Category: CBSE

CBSE Class 7 Maths Notes Chapter 6 The Triangle and its Properties Pdf free download is part of Class 7 Maths Notes for Quick Revision. Here we have given NCERT Class 7 Maths Notes Chapter 6 The Triangle and its Properties. https://www.cbselabs.com/the-triangle-and-its-properties-class-7-notes/

CBSE Class 7 Maths Notes Chapter 6 The Triangle and its Properties

Triangle And Its Properties Class 7 Notes Chapter 6

Triangle: A triangle is a simple closed curve made of three line segments. It has three vertices, three sides and three angles. Here in ΔABC, it has
Sides: \(\bar { AB }\), \(\bar { BC }\), \(\bar { CA }\).
Vertices: A, B, C.
Angles: ∠BAG, ∠ABC, ∠BCA.
The side opposite to the vertex A is BC. The angle opposite to the side AB is ∠BCA.

Classification of triangles based on sides

• A triangle having three unequal sides is called a scalene triangle.
• A triangle having two equal sides is called an isosceles triangle.
• A triangle having three equal sides is called an equilateral triangle.

Classification of triangles based on angles

• If each angle is less than 90°, then the triangle is called an acute-angled triangle.
• If anyone angle is a right triangle, then the triangle is called a right-angled triangle.
• If anyone angle is greater than 90°, then the triangle is called an obtuse-angled triangle.

Medians of a Triangle
The line-segment joining a vertex of a triangle to the mid-point of its opposite side is called a median of the triangle. Since there are three vertices in a triangle, therefore, a triangle has three medians.

Triangle And Its Properties Class 7 Notes Pdf

Altitudes of a Triangle
A line segment drawn from a vertex of a triangle perpendicular to its opposite side is called an altitude (height) of the triangle corresponding to the opposite side. Since there are three vertices in a triangle, therefore, a triangle has three altitudes.

Exterior Angle of a Triangle and its Property
An exterior angle of a triangle is formed when a side of a triangle is produced. At each vertex, we have two ways of forming an exterior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.

Class 7 Triangle And Its Properties Notes Chapter 6

Angle Sum Property of a Triangle
The sum of the measures of the three angles of a triangle is 180°.

Two Special Triangles: Equilateral and Isosceles
Equilateral Triangle: A triangle in which all three sides are of equal length is called an equilateral triangle.
In an equilateral triangle
(i) all sides have the same length and

(ii) each angle has measure 60°

Isosceles Triangle: A triangle in which two sides are of equal length is called an isosceles triangle.
In an isosceles triangle.
(i) two sides have the same length.

(ii) base angles opposite to the equal sides are equal.

Class 7 Maths Chapter 6 Notes

Sum of the Lengths of Two Sides of a Triangle
The sum of the lengths of any two sides of a triangle is greater than the third side.

Right-Angled Triangle and Pythagoras Property
A triangle whose one angle is a right-angle is called a right-angled triangle. The side opposite to the right angle is called the hypotenuse, the other two sides are known as the legs of the right-angled triangle.

Pythagoras Property: In a right-angled triangle, the square on the hypotenuse = sum of the squares on the legs. Conversely if the Pythagoras Property (holds) the triangle must be right-angled.

Triangle ∆ABC has

• three sides, AB, BC, CA.
• three angles, ∠BAC, ∠ABC, ∠BCA and also denoted by ∠A, ∠B, ∠C respectively.
• three vertices, A, B, C.

A triangle having all sides equal is called an equilateral triangle.

A triangle having two sides equal is called an isosceles triangle.

A triangle having all sides of different lengths is called a scalene triangle.

A triangle whose each angle measures less than 90° is called an acute-angled triangle.

A triangle one of whose angle measures 90° is called a right-angled triangle.

A triangle one of whose angle measures more than 90° is called an obtuse angled triangle.

Each angle of an equilateral triangle measures 60°.

The angles opposite to equal sides of an isosceles triangle are equal.

A scalene triangle has no equal angles.

A median connects a vertex of a triangle to the mid-point of the opposite side.

AD is the median of triangle ABC.

