraction. Improper fraction is opposite of proper fraction, since in improper fraction numerator is greater than its denominator

In generalise form for any two integer ‘a’ and ‘b’

where (a > b)

i.e Numerator > Denominator

Numerator = 7

Denominator = 10

Here, 7 < 10

i.e, Numerator < Denominator

Hence, is not an improper fraction.

Mixed Fraction is a sum of whole number and proper fraction. Improper fraction can be converted into mixed fraction and mixed fraction can be converted into improper fraction with the help of simple multiplication, division and addition.

To Convert Improper Fraction into Mixed Fraction, we follow the steps given below:-

**Step I**: Divide the Numerator by the Denominator.

**Step II**: Write the Mixed Fraction as: ( Quotient + )

**Example 1**

Convert into Mixed Fraction.

**Explanation**

Given Fraction:

Divide the Numerator by the Denominator

On dividing 3 by 2

We get Quotient = 1

Remainder = 1

For a Mixed Fraction we have to write ( Quotient + )

So, the Mixed Fraction is ( 1 + ) =

**Example 2**

Convert into Mixed Fraction.

**Explanation**

Given Fraction:

Divide the Numerator by the Denominator

On dividing 5 by 3

We get Quotient = 1

Remainder = 2

For a Mixed Fraction we have to write ( Quotient + )

So, the Mixed Fraction is ( 1 + ) =

To Convert Mixed Fraction into Improper Fraction, we follow the steps given below:-

**Step 1**: Multiply the Whole Number Part and Denominator of the Mixed Fraction

**Step 2**: Add the Numerator to the product obtained in Step 1

**Step 3**: Write the sum as the numerator and denominator would remain the same, as in the Mixed Fraction.

**Example 1**

Convert the Fraction into an Improper Fraction

**Explanation**

A combination of Whole Number and a Proper Fraction is called a Mixed Fraction

Here,

Mixed Fraction =

Whole Number Part = 2

Numerator = 2

Denominator = 3

Step 1: Multiply the Whole Number Part and Denominator of the Mixed Fraction

2 x 3 = 6

Step 2: Add the Numerator to the product obtained in Step 1

2 + 6 = 8

Write the sum as the numerator and denominator would remain the same, as in the Mixed Fraction.

Hence, =

**Example 2**

Convert the Fraction into an Improper Fraction

**Explanation**

A combination of Whole Number and a Proper Fraction is called a Mixed Fraction

Here,

Mixed Fraction =

Whole Number Part = 3

Numerator = 1

Denominator = 2

Step 1: Multiply the Whole Number Part and Denominator of the Mixed Fraction

3 x 2 = 6

Step 2: Add the Numerator to the product obtained in Step 1

1 + 6 = 7

Write the sum as the numerator and denominator would remain the same, as in the Mixed Fraction.

Hence, =

**Equivalent Fractions Examples****Fraction in Simplest Form****Like Fractions and Unlike Fractions****Comparing Fractions****Adding Fractions****Subtracting Fractions****Multiplying Fraction****Dividing Fractions**

Maths Class 4Maths Class 5Maths Class 6Maths Class 7

Properties of Integers, deals with various concepts which are as under:-

__Closure Property of Integers____Commutative Property of Integers____Associative Property of Integers____Distributive Property of Integers____Additive Identity____Multiplicative Identity of Integers__

__Closure Property under Addition of Integers__

If we add any two integers, the result obtained on adding the two integers, is always an integer. So we can say, that integers are closed under addition.

Let us say ‘a’ and ‘b’ are two integers, either positive or negative. When we add the two integers, their result would always be an integer, i.e (a + b) would always be an integer.

**Example –**

State whether (– 11) + 2 is closed under addition

**Solution**

– 11 + 2

– 9

Since both -11 and 2 are integers, and their sum, i.e (-9) is also an integer, we can say that integers are closed under addition.

__Closure Property under Subtraction of Integers__

If we subtract any two integers the result is always an integer, so we can say that integers are closed under subtraction.

Let us say ‘a’ and ‘b’ are two integers either positive or negative, their result should always be an integer, i.e (a + b) would always be an integer.

