Chapter 1: Knowing Our Numbers
 
1. Comparing numbers
• Numbers with same number of digits can be compared by placing them in a place value table and then comparing the digits at each place. For example, consider 2952 and 2748.

Thousands	Hundreds	Tens	Ones
2	9	5	2
2	7	4	8

The digits at thousands place are equal.
At hundreds place, 9 > 7
Therefore, 2952 > 2748
• Ascending order means arranging the numbers from smallest to greatest.
• Descending order means arranging the numbers from greatest to smallest.
• The greatest and smallest number without repetition can be formed using any number of digits by arranging them in descending and ascending order respectively.
• When repetition of digits is allowed, then the greatest number can be formed by writing the greatest digit as many times as the number is required.
• Smallest number can be formed by writing the smallest digit as many times as required.
• No number can have 0 as its first digit. If 0 is the smallest digit, then the smallest number can be formed by starting the number with the second-smallest digit and then placing the rest of the digits in the ascending order.
(i) The smallest 5-digit number is 10000.
10000 = 9999 + 1
Smallest 5-digit number = Greatest 4-digit number + 1
This pattern follows for higher numbers also.
(ii) A number, for example 48735, can be written in expanded form as
48735 = 4 × 10000 + 8 × 1000 + 7 × 100 + 3 × 10 + 5 × 1
(iii) There are two systems of numeration: the Indian system and the International system.
• In the Indian system, ones, tens, hundreds, thousands, lakhs and crores are used.
• In the International system, ones, tens, hundreds, thousands, millions and billions are used. 1 billion = 1000 millions
(iv) Commas are used to ease the reading and writing of larger numbers.
• According to the Indian system, the first comma is used after the hundreds place, the second comma comes after two digits from the first comma to the left, i.e. after ten thousands, then the third comma is placed after another two digits and so on.
• According to the International system, the first comma comes after the hundreds place, the next comma comes after three digits from the first comma to the left and the next comma is placed after another three digits and so on.
2. Conversion of units
• 10 millimetres = 1 centimetre
• 1 metre = 100 centimetres
• 1 kilometre = 1000 metres
• 1 gram = 1000 milligram
• 1 litre = 1000 millilitres
(i) Estimation is done to approximate a quantity.
For example, 53256
Rounding to nearest tens = 53260
Rounding to nearest hundreds = 53300
Rounding to nearest thousands = 53000
Rounding to nearest ten thousands = 50000
 
(ii) To ease calculations, sum, difference, or product can also be estimated.
For example, 5243 + 389 = 5200 + 400 = 5600
(iii) Brackets can also be used to ease calculations.
For example, 9 × 216 = 9 × (200 + 16) = 1800 + 144 = 1944
(iv) Roman numeral system: This is another system used for writing numerals.
• The standard symbols of this system are as follows:
1 = I    5 = V               10 = X             50 = L             100 = C
• Numbers can be written by combining these symbols.
For example: 56 = LVI; 84 = LXXXIV; 102 = CIIChapter 2: Whole Numbers
 
❖ Natural numbers: The counting numbers 1, 2, 3, 4 … are known as Natural numbers.
❖ Whole numbers
• The natural numbers, along with zero, form the collection of whole numbers. So, the whole numbers are 0, 1, 2, 3 …
• The smallest whole number is zero and there is no largest whole number.
• All the natural numbers are whole numbers. But all whole numbers are not natural numbers (since the whole number 0 is not a natural number).
• If we subtract 1 from a whole number, then we will get its predecessor and if we add 1 to a whole number, then we will find its successor.
• Each whole number has a successor. All whole numbers, except zero, has a predecessor.
• Basic operations of whole numbers like multiplication, division, addition and subtraction can be performed in the same way as for natural numbers.
❖ Number line:
To draw a number line, we take a line and mark a point on it, labeling it 0. Then, we mark the points to the right of zero at equal intervals and label them as 1, 2, 3 …, as follows:

On the number line, we can say that out of any two whole numbers, the number on the right of the other number is greater.
❖Basic operations on whole numbers with the help of the number line
• Addition
If we want to perform the addition of 7 and 2, first of all we need to mark point 7 on the number line. From there we move 2 places to the right of 7, i.e., up to 9.
 
∴ 7 + 2 = 9
Therefore, addition corresponds to moving to the right on the number line.
• Subtraction
Like addition, subtraction corresponds to moving to the left on the number line. For example, if we subtract 3 from 5, then first of all we have to mark 5 on the number line. From there, we jump 3 places to the left of 5, i.e., up to 2.

 ∴ 5 – 3 = 2
• Multiplication
Multiplication corresponds to moving equal distances, starting from zero. For example, if we have to find 2 × 5, then we move 2 equal units to the right of zero for 5 times.

∴ 2 × 5 = 10
❖Properties of whole numbers
• Closure property
⚬ Whole numbers are closed under addition i.e., the sum of two whole numbers is always a whole number.
⚬ Whole numbers are closed under multiplication i.e., the product of two whole numbers is always a whole number.  
⚬ Whole numbers are not closed under subtraction i.e., the difference between two whole numbers is not always a whole number.  
⚬ Whole numbers are not closed under division i.e., the division of two whole numbers does not always lead to a whole number.   
⚬ The division of whole numbers by zero is not defined.
• Identities
The addition of any whole number to zero gives the same whole number or we can say that 0 is the additive identity of whole numbers.
The multiplication of any whole number with 1 gives the same whole number or we can say that 1 is the multiplicative identity of whole numbers. 
• Commutative property
We can add or multiply two whole numbers in any order i.e., 12 + 5 = 5 + 12 = 17 and 9 × 8 = 8 × 9 = 72.
This property of addition and multiplication of whole numbers is known as the commutative property.
• Associative property
The addition and multiplication of whole numbers are associative.
For example: (17 + 19) + 25 = 17 + (19 + 25) = 61
Similarly, (6 × 13) × 19 = 6 × (13 × 19) = 1482
• Distributive property
Whole numbers show distributivity of multiplication over addition.
For example: 8 × (12 + 3) = 8 × 12 + 8 × 3 = 96 + 24 = 120
8 × (12 + 3) = 8 × 15 = 120
❖ Patterns in whole numbers
Different whole numbers have different patterns. Whole numbers can be represented through dots as lines, triangles, squares, pentagons, hexagons and so on.
For example:

❖ Patterns with numbers are useful for verbal calculations. It takes lesser time as compared to actual calculations.
For example, we can calculate 11 × 991 as:
$$\begin{gathered} 11\times 991=11\times \left(1000-9\right) \\ =11\times 1000-11\times 9 \left(\mathrm{Distributive}\right) \\ =11000-99 \\ =10901 \end{gathered}$$Chapter 3: Playing with Numbers
 
❖ Factors of numbers
• A factor of a number is an exact divisor of that number.
For example: 3 is an exact divisor of 15 (15 = 3 × 5). So, 3 is a factor of 15.
• 1 is a factor of every number.
• Every number is a factor of itself.
• Every factor of a number is less than or equal to the number. For example, factors of 12 are 1, 2, 3, 4, 6 and 12. Here, 1, 2, 3, 4, 6 and 12 are less than or equal to 12.
• Though we may find difficulties in factorising bigger numbers. But the number of factors of a number is finite.
For example:­ The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. Here, we may observe that there are only 8 factors of 30. Similarly, we can find a fixed number of factors of bigger numbers.
❖ Multiples of numbers
• We can find the multiples of a given number by multiplying 1, 2, 3…, to the number.
• A number is a multiple of each of its factors.
• Every multiple of a number is greater than or equal to that number.
For example, the multiple of 6 is 6, 12, 18 …
Here we may observe that the multiples of 6 are greater than or equal to 6.
• Every number is a multiple of itself.
• The number of multiples of a given number is infinite.
For example, the multiples of 11 are 11, 22, 33, 44 …. So, the number of multiples of 11 is infinite.
❖ Even and odd numbers
• Numbers that are divisible by 2 are called even numbers.
For example: 2, 4, 6, 8, 10… and so on
• Numbers that are not divisible by 2 are called odd numbers.
For example: 1, 3, 5, 7, 9, 11… and so on
❖ Prime and composite numbers
• Prime numbers are numbers having exactly two factors: 1 and the number itself.
For example, the factors of 23 are 1 and 23 only. So, 23 is a prime number.
• Composite numbers are numbers having more than two factors.
For example, the factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. Since there 8 factors of 42, it is a composite number.
• 1 is neither prime nor composite as it has exactly one factor.
• The smallest prime number is 2.
• The smallest even prime number is 2 and the smallest odd prime number is 3.
• All even numbers except 2 are composite.
• The pairs of prime numbers whose difference is 2 are known as twin primes. For example, 11 and 13 are twin primes.
Example:
Express 44 as the sum of four different primes.
Solution:
44 = 3 +11 + 13 + 17
44 = 5 +7 + 13 + 19
44 = 3 +5 + 17 + 19
44 = 3 +7 + 11 + 23
In this way we can express 44 as the sum of four different primes.
❖ Divisibility rules
• A number is divisible by 10, if the digit in ones place is zero.
• A number is divisible by 5, if the digit in ones place is either 0 or 5.
• A number is divisible by 2, if the digit in ones place is either 0, 2, 4, 6, or 8.
• A number is divisible by 3, if the sum of the digits in the number is a multiple of 3. For example, the sum of the digits of 9528 is 9 + 5 + 2 + 8 = 24, which is a multiple of 3. Hence, 9528 is divisible by 3. But the sum of the digits of 3725 is 3 + 7 + 2 + 5 = 17, which is not a multiple of 3. Hence, 3725 is not divisible by 3.
• Divisibility by 6
A number is divisible by 6, if it is divisible by both 2 and 3. For example, 39612 is divisible by 2. Sum of the digits of 39612 is 3 + 9 + 6 + 1 + 2 = 21, which is a multiple of 3. So, 39612 is divisible by 3. Now, 39612 is divisible by both 2 and 3. So, it is divisible by 6.
• Divisibility by 9
A number is divisible by 9, if the sum of the digits in the number is a multiple of 9.
For example, the sum of the digits in the number 9567 is 9 + 5 + 6 + 7 = 27, which is a multiple of 9. So, 9567 is a multiple of 9.
• Divisibility by 4
A number with 3 or more digits is divisible by 4, if the number formed by its last two digits (one’s and ten’s) is divisible by 4. For example, the last two digits of 9584 is 84, which is divisible by 4. So, 9584 is divisible by 4.
• Divisibility by 8
A number with 4 or more digits is divisible by 8, if the number formed by its last three digits (one’s, ten’s and hundred’s) is divisible by 8. For example, the last three digits of 9368 is 368, which is divisible by 8. So, 9368 is divisible by 8.
• Divisibility by 11
A number is divisible by 11, if the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) is either 0 or divisible by 11. For example, for the number 82918, the sum of the digits at odd places = 8 + 9 + 8 = 25 and the sum of the digits at even places = 1 + 2 = 3. Now, 25 – 3 = 22, which is a multiple of 11. Hence, 82918 is a multiple of 11.
❖ Two numbers are called co-prime, if they have 1 as the only common factor. For example, the factors of 8 are 1, 2, 4 and 8. The factors of 15 are 1, 3, 5, and 15. The common factor to both 8 and 15 is 1. So, they are co-prime numbers.
❖ If a number is divisible by another number, then it is divisible by each factor of that number.
❖ If a number is divisible by two co-prime numbers, then it is divisible by their product as well.
❖ If two numbers are divisible by a number, then their sum is also divisible by that number.
❖ If two numbers are divisible by a number, then their difference is also divisible by the number.
❖ Prime factorisation
We can express a number as the product of prime numbers and expressing a number as the product of prime numbers is known as prime factorisation. To find the prime factorisation of a number, we need to divide it by prime numbers that are factors of the given number, till we get 1.
Example:  Find the prime factorisation of 840.
We can proceed as follows:
 
$$\begin{gathered} \overline{)\underline{ 2} } \underline{840} \\ \overline{)\underline{ 2} }\underline{ 420} \\ \overline{)\underline{ 2} }\underline{ 105} \\ \overline{)\underline{ 3} }\underline{ 35 } \\ \overline{)\underline{ 5} }\underline{ 7 } \\ \overline{)\underline{ 7} }\underline{ 1 } \\ | \end{gathered}$$
So, the prime factorisation of 840 is
840 = 2 × 2 × 2 × 3 × 5 × 7
 