An altitude has one endpoint at a vertex of the triangle and the other on the line containing the opposite side.

AD is the altitude of triangle ABC.
∠ADB = ∠ADC = 90°.

A triangle has 3 medians and 3 altitudes.

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

∠ACD = ∠A + ∠B.

Angle Sum Property: The total measure of the three angles of a triangle is 180°.

∠A + ∠B + ∠C = 180°.
In right angled triangle, the square on the hypotenuse = sum of the squares on the legs.

AC² = AB² + BC²
It is known as Pythagoras Property.

We hope the given CBSE Class 7 Maths Notes Chapter 6 The Triangle and its Properties Pdf free download will help you. If you have any query regarding NCERT Class 7 Maths Notes Chapter 6 The Triangle and its Properties, drop a comment below and we will get back to you at the earliest.

CBSE Class 7 Maths Notes Chapter 2 Fractions and Decimals Pdf free download is part of Class 7 Maths Notes for Quick Revision. Here we have given NCERT Class 7 Maths Notes Chapter 2 Fractions and Decimals. https://www.cbselabs.com/fractions-and-decimals-class-7-notes/

CBSE Class 7 Maths Notes Chapter 2 Fractions and Decimals

Fractions And Decimals Class 7 Notes Chapter 2

The number of the \(\frac { a }{ b }\), where a and b are natural numbers is known as fraction.
e.g. \(\frac { 3 }{ 5 }\) is a fraction, where 3 is numerator and 5 is denominator.

A fraction whose numerator is less than its denominator is called a proper fraction.

A fraction whose numerator is more than or equal to its denominator is called an improper fraction.

A number which can be expressed as the sum of a natural number and a proper fraction is called a mixed fraction.

Fractions having the same denominator but different numerators are called like fractions.

Fractions having different denominator are called, unlike fractions.

To multiply a whole number with a proper or an improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator same.

To multiply a mixed fraction to whole numbers, first convert the mixed fraction to an improper fraction and then multiply.

The value of the product of two proper fractions is smaller than each of the two fractions.

The value of the product of two improper fractions is more than each of the two fractions.

To divide a whole number by any fraction, multiply that whole number by the reciprocal of that fraction.

To divide two fractions, multiply the first fraction by the reciprocal of the second fraction.

The non – zero numbers whose product with each other is 1, are called the reciprocals of that number.
e.g. Reciprocal of \(\frac { 5 }{ 9 }\) is \(\frac { 9 }{ 5 }\)

While dividing a whole number by a mixed fraction, first convert the mixed fraction into an improper fraction and then solve.

Place – value expansion of decimal number.
e.g. 253.417

When a decimal number is multiplied by 10, 100 or 1000, the digits in the product are same as in the decimal number but the decimal point in the product is shifted to the right by as, many of places as there are zeroes over one.
e.g. 0.07 × 10 = 0.7
0.07 × 100 = 7
0.07 × 1000 = 70

To multiply two decimal numbers, first, multiply them as whole numbers. Count the number of digits to the right of the decimal point in both the numbers. Add the number of digits counted. Put the decimal point in the product by counting the number of digits equal to the sum obtained from its rightmost place.
e.g. 1.2 × 2.43 = 2.916.

While dividing a number by 10,100 or 1000, the digits of the number and the quotient are same but the decimal point in the quotient shifts to the loft by as many places as there are zeroes over 1.
e.g. 2.38 ÷ 10 = 0.238
2.38 ÷ 100 = 0.0238
2.38 ÷ 1000 = 0.00238

While dividing one decimal number by another, first shift the decimal points to the right by equal number of places in both, to convert the divisor to a natural number and then divide.
e.g. \(\frac { 1.44 }{ 1.2 }\) = \(\frac { 14.4 }{ 12 }\) = 1.2

To divide a decimal number by a whole number, we first divide them as whole numbers. Then place the decimal point in the quotient as in the decimal number.
e.g. \(\frac { 8.4 }{ 4 }\) = 2.1

Proper Fraction: A proper fraction is a number that represents a part of a whole.
For example: \(\frac { 4 }{ 7 }\) is a proper fraction. Here, 4 is the numerator and 7 is the denominator.
In a proper fraction, the denominator is bigger than the numerator.