**Example –**

State whether (24 – 12) is closed under subtraction

**Solution –**

24 – 12

12

Since both 24 and -12 are integers, and their difference, i.e (12) is also an integer, we can say that integers are closed under subtraction.

__Closure Property under Multiplication of Integers__:

If we multiply any two integers the result is always an integer, so we can say that integers are closed under multiplication.

Let us say ‘a’ and ‘b’ are two integers either positive or negative, and if multiply it, their result should always be an integer, i.e [(-a) x b] and [a x (–b)] would always be an integer.

**Example –**

Show that (-30) x 11 closed under multiplication

**Solution – **

– 30 x 11

-330

Since both -30 and 11 are integers, and their product, i.e (-330) is also an integer, we can say that integers are closed under multiplication.

__Closure Property under Division of Integers__:

If we divide any two integers the result is not necessarily an integer, so we can say that integers are not closed under division.

Let us say ‘a’ and ‘b’ are two integers, and if we divide them, their result ( a ÷ b ) is not necessarily an integer.

**Example – **

State whether (14) ÷ 5 is closed under division.

**Solution – **

(14) ÷ 5

8

Since both 14 and 5 are integers, but (14) ÷ 5 = 2.8 which is not an integer. Hence, we can say that integers are not closed under division.

__Commutative Property under Addition of Integers:__

- If we add two whole numbers say ‘a’ and ‘b’ the answer will always same, i.e if we add (2+3) = (3+2) = 5. So whole numbers are commutative under addition. Similarly if we apply this to integers, (-5+3) = (3+(-5))= -2, it also hold for all integers. So we can say that commutative property holds under addition for all integers.
- In generalise form for any two integers ‘a’ and ‘b’

a + b = b + a

**Example –**

Show that -32 and 23 follow commutative property under addition.

**Solution –**

L.H.S = -32 + 23 = – 9

R.H.S = 23 + (-32) = 23 – 32 = – 9

So, L.H.S = R.H.S, i.e a + b = b + a

This means the two integers follow commutative property under addition.

__Commutative Property under Subtraction of Integers__:

- On contradictory, commutative property will not hold for subtraction of whole number say (5 – 6) is not equal to (6 – 5). Let us consider for integers (4) and (-1), the difference of two numbers are not always same.

{4 – (-1) = 4 + 1= 5} and {(-1) – 4 = – 1 – 4 = -5}, so the difference of two integers are 5 and (-5) which are not equal so we can say that commutative property will not hold for subtraction of integers.

- In generalise form for any two integers ‘a’ and ‘b’

(a – b) ≠ (b – a)

**Example –**

Check whether -88 and 22 follow commutative property under subtraction.

**Solution –**

L.H.S = -88 – 22 = – 110

R.H.S = 22 – (-88) = 22 + 88 = 110

So, L.H.S ≠ R.H.S

This means the two integers do not follow commutative property under subtraction.

__Commutative Property under Multiplication of Integers__:

- If we multiply two whole numbers say ‘a’ and ‘b’ the answer will always same, i.e if we multiply (2×3) = (3×2) = 6. So whole numbers are commutative under multiplication. Similarly if we apply this to integers, (-5×3) = (3x(-5))= -6, it also hold true for all integers. So we can say that commutative property holds under multiplication for all integers.
- In generalise form for any two integers ‘a’ and ‘b’

a x b = b x a

**Example –**

Show that any two integers follow commutative property under multiplication.

Let us assume that the two integers are (-12) and 20,

L.H.S = (-12) x 20

= -240

R.H.S = 20 x (-12)

= -240

So, L.H.S = R.H.S, i.e ‘a x b’ = ‘b x a’

This means the two integers follow commutative property under multiplication.

__Commutative Property under Division of Integers__:

- Commutative property will not hold true for division of whole number say (12 ÷ 6) is not equal to (6 ÷ 12). Let us consider for integers say, (-14) and (7), the division of two numbers are not always same.

[(-14) ÷ 7 = -2] and [7 ÷ (-14) = -0.5}, so the result of division of two integers are not equal so we can say that commutative property will not hold for division of integers.