❖ HCF and LCM
• The highest common factor (HCF) of two or more given numbers is the highest of their common factors. To find the HCF of given numbers, first of all we need to prime factorise the given numbers.
For example, the HCF of 90, 120, 150 can be found as:

Thus, the HCF of 90, 120 and 150 = 2 × 3 × 5 = 30
• The lowest common multiple (LCM) of two or more given numbers is the least of their common multiples. To find the LCM of 90, 120 and 150, we may proceed as follows.

$$\begin{gathered} \underline{\overline{)\begin{array}{ccc}& & 2\end{array}}}\underline{\begin{array}{ccc}90,& 120,& 150\end{array}} \\ \underline{\overline{)\begin{array}{ccc}& & 2\end{array}}}\underline{\begin{array}{ccc}45,& 60,& 75 \end{array}} \\ \underline{\overline{)\begin{array}{ccc}& & 2\end{array}}}\underline{\begin{array}{ccc}45,& 30,& 75 \end{array}} \\ \underline{\overline{)\begin{array}{ccc}& & 3\end{array}}}\underline{\begin{array}{ccc}45,& 15,& 75 \end{array}} \\ \underline{\overline{)\begin{array}{ccc}& & 3\end{array}}}\underline{\begin{array}{ccc}15,& 5,& 25 \end{array}} \\ \underline{\overline{)\begin{array}{ccc}& & 5\end{array}}}\underline{\begin{array}{ccc}5,& 5,& 25 \end{array}} \\ \underline{\overline{)\begin{array}{ccc}& & 5\end{array}}}\underline{\begin{array}{ccc}1,& 1,& 5 \end{array}} \\ \underline{\overline{)\begin{array}{ccc}& & \end{array}}}\underline{\begin{array}{ccc}1,& 1,& 1 \end{array}} \\ \overline{)\begin{array}{ccc}& & \end{array}} \end{gathered}$$
∴ LCM of 60 and 120 = 2 × 2 × 2 × 3 × 3 × 5 × 5 = 1800Chapter 4: Basic Geometrical ideas
 
❖ Point
A point determines a location. The tip of a compass, the sharpened end of a pencil, the pointed end of a needle, etc., are the examples of points. Generally, points are denoted by capital letters.
❖ Lines
• A line segment corresponds to the shortest distance between two points. The line segment joining the points P and Q is denoted as $\overline{\mathrm{PQ}}$

• When a line segment $\overline{\mathrm{PQ}}$ is extended indefinitely on both sides of points P and Q, then it is called a line. It is denoted by small letters l, m, n.


• Two lines l and m are said to be intersecting lines, if they intersect at a point.

• Two lines are said to be parallel lines, if they never intersect each other.

Here, lines l and m never intersect each other, so these lines are parallel lines. We represent it as l||m.

• A ray is a portion of a line, which starts at one point and goes endlessly in a direction.

This ray is denoted as $\underset{\mathrm{PQ}}{\to }$. Arrow head is towards Q since it is extended along Q.
❖ Curve
Any drawing (straight or non-straight) done without lifting the pencil is called a curve. Line is also a curve.
• The curve which does not intersect itself is called a simple curve.



• A curve is said to be closed, if it has no starting or ending point.

• A curve is said to be opened, if its ends are not joined.


• In a closed curve there are three parts. They are interior (inside of the curve), boundary (on the curve), and exterior (outside the curve).
In the following figure, P and Q lie in the interior of the curve, R and S lie on the exterior of the curve, and X and Y lie on the boundary of the curve.

 
❖ Polygon
A polygon is a simple closed curve made up of line segments. ABCDEF is a polygon.

 
• The line segments AB, BC, CD, DE, DF, and FA are known as the sides of the polygon ABCDEF.
• Any two sides with common end points are called adjacent sides. AB and BC are adjacent sides with common end point B.
• The meeting point of a pair of sides of a polygon is known as vertex. In the polygon ABCDEF, sides AB and BC meets at point B. So, point B is called the vertex of the polygon. Similarly, the other vertices are A, C, D, E, and F.
• The line joining any two non-adjacent vertices of a polygon is known as its diagonal.
In the following polygon ABCDEF, the diagonals are AC, AD, AE, BD, BE, BF, CE, CF, and DF.

❖ Angle
An angle is made up of two rays starting from a common end point.

In this figure rays have one common end point, that is, B. The rays and are called the arms or sides of the angle. The common end point B is the vertex of the angle.
We name the above angle as ∠BAC.
❖ Triangle
A triangle is a three-sided polygon. It is the polygon with the least number of sides.

We denote this triangle as DPQR. Here, are the sides of DPQR. The points P, Q and R are the vertices of DPQR and the angles are ∠RPQ, ∠PQR and ∠QRP.
❖ Circle
A circle is the path of points moving at the same distance from a fixed point.

 
• The fixed point O is the centre of the circle.
• The fixed distance OP = OQ is the radius of the circle.
• The distance around the circle is its circumference.
• A line joining any two points on a circle is known as chord. In the given figure, RS and PQ are the chords.
• The chord passing through the centre of a circle is called diameter. The diameter of a circle divides it into two semicircles.
• The diameter of a circle is the longest chord of the circle and it is twice the radius.
• The portions on a circle are known as arcs. In the figure, XY and AB are arcs.

• The region in the interior of a circle enclosed by a chord and an arc is known as segment.
• The region in the interior of a circle enclosed by an arc on one side and a pair of radii on the other side is called sector.
 

❖ Quadrilateral
A quadrilateral is a four-sided polygon.