Improper Fraction: An improper fraction is a combination of whole and a proper fraction.
For example: \(\frac { 7 }{ 4 }\) is an improper fraction. Here, 7 is the numerator and 4 is the denominator.
In an improper fraction, the numerator is bigger than the denominator.

Mixed Fraction: An improper fraction can be expressed as a mixed fraction by dividing the numerator by denominator to obtain the quotient and the remainder. Then, the mixed fraction will be written as

Fractions And Decimals Class 7 Notes Pdf Chapter 2

Multiplication of Fractions
There arise many situations when we have to multiply two fractions. For example: to calculate the area of a rectangle with given length and breadth.

Multiplication of a Fraction by a Whole Number
To multiply a whole number with a proper or an improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator same.

Multiplication of a mixed fraction to a whole number: To multiply a mixed fraction by a whole number, we first convert the mixed fraction to an improper fraction and then multiply.

Fraction as an operator ‘of’ indicates multiplication.

Product of two whole numbers: The product of two whole numbers is bigger in size than each of the two whole numbers. For example, 3 x 4 = 12 and 12 > 4, 12 > 3.

Fraction And Decimals Class 7 Notes Chapter 2

Product of two proper fractions: When two proper fractions are multiplied, the product is less than both the fractions or, we say the size of the product of two proper fractions is smaller than the size of each of the two fractions.
Examples: We have,

Product of two Improper Fractions: The product of two improper fractions is greater than each of the two fractions. In this case, the size of the product is more than the size of each of the two fractions.
Examples: We have,

Division of Fractions
There arise situations when we have to divide a whole number by a fraction or a fraction by another fraction. For example: to cut a paper strip of given length into smaller strips of a certain length.

Notes On Fractions And Decimals Class 7 Chapter 2

Division of the Whole Number by a Fraction
Reciprocal of a fraction
Observe these products:

To divide a whole number by a fraction
To divide a whole number by any fraction, we multiply that whole number by the reciprocal of that fraction. Thus,

Class 7 Fractions And Decimals Notes Chapter 2

To divide a whole number by a mixed fraction
To divide a whole number by a mixed fraction, we first convert the mixed fraction into an improper fraction and then perform the division. Thus,

Division of a Fraction by a Whole Number
To divide a fraction by a whole number, we multiply that fraction by the reciprocal of that whole number. Thus,

Notes On Fractions Class 7 Chapter 2

To divide a mixed fraction by a whole number
While dividing mixed fractions by whole numbers, we convert the mixed fractions into improper fractions. Thus,

Division of a Fraction by Another Fraction

So, to divide a fraction (dividend) by a fraction (divisor), we multiply the fraction (dividend) by the reciprocal of the fraction (divisor).

Fractions And Decimals Class 7 Pdf Chapter 2

Decimal number and its expanded form

Using this table, we can write a decimal number in its expanded form also. For example:

So, a decimal number and its expanded form can be written if the place values of the digits are known to us.

Comparison of decimal numbers: To compare the decimal numbers, we first compare the digits on the left of the decimal point, starting from the leftmost digit.
If all the digits on the left of the decimal point are exactly the same, then we compare the digits on the right of the decimal point starting from the tenths place.
If the digits at the tenths place are also the same, then we compare the digits at the hundredths place and so on.
Example:
John has ₹ 15.50 and Salma has ₹ 15.75.
To find who has more money, we shall compare the two decimal numbers 15.50 and 15.75.
Here, both the digits 1 and 5, on the left of the decimal point are the same.
Also, comparing digits at the tenths place, we find that 5 < 7. So, Salma has more money than John.

Need of decimals: While converting lower units of money, length, and weight, to their higher units, we are required to use decimals. For example,
3 paise = ₹ 0.03
5 g = 0.005 kg
7 cm = 0.07 m.