- In generalise form for any two integers ‘a’ and ‘b’

(a ÷ b) ≠ (b ÷ a)

**Example – **

State whether (-12) and (-3) follow commutative law under division?

**Solution – **

L.H.S = (-12) ÷ (-3) = 4

R.H.S = (-3) ÷ (-12) = 25

So, L.H.S ≠ R.H.S; i.e (a ÷ b) ≠ (b ÷ a)

This means the two integers do not follow commutative property under division.

__Associative Property under Addition of Integers__:

- As commutative property hold for addition similarly associative property also holds for addition.
- In generalize form for any three integers say ‘a’, ’b’ and ‘c’

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

**Example – **

Show that (-6), (-1) and (3) are associative under addition.

**Solution – **

L.H.S = – 6 + ( -1 + 3)

= – 6 + 2

= – 4

R.H.S = (- 6 + (-1)) + 3

= (- 6 – 1) + 3

= -7 + 3

= -4

So, L.H.S = R.H.S, i.e a + (b + c) = (a + b) + c

This means all three integers follow associative property under addition.

__Associative Property under Subtraction of Integers__:

- On contradictory, as commutative property does not hold for subtraction similarly associative property also does not hold for subtraction of integers.
- In generalize form for any three integers say ‘a’, ’b’ and ‘c’

a – (b – c) ≠ (a – b) – c

**Example –**

Check whether (-6), (-1) and 3 follow associative property under subtraction.

**Solution –**

L.H.S = -6 – (-1 -3)

= -6 – (-4)

= -6 +4

= -2

R.H.S = (-6 – (-1)) – 3

= (-6 +1) – 3

= -5 -3

= -8

So, L.H.S ≠ R.H.S

This means all three integers do not follow associative property under subtraction.

__Associative Property under Multiplication of Integers__:

- As commutative property hold true for multiplication similarly associative property also holds true for multiplication.
- The associative property of multiplication does not depend on the grouping of the integers.
- In generalize form for any three integers say ‘a’, ’b’ and ‘c’

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

**Example –**

Show that (-2), (-3) and 4 hold associative property for multiplication

**Solution – **

L.H.S = (-2) x ( -3 x 4)

= -2 x -12

= 24

R.H.S = (-2 x -3) x 4

= 6 x 4

= 24

So, L.H.S = R.H.S; i.e a x (b x c) = (a x b) x c

This means all three integers follow associative property under multiplication.

**Distributive properties of multiplication of integers are divided into two categories, **over** addition **and** over subtraction.**

1. Distributivity of multiplication over addition hold true for all integers.

In generalize form for any three integers say ‘a’, ’b’ and ‘c’

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

**Example –**

Show that (-2), 3 and 5 follow distributive property of multiplication over addition.

**Solution – **

L.H.S = -2 x (3 + 5)

= -2 x 8

= -16

R.H.S = (-2 x 3) + (-2 x 5)

= -6 + (-10)

= -6 -10

= -16

So, L.H.S = R.H.S; i.e a x (b + c) = (a x b) + (a x c)

This means that distributive property of multiplication over addition holds true for all integers.

2. Distributivity of multiplication over subtraction hold true for all integers.

In generalize form for any three integers say ‘a’, ’b’ and ‘c’

a x (b – c) = (a x b) – (a x c)

**Example – **

Show that (-5), (-4) and (-2) follow distributive property of multiplication over subtraction.

L.H.S = -5 x (-4 – (-2))

= -5 x (-4 + 2)

= -5 x -2

= 10

R.H.S = (-5 x -4) – (-5 x -2)

= 20 – 10

= 10

So, L.H.S = R.H.S; i.e a x (b – c) = (a x b) – (a x c)

This means that distributive property of multiplication over subtraction holds true for all integers.

When we add zero to any whole number we get the same number, so zero is additive identity for whole numbers. Similarly if we add zero to any integer we get the back the same integer whether the integer is positive or negative. In general for any integer ‘a’

a + 0 = 0 + a = a

- When we multiply any positive or negative integer by 1 the answer will remains same the integer only.
- In generalise form, for any integer ‘a’

a x 1 = 1 x a = a

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