AC and BC are the diagonals of quadrilateral ABCD. ∠A and ∠C, and ∠B and ∠D are the opposite angles. ∠B and ∠C are the adjacent angles. Similarly, the other adjacent angles are ∠A and ∠B, ∠C and ∠D, ∠D and ∠A.Chapter 5: Understanding Elementary Shapes
 
❖ Comparison of line segments
• The distance between the end points of a line segment is its length.
• The length of line segments can be compared by three methods. They are: comparison by observation, comparison by tracing, and comparison using a ruler and a divider.
• Comparison by observation
In the method of comparison by observation, we compare the length of the line segments by just looking at them.


In the above figure, we may observe that the length of AB is greater than CD. But, the lengths of line segments XY and PQ cannot be compared just by this observation. For this, we follow another method called comparison by tracing method.
• Comparison by tracing
In the method of comparison by tracing, we use trace papers to compare the lengths of the line segments. This method is dependent upon the accuracy in tracing the line. If we want to compare with another length, we have to trace another line segment. Thus, it is a difficult method as we cannot trace the line segments every time we want to compare them.
• Comparison using ruler and divider
An accurate method to compare the length of the line segments is by comparing using a ruler and a divider. In this method, we measure the length of the line segments with a ruler and a divider and then, compare them easily.
❖ Angles
• Complete angle
One complete turn of the hand of a clock is one revolution. The angle of one revolution is called a complete angle.


• Right and straight angles
A right angle is ${\frac{1}{4}}^{\mathrm{th}}$ of a revolution and a straight angle is $\frac{1}{2}$ of a revolution.

1 complete angle = 2 straight angles = 4 right angles
1 straight angle = 2 right angles
• Acute angle
If an angle measures less than a right angle, then it is known as an acute angle.
The following angles are acute.

• Obtuse angle
If an angle measures more than a right angle but less than a straight angle, then it is an obtuse angle.
The following angles are obtuse.


• Reflex angle
If an angle measures more than a straight angle, then it is known as a reflex angle.
The following angles are reflex. 
 
 
❖ Measuring angles by using protractor
• One complete revolution is divided into 360 equal parts. Each part is called a degree. Thus, the unit of angle is degree (°).
• Right angle measures 90°, complete angle measures 360°, and straight angle measures 180°.
• Acute angle is less than 90°, obtuse angle is more than 90° but less than 180°, and reflex angle is more than 180° but less than 360°.
❖ Two lines are perpendicular, if the angle between them is 90°.
❖ Classification of triangles
A triangle can be classified on the basis of the measures of its angles and sides.
• Classification of triangles on the basis of the measures of its angles:
 

Name	Nature of the angle
Acute-angled triangle obtuse-angled triangle Right-angled triangle	Each angle is acute One angle is obtuse One angle is a right angle

 
• Classification of triangles on the basis of the lengths of its sides:
 

Name	Nature of the angle
Scalene triangle Isosceles triangle Equilateral triangle	All three sides are of unequal length Any two sides are of equal length All sides are of equal length

 
❖ Polygons
A polygon’s name is based on the number of sides it has.
 

Number of sides	Figure	Name
3		Triangle
4		Quadrilateral
5		Pentagon
6		Hexagon
8		Octagon

 
❖ Classification of quadrilaterals
Quadrilaterals are further classified according to their properties.
 

Name of the quadrilateral	Figure	Properties
Rectangle		(1)  Opposite sides are equal. (2)  Each angle is 90°. (3)  Diagonals are equal. (4)  Opposite sides are parallel.
Square		(1)  All sides are equal. (2)  Each angle is 90°. (3)  Diagonals are equal. (4)  Opposite sides are parallel.
Parallelogram		(1)  Opposite sides are parallel. (2)  Opposite sides are equal. (3)  Diagonals may or may not be equal.
Rhombus		(1)  Opposite sides are parallel. (2)  All sides are equal. (3)  Diagonals may or may not be equal.
Trapezium		One pair of opposite sides is parallel.

 
• A parallelogram is a rhombus if all sides are equal.
• A parallelogram is a rectangle if all angles are 90°.
• A parallelogram is a square if all sides are equal and all angles are 90°.
• A rhombus is a square if all angles are 90°.
• A Rectangle is a square if all sides are equal.
 
❖ Three dimensional shapes
We see a few shapes in our day- to-day life which are not flat. Some of these shapes are solids.


❖ Each face of a solid is flat, called a flat surface (or surface). Two faces meet at a line segment called an edge. Three edges meet at a point called the vertex.


 
❖ Properties of three dimensional shapes
                 

Solid	Figure	Properties
Cube		6 faces  12 edges  8 vertices (corners)
Cuboid		6 faces  12 edges  8 vertices
Cylinder		2 flat faces (circles)  1 curved face
Cone		1 flat surface.  1 curved surface  1 vertex
Triangular Pyramid		4 faces 6 edges 4 vertices
Square pyramid		5 faces 8 edges 5 vertices
Triangular prism		5 faces 9 edges 6 vertices

           Chapter 6: Integers
 
❖ Integers
• When we move below zero, we assign negative signs to those numbers on the number line. These numbers are called negative numbers.
For example, if we go 10 km below the sea level, we represent it as –10.
• The collection of numbers … –3, –2, –1, 0, 1, 2, 3 …, are called integers.
–1, –2, –3 … are called negative integers whereas 1, 2, 3 … are called positive integers.
❖ Representation of integers on number line
Integers can be represented on a number line. For this, a line has to be drawn and a point, 0, has to be marked on it. Towards the right of zero, the points, 1, 2, 3 …, are marked at equal gaps. Similarly, to the left of zero, the points, –1, –2, –3 …, are marked at equal gaps as shown below.

 
• To represent a negative number, steps equal to the number have to be jumped to the left of zero and for a positive number, the steps equal to the number have to be jumped to the right of zero.
–2 and 4 can be represented on the number line as shown below.

 
• As we move to the right of the number line, the numbers increase.
For example, –2 > –6 since –2 is to the right of –6 on the number line.