Addition and subtraction of decimals: We are already familiar with how to add and subtract decimals.
Thus, 21.36 + 37.35 is 58.71

Multiplication of Decimal Numbers by 10, 100 and 1000
(i) Decimal numbers as a fraction with denominator 10 or 100 or 1000, etc.
We observe that,
2.3 = \(\frac { 23 }{ 10 }\) whereas 2.35 = \(\frac { 235 }{ 100 }\) Thus, depending upon the position of the decimal point, the decimal number can be converted to a fraction with denominator 10 or 100 or 1000, etc.

(ii) Multiplication of a decimal by 10 or 100 or 1000
Let us look at the following table:
1.7 × 10 = 17.0 or 17
1.7 × 100 = 170.0 or 170
1.7 × 1000 = 1700.0 or 1700

2.35 × 10 = 23.5
2.35 × 100 = 235.0 or 235
2.35 × 1000 = 2350.0 or 2350

We observe that in 1.7 × 10 = 17.
The digits in the number 1.7 and product 17 are same,
i.e., 1 and 7 and 0 is decimal in 17,
i.e., 17.0 is placed one digit to the right as compared to its placement in 17.
Note that 10 has one zero.
The observation is the same in the other products involving 10, in the table.

In 1.7 × 100 = 170, the digits in 1.7 (i.e., 1.70) and 170 are same,
i.e., 1, 7 and 0.
The decimal in the product 170 (i.e., 170.0) is placed two digits to the right as compared so its position in 1.70.
Note that 100 has two zeros.
The observation is the same in the other products involving 100 in the table.

In 1.7 × 1000 = 1700, the digits in 1.7 (i.e., 1.700) and 1700 are same,
i.e., 1, 7 and 0.
The decimal in the product 1700 (i.e., 1700.0) is placed three digits to the right as compared to its position in 1.700.
Note that 1000 has three zeros.
The observation is the same in the other products involving 1000 in the table.

Result: When we multiply a number by 10 or 100 or 1000, we get a product in which the digits are the same as in the number itself, but the decimal is shifted to the right by one digit (for multiplication by 10), by two digits (for multiplication by 100) and by three digits (for multiplication by 1000).
Thus, 1.92 × 10 = 19.2
1.92 × 100 = 192
1.92 × 1000 – 1.920 × 1000 = 1920.

We hope the given CBSE Class 7 Maths Notes Chapter 2 Fractions and Decimals Pdf free download will help you. If you have any query regarding NCERT Class 7 Maths Notes Chapter 2 Fractions and Decimals, drop a comment below and we will get back to you at the earliest.

CBSE Class 7 Maths Notes Chapter 1 Integers Pdf free download is part of Class 7 Maths Notes for Quick Revision. Here we have given NCERT Class 7 Maths Notes Chapter 1 Integers. https://www.cbselabs.com/integers-class-7-notes/

CBSE Class 7 Maths Notes Chapter 1 Integers

Class 7 Maths Chapter 1 Integers Notes

Representation of integers on the number line.

Integers are closed under addition. In general, for any two integers a and b, a + b is an integer.

Integers are closed under subtraction. Thus, if a and b are two integers then a – b is also an integer.

Addition is commutative for integers. In general, for any two integers a and b, we can say a + b = b + a

Subtraction is not commutative for integers.

Addition is associative for integers.

In general for any integers a, b and c, we can say a + (b + c) = (a + b) + c

Zero is an additive identity for integers. In general, for any integer a
a + 0 = a = 0 + a

While multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (-) before the product. We thus get a negative integer. In general, for any two positive integers a and b we can say a × (-b) = (-a) × b = -(a × b)

Product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put positive sign before the product. In general, for any two positive integers a and b, (-a) × (-b) = a × b

Integers are closed under multiplication. a × b is an integer, for all integers a and b,

Multiplication is commutative for integers. In general, for any two integers a and b, a × b = b × a

The product of a negative integer and zero is zero a × 0 = 0 × a=0

1 is the multiplicative identity for integers.
a × 1 = 1 × a = a

Multiplication is associative for integers, (a × b) × c = a × (b × c)

The distributivity of multiplication over addition is true for integers.
a × (b + c) = a × b + a × c

The distributivity of multiplication over subtraction is true for integers.
a × (b – c) = a × b – a × c

When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (-) before the quotient.
a ÷ (-b) = (-a) ÷ b where b ≠ 0

When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+).
(-a) ÷ (-b) = a ÷ b where b ≠ 0

Any integer divided by 1 gives the same number.
a ÷ 1 = a

For any integer a, we have a ÷ 0 is not defined.

Natural numbers, whole numbers and integers: The numbers 1, 2, 3,……… which we use for counting are known as natural numbers. The natural numbers along with zero forms the collection of whole numbers.
The numbers……., -3, -2, -1, 0, 1, 2, 3, form the collection of integers.

Integers	Whole numbers
1. The integers form a bigger group which contains whole numbers and negative numbers.	1. The whole numbers do not form a group as big as integers because they do not contain negative numbers.
2. The group of integers includes all the whole numbers.	2. The group of whole numbers does not include all the integers.
3. There is no smallest integer.	3. 0 is the smallest whole number.
4. Integers are closed under subtraction.	4. Whole numbers are not closed under subtraction.

In this chapter, we shall learn more about integers, their properties and operations.

Integers Class 7 Notes Chapter 1

Properties of Addition and Subtraction of Integers
Closure Under Addition
We know that the addition of two whole numbers is again a whole number. For example, 17 + 24 = 41 which is a whole number. This property is known as the closure property for the addition of whole numbers.

This property is true for integers also, i.e., the sum of two integers is always an integer. We cannot find a pair of integers whose addition is not an integer. Since additions of integers give integers, we can say integers are closed under’addition just like whole numbers. In general, for any two integers a and b, a + b is also an integer.

Integers Notes For Class 7 Chapter 1

Closure Under Subtraction
If we subtract two integers, then their difference is also an integer. We cannot find any pair of integers whose difference is not an integer. Since subtraction of integers gives integers, we can say integers are closed under subtraction. In general, for any two integers a and b, a – b is also an integer.

Note: The whole numbers do not satisfy this property.
For example: 5 – 7 = -2 which is not a whole number.

Commutative Property
Commutativity of Addition: We know that 3+ 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In other words, addition is commutative for whole numbers. Similarly, the addition is commutative for integers.
We cannot find any pair of integers for which the sum is different when the order is changed. So, we conclude that addition is commutative for integers also. In general, for any two integers a and b, we can say that a + b = b + a.

7th Standard Maths Notes Integers Chapter 1

Commutativity of Subtraction: We know that the subtraction is not commutative for whole numbers.
For example, 10 – 20 = -10 and 20 – 10 = 10
So, 10 – 20 ≠ 20 – 10
Similarly, the subtraction is not commutative for integers.

Associative Property
We cannot find any example for which sum is different when the order of addition is changed. This shows that addition is associative for integers.
In general, for any integers a, b and c, we can say that a + (b + c) = (a + b) + c

Additive Identity
When we add zero to any whole number {i.e., zero and positive integer), we get the same whole number. So, zero is an additive identity for whole numbers. In particular, we can say that zero is an additive identity for positive integers.
Consider the following examples:
(-8) + 0 = -8
(-23) + 0 = -23
0 + (-37) = -37
0 + (-59) = -59
0 + (-43) = -43
-61 + 0 = -61
-50 + 0 = -50
These examples show that zero is an additive identity for negative integers also. Thus, we can say that zero is an additive identity for integers. In general, for any integer a, a + 0 = a = 0 + a

Class 7 Integers Notes Chapter 1

Product of Three or More Negative Integers
We find that if the number of negative integers in a product is even, the product is a positive integer; if the number of negative integers in a product is odd, the product is a negative integer.

Properties of Multiplication of Integers
Closure Under Multiplication
Closure: Let us observe the following table:

We observe that the product of two integers is an integer. We cannot find a pair of integers whose product is not an integer. This gives an idea that the product of two integers is again an integer. So, we say that integers are closed under multiplication. In general, a × b is an integer, for all integers a and b.

Notes On Integers Class 7 Chapter 1

Commutativity of Multiplication
We know that multiplication is commutative for whole numbers (i.e., zero and positive
integers). Now, let us observe the following table:

We observe that two integers can be multiplied in any order. The above examples suggest commutativity of multiplication of integers. So, in general, we can say that for any two integers a and b, a × b = b × a.