 
 
❖ Addition of integers
• To add two integers with same sign, the integers are first added as whole numbers and then the same sign is put.
For example: (–9) + (–6) = –(9 + 6) = –15
• To add one positive integer and one negative integer, the smaller integer is subtracted from the larger integer without any sign and then the sign of the larger number is put.
Example: (– 15) + (+ 8) = – (15 – 8)   [15 is larger and it has – sign]
= – 7
• Integers can also be added using number line. To add a positive integer, we move towards right and to add a negative integer, we move towards left.
Example: To add (– 3) and (– 4), first of all, (– 3) is marked on the number line. Since (–4) has to be added to (–3), 4 steps are moved to the left of (–3) to reach (–7).


• Integers can be added using concrete materials. For this, two items are taken. In this method, the combination of one item is considered from each category.
For example: If (–5) and (+ 3) have to be added, 5 yellow buttons (each yellow button represents (–1)) and 3 blue buttons (each blue button represents (+ 1)) can be taken.

 
= (–2) + 0 = –2
❖ Additive inverse
• If two integers are added such that their sum is zero, then these integers are called additive inverses of each other.
Example: (–8) + (+8) = 0
Therefore, (–8) and (+ 8) are additive inverses of each other.
• Additive inverse of an integer can be found just by changing its sign.
❖ Subtraction of integers
• The subtraction of an integer is same as the addition of its additive inverse.
Example: (–8) – (–5) = (–8) + Additive inverse of (–5)
                                  = – 8 + (+ 5) = –3
• The subtraction of an integer can be represented on a number line.
Example: If (–5) has to be subtracted from (–2), then
(–2) – (–5) = (–2) + 5
It can be represented on the number line as:


    (–2) – (–5) = 3Chapter 7: Fractions
 
❖ A fraction is a number representing a part of a whole. The whole may be a single object or a group of objects.
• Example:



• While expressing a situation where parts have to be counted to write a fraction, it must be ensured that all parts are equal.
• For a fraction $\frac{2}{13}$, 2 is called its numerator and 13 is called its denominator.
❖ Representation of fractions on number line
To represent $\frac{2}{7}$ on the number line, first of all, the gap between 0 and 1 is divided into 7 equal parts. Then, each part represents the fraction $\frac{1}{7}$.
Therefore, $\frac{2}{7}$ can be easily represented on the number line as:

 
❖ Types of Fractions
Fractions are categorized into three types: proper, improper, and mixed fraction.
• Proper fractions are those fractions in which the numerator is less than the denominator. These fractions are always less than 1.Example: $\frac{17}{24}$
• Improper fractions are those fractions in which the numerator is greater than the denominator. These fractions are always greater than 1. Example: $\frac{15}{7}$
• A mixed fraction is a combination of a whole number and a part. Example: $9\frac{5}{13}$
• If numerator and denominator of a fraction are equal, then the fraction is 1.

Example: $\frac{5}{5}=1$
❖ Conversion of fractions
• From mixed to improper
A mixed fraction can be converted into an improper fraction as
$\frac{(\text{Whole}\times \text{Denominator})+\text{Numerator}}{\text{Denominator}}$

Example: $8\frac{2}{23}=\frac{\left(8\times 23\right)+2}{23}=\frac{184+2}{23}=\frac{186}{23}$
• From improper to mixed
To convert an improper fraction into a mixed fraction, first of all, the quotient and remainder are obtained by just dividing the numerator by the denominator. Then, the mixed fraction corresponding to the given improper fraction is written as
$\text{Quotient}\frac{\text{Remainder}}{\text{Divisor}(=\text{Denominator})}$
Example: To find the mixed fraction corresponding to the improper fraction $\frac{182}{7},$ first of all, 182 is divided by 17.

Here, divisor = 17, quotient = 10, and remainder = 12
$\therefore \frac{182}{17}=10\frac{12}{17}$
 
❖ Representing fractions with the help of figures
• To represent a proper fraction in the form of a figure, an object or a group of objects are divided into equal number of parts related to the denominator. The parts equal to the numerator are shaded.
Example: $\frac{3}{5}$ can be represented as



 • To represent a mixed fraction in the form of a figure, both the figures denoting whole part and the proper fraction part are shaded.
Example: $2\frac{1}{4}$ can be represented as

• To represent an improper fraction in the form of a figure, firstly, it is converted into mixed fraction and then, the same procedure is followed as above.
Example: $\frac{7}{4}=1\frac{3}{4}$ can be represented as:

 
❖ Equivalent fractions
• To find an equivalent fraction of a given fraction, both the numerator and denominator of the given fraction are multiplied or divided by the same number.
Example:

$$\begin{gathered} \frac{12}{18}=\frac{12\times 2}{18\times 2}=\frac{24}{36}, \frac{12}{18}=\frac{12\times 5}{18\times 5}=\frac{60}{90} \\ \frac{12}{18}=\frac{12÷2}{18÷2}=\frac{6}{9}, \frac{12}{18}=\frac{12÷3}{18÷3}=\frac{4}{6} \\ \therefore \frac{12}{18}=\frac{24}{36}=\frac{60}{90}=\frac{6}{9}=\frac{4}{6} \end{gathered}$$

• Two fractions are equivalent, if the product of the numerator of the first fraction and the denominator of the second fraction is equal to the product of the numerator of the second fraction and the denominator of the first fraction.
Example: To check the equivalence of $\frac{3}{5}$ and $\frac{18}{30}$, the following calculation is carried out.
3 × 30 = 90 and 5 × 18 = 90
Since 3 × 30 = 5 × 18,  $\frac{3}{5}=\frac{18}{30}$
❖ Simplest form of fractions
• A fraction is said to be in its simplest or lowest form, if its numerator and denominator have no common factor other than 1.
• A fraction can be reduced to its simplest form or lowest form by dividing both numerator and denominator by the HCF of the numerator and denominator.
Example: Convert $\frac{49}{91}$ into its simplest form.
Solution:
HCF of 49 and 91 is 7.
$\therefore \frac{49}{91}=\frac{49÷7}{91÷7}=\frac{7}{13}$
Therefore, $\frac{7}{13}$ is the simplest form of $\frac{49}{91}$.
❖ Like and unlike fractions
• Fractions with same denominators are called like fractions.
Example: $\frac{9}{5},\frac{2}{5},\frac{11}{5},\frac{7}{5}$ are like fractions.
• Fractions with different denominators are called unlike fractions.
Example: $\frac{9}{8},\frac{11}{3},\frac{12}{7}$ are unlike fractions.
❖ Comparison of fractions
• If two or more fractions are like fractions, then greater the numerator, greater is the fraction.
Example: Among the fractions, $\frac{9}{17},\frac{25}{17},\frac{21}{17}$, and $\frac{6}{17}$, it can be observed that,
25 > 21 > 9 > 6
$\therefore \frac{25}{17}>\frac{21}{17}>\frac{9}{17}>\frac{6}{17}$
• If two or more fractions have the same numerator, then smaller the denominator, greater is the fraction.
Example: Among the fractions, $\frac{17}{5},\frac{17}{3}$, and $\frac{17}{11}$, it can be observed that,
3 < 5 < 11
$\therefore \frac{17}{3}>\frac{17}{5}>\frac{17}{11}$
• To compare two unlike fractions (without same numerator), first of all, these fractions are converted into their equivalent fractions of same denominator, which is the LCM of the denominators of the fractions. Then, like fractions are obtained, which can be compared easily.
Example: $\frac{5}{6}$ and $\frac{20}{21}$ can be compared as
LCM of 6 and 21 = 42
$\therefore \frac{5}{6}=\frac{5\times 7}{6\times 7}=\frac{35}{42},\frac{20}{21}=\frac{20\times 2}{21\times 2}=\frac{40}{42}$
Here, $\frac{35}{42}$ and $\frac{40}{42}$ are like fractions.
Since $\frac{40}{42}>\frac{35}{42}$, we obtain
$\frac{20}{21}>\frac{5}{6}$
 
❖ Basic operations on fractions
• Addition of two like fractions can be performed just by adding the numerators and retaining the denominator of the fractions.
Example: $\frac{17}{25}+\frac{3}{25}=\frac{17+3}{25}=\frac{20}{25}=\frac{20÷5}{25÷5}=\frac{4}{5}$
• Subtraction of two like fractions can be performed just by subtracting the numerators and retaining the denominator of the fractions.
Example: $\frac{31}{15}-\frac{4}{15}=\frac{31-4}{15}=\frac{27}{15}=\frac{27÷3}{15÷3}=\frac{9}{5}$
• To perform the addition and subtraction of unlike fractions, first of all, they are converted into their equivalent fractions with the denominator as the LCM of their denominators. Then, addition or subtraction can be performed easily.
Example: Find the sum of $\frac{4}{3}$and $\frac{5}{12}$.
Solution:
LCM of 3 and 12 = 12
$\therefore \frac{4}{3}+\frac{5}{12}=\frac{4\times 4}{3\times 4}+\frac{5\times 1}{12\times 1}=\frac{16}{12}+\frac{5}{12}=\frac{21}{12}=\frac{21÷3}{12÷3}=\frac{7}{4}$
• To add or subtract mixed fractions, first of all, they are converted into improper fractions. Then, they can be added or subtracted easily.
 
Example:
$$\begin{gathered} 7\frac{2}{5}+3\frac{4}{9}=\frac{37}{5}+\frac{31}{9} \\ =\frac{37\times 9}{5\times 9}+\frac{31\times 5}{9\times 5} \left(\mathrm{LCM} \mathrm{of} 5 \mathrm{and} 9 \mathrm{is} 45.\right) \\ =\frac{333}{45}+\frac{155}{45} \\ =\frac{333+155}{45} \\ =\frac{488}{45} \\ =10\frac{38}{45} \end{gathered}$$Chapter 8: Decimals
 
❖ Place value tables for decimals
The place value table of the number 8570.216 can be compiled as
     

Thousands (1000)	8
Hundreds (100)	5
Tens (10)	7
Ones (1)	0
Tenths $\left(\frac{1}{10}\right)$	2
Hundredths $\left(\frac{1}{100}\right)$	1
Thousandths $\left(\frac{1}{1000}\right)$	6

                       
Using this table, we can expand 8570.216 according to its place value as
$8570.216=8\times 1000+5\times 100+7\times 10+2\times \frac{1}{10}+1\times \frac{1}{100}+6\times \frac{1}{1000}$
 
❖ Representation of decimals on the number line
To represent a decimal number on number line, the unit length is divided into 10 equal parts. Each equal part represents 0.1.
To represent 3.6 on the number line, the distance between 3 and 4 is divided into 10 equal parts. The 6^th part after 3 represents 3.6.
 

 
❖ Representation of decimals using figures
We can represent the number 129.56 by using blocks as follows:
 
 
❖ Conversion of decimals into fractions
Every decimal can be written as a fraction.
Example:
$2.96=2+\frac{96}{100}=2+\frac{96÷4}{100÷4}=2+\frac{24}{25}=2\frac{24}{25}=\frac{74}{25}$
❖ Conversion of fractions into decimals
• Every fraction with denominator 10 or 100 can be converted into decimal form easily.
Example:
$\frac{56}{10}=\frac{50+6}{10}=5+\frac{6}{10}=5.6$
• A fraction whose denominator is 10 or 100 can be converted into decimal form by multiplying the numerator and denominator by the same number such that the denominator is 10 or 100.
Example:
$\frac{41}{20}=\frac{41\times 5}{20\times 5}=\frac{205}{100}=\frac{200+5}{100}=2+\frac{5}{100}=2.05$
 
❖ Any two decimal numbers can be compared among themselves. The comparison can start with the whole part. If the whole parts are equal, then the tenth part is compared and so on.
 