Multiplication by Zero
We know that any whole number [i.e., zero and positive integers] multiplied by zero gives zero. Let us observe the following table showing the product of a negative integer and zero.
(-3) × 0 = 0
0 × (-4) = 0
(-5) × 0 = 0
0 × (-6) = 0
This table shows that the product of a negative integer and zero is again zero.
In general, for any integer a, a × 0 = 0 × a = 0

Definition Of Integers For Class 7 Chapter 1

Multiplicative Identity
We know that 1 is the multiplicative identity for whole numbers (i.e., zero and positive integers). Let us observe the following table showing the product of a negative integer and 1.
(-3) × 1 = -3
(-4) × 1 = -4
1 × (-5) = -5
1 × (-6) = -6
This table shows that 1 is the multiplicative identity for negative integers also. In general, for any integer a, we have,
a × 1 = 1 × a = a

Integers Class 7 Notes Pdf Chapter 1

Multiplication with (-1): Let us observe the following table showing the product of an integer and (-1).
(-3) × (-1) = 3
3 × (-1) = – 3
(-6) × (-1) = 6
(-1) × 13 = -13
(-1) × (-25) = 25
18 × (-1) = -18.
This table shows that (-1) is not the multiplicative identity for integers because when we multiply an integer with (-1) or (-1) with an integer, the result is the integer with the sign changed, i.e., we do not get the same integer. Therefore, for any integer a, we have, a × (-1) = (-1) × a = -a ≠ a

Note: 0 is the additive identity whereas 1 is the multiplicative identity for integers. We get additive inverse of an integer a when we multiply (-1) to a,
i.e., a × (-1) = (-1) × a = -a.

Integers Notes Class 7 Chapter 1

Associativity for Multiplication
Take the integer (- 3). Multiply it with (- 2) to get 6, i.e., (-3) × (-2) = 6.
Then, multiply the product 6 with 5 to get 30, i.e., [(-3) × (-2)] × 5 = 6 × 5 = 30.
Also, (-2) × 5 = (-10).
Multiply integer (-3) with (-10) to get 30.
i.e., (-3) × [(-2) × 5] = (-3) × (-10) = 30.
So, we get the same answer in both the processes, i.e., we get [(-3) × (-2)] × 5 = (-3) × [(-2) × 5]
We observe that the arrangement of integers does not affect the product of integers.
In general, for any three integers a, b and c, (a × b) × c = a × (b × c)
Thus, like whole numbers, the product of three integers does not depend upon the arrangement of integers and this is called associative property for multiplication of integers.

Notes Of Integers Class 7 Chapter 1

Distributive Property
(i) Distributivity of Multiplication Over Addition: We know that the property of distributivity of multiplication over addition is true for whole numbers.
For example: 16 × (10 + 2) = (16 × 10) + (16 × 2).

(ii) Distributivity of Multiplication Over Subtraction: We know that the property of distributivity of multiplication over subtraction is true for whole numbers (i.e. zero and positive integers).
For example: 4 × (3 – 8) = 4 × 3 – 4 × 8
This property is also true for integers.
For example:
(-9) × [10-(-3)] = (-9) × 13 = -117
and, -9 × 10 – (-9) × (-3) = -90 – 27 = -117
So, (-9) × [10-(-3)]=(-9) × 10 – (-9) × (-3).
We find that these are also equal.
In general, for any three integers a, b and c, a × (b – c) = a × b – a × c.

Division of Integers
1. The division is the inverse operation of multiplication.

Observing the entries in the above table, we find that

• When we divide a negative integer by a positive integer, we get a negative integer.
• When we divide a positive integer by a negative integer, we get a negative integer.
• When we divide a negative integer by a negative integer, we get a positive integer.

2. Division of a negative integer by a positive integer
We observe that
(-12) ÷ 6 = -2 = -(12 ÷ 6)
(-32) ÷ 4 = -8 = -(32 ÷ 4)
(-45) ÷ 5 = -9 = -(45 ÷ 5)
(-12) ÷ 2 = -6 = -(12 ÷ 2)
(- 20) ÷ 5 = -4 = -(20 ÷ 5)
So, we find that while dividing a negative integer by a positive integer, we divide them as whole numbers and put a minus sign (-) before the quotient (i.e. we get a negative integer).