❖ We use decimals in our day to day lives in many ways, for example, in representing units of money, weight, length, volume, etc.
Example: If we want to represent 6 kg 5g into kg, then we may proceed as follows.
$6 \mathrm{kg} 5 \mathrm{g}=6 \mathrm{kg}+5\times \frac{1}{1000}\mathrm{kg} \left(1 \mathrm{g}=\frac{1}{1000}\mathrm{kg}\right)=\left(6+\frac{5}{1000}\right)\mathrm{kg}=6.005 \mathrm{kg}$

❖ We can add or subtract decimals in the same way as whole numbers by placing decimal points one above the other.
Example:
If 9.56 and 17.15 are to be added, then we proceed as:
 

Tens	Ones	Tenths	Hundredths
+ 1	9 7	5 1	6 5
2	6	7	1

 
∴ 9.56 + 17.15 = 26.71Chapter 9: Data Handling
 
❖ Data is a collection of numbers. It is gathered to obtain specific types of information.
For example, let us consider the following data.
 

Name of The Student	Marks Obtained (out of 100)
Manasi	81
Praveen	73
Pradeep	98
Kartik	61
Mamta	96
Vinod	83
Salma	69
Jyoti	83
Amardeep	67
Suraj	52

 
This data gives information about the marks obtained (out of 100) by 10 students.
By observing this data, we can say that Mamta obtained the highest marks and Suraj obtained the least marks among all the students.
We can also say that Jyoti and Vinod obtained the same marks.
 
❖ Tally marks
We arrange any data in tabular form using tally marks to obtain particular information in very little time.
For 1, we use the tally mark 
For 2, we use the tally mark 
For every 5, we use the tally mark
For 6, we use
For 19, we use the tally marks
And so on.
 
❖ Pictographs
In a pictograph, pictures of objects are used for representing data. Tally marks cannot be used for representing huge numbers. However, these numbers can be represented with the help of pictographs.
Example: The given pictograph represents the number of mobile users in six different cities of a country. We can interpret a pictograph by reading and analyzing it.
 
 

City	Number of mobile users
A.
B.
C.
D.
E.
F.

 
 
❖ Bar graph
Data can also be represented by using bar diagram or bar graph.
In a bar graph, bars of uniform width are drawn horizontally or vertically. These bars are placed at equal distance from each other. The length of each bar gives the required information.
Example: The given bar graph represents the number of bikes sold by a retailer in the first five months of a year. We interpret a bar graph by reading and analyzing it.
 
Chapter 10: Mensuration
 
❖ Perimeter
The perimeter of a closed figure is the distance covered along the boundary of the figure when we go around it once. Its units are cm, m etc.
Example: Let us find the perimeter of the following figure.

Perimeter of ABCDEFGHIJA = AB + BC + CD + DE + EF + FG + GH + HI + IJ + JA
= (5 + 6 + 6 + 5 + 7 + 9 + 7 + 4 + 4 + 25) cm
= 78 cm
 
❖ Perimeters of different figures
• Perimeter of a rectangle = 2 (length + breadth)
• Perimeter of an equilateral triangle = 3 × length of a side
• Perimeter of a regular polygon = Number of sides of the polygon × length of each side
❖ Area
• The region enclosed by a closed figure is called its area.


• The units of area are square cm, square m etc.
• We can estimate the area of a surface by drawing it on a square graph paper, where every square measures 1 cm × 1 cm. For this, we have to adopt the following conventions.
1. The area of 1 full square is taken as 1 square unit.
2. The area of a region which is more than half the square is taken as 1 square unit.
3. The area of half the square is taken as $\frac{1}{2}$ square unit.
4. We have to ignore the portions of area that are less than half a square.
 
Example: Find the area of the following figure.


Solution: We can represent the number of full-filled squares, half filled squares etc. in a tabular form as follows:
 

Area covered	Number	Area estimate (square unit)
Full-filled squares	6	6
Half-filled squares	2	$\frac{1}{2}\times 2=1$
More than half-filled squares	2	2
Less than half-filled squares	2	0

 
Total area = (6 + 1 + 2) square units = 9 square units
 
❖ Areas of different figures
• Area of a rectangle = length × breadth
• Area of a square = side × sideChapter 11: Algebra
 
❖ The branch of mathematics in which numbers are studied is called arithmetic.
❖ The branch of mathematics in which shapes are studied is called geometry.
❖ The branch of mathematics in which numbers, shapes, and different patterns are studied together is known as algebra.
❖ Variables
• A variable can take different values whereas a constant has a fixed value.
For example, the lengths of an equilateral triangle can have any values. Therefore, length of an equilateral triangle is a variable. The number of sides of a hexagon is 6, which is fixed. Therefore, the number of sides of a hexagon is a constant.
• The letters, l, m, n, p, x, y, z, etc., are used to denote variables.
• In our day to day life, we come across many examples of variables.
Consider the following pattern made up of match sticks.

In this case, the number of match sticks required are given by the rule, 3n, where n is the number of .
If patterns of 15 numbers of have to be made, then the number of match sticks required is 3 × 15 = 45
 
❖ Use of variables to represent different rules and formulae
• The formulae and rules in geometry can also be expressed by variables.
1. Perimeter of a square = 4l, where l is the side of the square
2. Perimeter of a rectangle = 2l + 2b, where l is the length and b is the breadth of the rectangle
• Variables can be used to express rules from arithmetic as follows.
1. Commutative property of addition
For two variables, a and b, this rule can be expressed as $\boxed{a+b=b+a}$
Here, the variables, a and b, can represent any number.
2. Commutative property of multiplication
For any two variables, a and b, this rule can be expressed as $\boxed{a\times b=b\times a}$
3. Associative property of addition
For any three variables, a, b and c, this rule can be stated as
$\boxed{a+\left(b+c\right)=\left(a\times b\right)\times c}$
4. Associative property of multiplication
For any three variables, a, b and c, this rule can be stated as
$\boxed{a\times \left(b\times c\right)=\left(a\times b\right)\times c}$

5. Distributive property
For any three variables, a, b, and c, this rule can be stated as
$\boxed{a\times \left(b+c\right)=a\times b+a\times c}$
 