3. Division of a positive integer by a negative integer
We also observe that
72 ÷ (- 8) = -9 = – (72 ÷ 8)
21 ÷ 7 = -3 = -(27 ÷ 7)
This shows that while dividing a positive integer by a negative integer, we divide them as whole numbers and put a minus sign (-) before the quotient (i.e., we get a negative integer).

4. If the dividend and divisor are of opposite sign, then the quotient is negative integer.
Wehave, (—48) ÷ 8= -(48 ÷ 8) = -6
(48) ÷ (-8) = -(48÷8) = -6
So, (-48) ÷ 8 = -6 = 48 ÷ (-8)

5. Division of a negative integer by a negative integer
Lastly we observe that
(-20) ÷ (-4) = 5 = 20 ÷ 4
(-12) ÷ (-6) = 2 = 12 ÷ 6
(-32) ÷ (-8) = 4 = 32 ÷ 8
(-45) ÷ (-9) = 5 = 45 ÷ 9
Here, we notice that while dividing a negative integer by a negative integer, we divide them as whole numbers and put a positive sign i.e. we get a positive integer. We can say that if dividend and divisor are of same signs, then the quotient is a positive integer.

Properties of Division of Integers
(i) Closure: We know that integers are closed under addition, subtraction and multiplication. However, the integers are not closed under division. It can be observed from the following table:

(ii) Commutativity: We know that division is not commutative for whole numbers. For example 16 ÷ 4 ≠ 4 ÷ 16.
Similarly, the division is not commutative for integers.
Note: The division is commutative for integers when the dividend and divisor are equal.

(iii) Like whole numbers, any integer divided by zero is meaningless and zero divided by any integer (other than zero) is equal to zero, i.e., for any integer a, a + 0 is not defined but 0 ÷ a (≠0) = 0.

(iv) When we divide a whole number (i.e., zero and positive integers) by 1, it gives the same whole number.
It is true for negative integers also. For example:
(-8) ÷ 1 = -8
(-11) ÷ 1 = -11
These examples show that negative integer divided by one gives the same negative integer. So, any integer divided by 1 gives the same integer. In general, we can say that for any integer a, a ÷ 1 = a.

We hope the given CBSE Class 7 Maths Notes Chapter 1 Integers Pdf free download will help you. If you have any query regarding NCERT Class 7 Maths Notes Chapter 1 Integers, drop a comment below and we will get back to you at the earliest.

Class 6 Maths Notes students can refer to the Data Handling Class 6 Notes Maths Chapter 9 https://www.cbselabs.com/data-handling-class-6-notes/ Pdf here. They can also access the CBSE Class 6 Data Handling Chapter 9 Notes while gearing up for their Board exams.

CBSE Class 6 Maths Notes Chapter 9 Data Handling

Data Handling Class 6 Notes Chapter 9

In our day to day life, we see various kinds of tables which consist of numbers, figures, names, etc. These tables provide ‘Data’. Data is a collection of numbers gathered to give some information.

Recording Data
Sometimes some information is required very quickly. It is possible only when we adopt some suitable system of collecting data.

Raghav	Banana
Preeti	Apple
Amar	Guava
Bawana	Apple
Manoj	Banana
Donald	Apple
Fatima	Orange
Raman	Banana
Radha	Orange
Rarida	Guava
Anuradha	Banana
Rati	Banana
Maria	Banana
Akhtar	Orange
Ritu	Apple
Salma	Banana
Kavita	Guava
Javed	Banana

Data Handling Class 6 Chapter 9

Organising Data
If we want to get particular information from the given data quickly, die data can be arranged in a tabular form using tally marks.

Pictograph
A pictograph represents data in the form of pictures, objects or parts of objects. The picture usually helps to understand the data. It helps answer the questions on the data at a glance. Pictographs are often used by dailies and magazines to attract the attention of the readers. However, it requires some practice to understand the information given by a pictograph.

Interpretation of a Pictograph
Simply by observing a pictograph, one can answer the related questions very quickly.

Drawing a Pictograph
We draw pictographs using symbols to represent a certain number of items or things.

Data Handling Class 6 Notes Pdf Chapter 9

A Bar Graph
Representing data by pictograph is tedious and time-consuming. So, a bar graph supplies with another way of representing data visually.
In a bar graph or a bar diagram, bars of uniform width is erected horizontally or vertically with equal spacing between them. The length of the bar gives the required information.

Interpretation of a bar graph
Looking at the bar graph; we can answer the related questions quite easily.

Data Handling Class 6 Introduction Chapter 9

Drawing a Bar graph
First of all, we draw a horizontal line and a vertical line on the horizontal line, we drawbars and on vertical line, we write the numerals. However, we choose a proper scale. For example, 1 unit =100 students. Good practice helps reading a given bar graph a lot.

Class 6 Maths Notes students can refer to the Decimals Class 6 Notes Maths Chapter 8 https://www.cbselabs.com/decimals-class-6-notes/ Pdf here. They can also access the CBSE Class 6 Decimals Chapter 8 Notes while gearing up for their Board exams.

CBSE Class 6 Maths Notes Chapter 8 Decimals

Decimals Class 6 Notes Chapter 8

Dot represents a decimal point.
Thus, five rupees seventy-five paise = ₹ 5.75
Seven rupees fifty paise = ₹ 7.50.

Tenths
To understand the parts of one whole (i.e., a unit) we show a unit by a block. Thus two units are two blocks and so on. One block divided into 10 equal parts means each part is \(\frac { 1 }{ 10 }\) (one- tenth) of a unit, two parts are 2-tenths, five parts are 5-tenths and so on. \(\frac { 1 }{ 10 }\) in decimal notation is written as 0.1. The dot represents the decimal point and it comes between the units place and tenths place.

Every fraction with denominator 10 can be written in decimal notation and vice-versa.
For example:

Hundredths
One block divided into 100 equal parts means each part is \(\frac { 1 }{ 100 }\) (one-hundredth) of a unit. It can be written as 0.01 in decimal notation.

Every fraction with denominator 100 can be written in decimal notation and vice-versa.

In place value table, as we go from left to right, then at every step the multiplying factor becomes \(\frac { 1 }{ 10 }\) of the previous factor. The place value table can be further extended from hundredths to \(\frac { 1 }{ 10 }\) of hundredths, i.e., thousandths (\(\frac { 1 }{ 100 }\)), which is written as 0.001 in decimal notation.

Decimals Class 6 Notes Pdf Chapter 8

All the decimals can be represented on a number line.

Every decimal can be written as a fraction.

We can compare any two decimals. The process of comparison starts with the whole part. If the whole parts are equal then the tenths parts are compared. If the tenths parts are also equal then hundredths parts arc compared and so on.

Decimals are used in numerous ways in our lives. For example in representing units of money, length and weight.
Money
1 paisa = ₹ \(\frac { 1 }{ 100 }\) = ₹ 0.01
85 paise = ₹ \(\frac { 85 }{ 100 }\)— = ₹ 0.85
125 paise = ₹ 1 and 25 paise = ₹ 1.25

Length
1 cm = \(\frac { 1 }{ 100 }\) m = 0.01 m
46 cm = \(\frac { 46 }{ 100 }\) m = 0.46 m
126 cm = 100 cm + 26 cm = 1 m + \(\frac { 26 }{ 100 }\) m = 1.26 m

Weight
1 g = \(\frac { 1 }{ 1000 }\) kg = 0.001 kg
2650 g = 2000 g + 650 g
= \(\frac { 2000 }{ 1000 }\) kg + \(\frac { 650 }{ 1000 }\) kg
= 2 kg + 0.650 kg
= 2.650 kg

Notes On Decimals For Class 6 Chapter 8

We can add decimals in the same way as whole numbers.

Subtraction of decimals can be carried out by subtracting hundredths from hundredths, tenths from tenths, ones from ones and so on.