❖ Expressions
Using different operations on variables and numbers, expressions such as  $\frac{1}{7}a-4, 9x-5,$ etc. can be formed.
Example: Meena’s age is 4 years less than 7 times the age of Ravi. Express it using variables.
Solution: Let the age of Ravi be x.
∴ Age of Meena = (7x – 4)
 
❖ Equations
An equation is a condition on one or more variables. The left portion of the equation is known as left hand side (LHS) and the right portion as right hand side (RHS).
For example, 2x – 5 = 7
• The definite value of variable which satisfies the equation is known as the solution of the equation.
For example, consider the equation 2x – 5 = 7
Since LHS = 2 × 6 – 5 = 12 – 5 = 7 = RHS,
x = 6 is the solution of the equation, 2x – 5 = 7
• The solution of an equation is calculated by a method known as trial and error method. In this method, some value of the variable is substituted and it is checked whether it satisfies the equation. Different values of variables are substituted until the right value is found, which satisfies the equation.
For example, for the equation 7x – 2 = 19,
Put x = 1
∴ LHS = 7 × 1 – 2 = 7 – 2 = 5 ≠ 19
⇒ LHS ≠ RHS
Therefore, x = 1 is not the solution.
 
Put x = 2
LHS = 7 × 2 – 2 = 14 – 2 = 12 ≠ 19
∴ LHS ≠ RHS
Therefore, x = 2 is not the solution.
 
Put x = 3
 
LHS = 7 × 3 – 2 = 21 – 2 = 19 = RHS
Therefore, x = 3 is the solution of the equation, 7x – 2 = 19Chapter 12: Ratio and Proportion
 
❖ Ratio
• In many situations, a meaningful comparison between quantities is made by using division i.e., by observing how many times one quantity is in relation to the other quantity. This comparison is known as ratio. We denote it by using the symbol ‘:’.
• A ratio may be treated as a fraction. For example, 3:11 can be treated as $\frac{3}{11}$.
• We can compare two quantities in terms of ratio, if these quantities are in the same unit. If they are not, then they should be expressed in the same unit before the ratio is taken.
 For example: If we want to compare 70 paise and Rs 3 in terms of ratio, then we have to convert Rs 3 into paise.
 Rs 3 = 300 paise
Hence, required ratio = 70: 300 = 7: 30
• The same ratio may occur in different situations.
To understand this concept, let us consider the following situations.
1. Distances of Lata’s home and Ravi’s home from their school are 12 km and 21 km respectively. Therefore, the ratio of the distance of Lata’s home to the distance of Ravi’s home from their school is $\frac{12}{21}=\frac{4}{7}=4 : 7$
2. Neha has Rs 20 and Saroj has Rs 35. Therefore, the ratio of the amount of money that Neha has to that of Saroj is $\frac{20}{35}=\frac{4}{7}=4 : 7$
In this way, we can come across many situations where the ratio would be 4:7.
• The order of ratio cannot be changed.
For example, the ratio 8: 3 cannot be written as 3: 8.
 
❖ Equivalent Ratio
We can find equivalent ratios by multiplying or dividing the numerator and denominator by the same number.
Example: To find the equivalent ratios of 12:20, we proceed as follows.
$$\begin{gathered} 12 : 20=\frac{12}{20}=\frac{12÷2}{20÷2}=\frac{6}{10}=6 : 10 \\ 12 : 20=\frac{12\times 3}{12\times 3}=\frac{36}{60}=36 : 60 \end{gathered}$$

• We can say that two ratios are equivalent, if the product of the numerator of the first ratio and the denominator of the other ratio is equal to the product of the denominator of first ratio and the numerator of the other ratio.
Example: To check the equivalence of the ratios, 14:49 and 6:21, we have to check whether $\frac{14}{29}$ and $\frac{6}{21}$ are equivalent or not.
14 × 21 = 294 = 6 × 49
Therefore, $\frac{14}{29}$ and $\frac{6}{21}$are equivalent fractions.
Hence, 14:49 and 6:21 are equivalent ratios.
 
❖ Lowest Form of Ratio
A ratio can be expressed in its lowest form. For this, we have to find the lowest form of the fraction corresponding to the given ratio.
For example,
The lowest form of 45:72 is given by,
$45 : 72=\frac{45}{72}=\frac{45÷9}{72÷9} (HCF\mathrm{of} 45 \mathrm{and} 72 \mathrm{is} 9.)$
$=\frac{5}{8}=5 : 8$
∴ 45 : 72 = 5 : 8
 
❖ Proportion
• Four quantities are said to be in proportion, if the ratio of first and second quantities is equal to the ratio of third and fourth quantities.
For example:
To check whether 8, 22, 12, and 33 are in proportion or not, we have to find the ratio of 8 to 22 and the ratio of 12 to 33.
$$\begin{gathered} 8 : 22 =\frac{8}{22}=\frac{4}{11}=4 : 11 \\ 12 : 33 =\frac{12}{33}=\frac{4}{11}=4 : 11 \end{gathered}$$
Therefore, 8, 22, 12, and 33 are in proportion as 8:22 and 12:33 are equal.
• When four terms are in proportion, the first and fourth terms are known as extreme terms and the second and third terms are known as middle terms.
In the above example, 8, 22, 12, and 33 were in proportion. Therefore, 8 and 33 are known as extreme terms while 22 and 12 are known as middle terms.
 
❖ Relation between Ratio and Proportion
• If two ratios are equal, then we say that they are in proportion and use the symbol ‘::’ or ‘=’ to equate the two ratios.
For example: 8:36 and 14:63 are equal since $8 : 36=\frac{8}{36}=\frac{2}{9}$ and $14 : 63 =\frac{14}{63}=\frac{2}{9}$
Since 8:36 and 14:63 are in proportion, we write it as 8: 36:: 14: 63 or 8: 36 = 14: 63
 
❖ Unitary method
The method in which we first find the value of one unit and then the value of the required number of units is known as unitary method.
Example:
What is the cost of 9 bananas if the cost of a dozen bananas is Rs 20?
Solution:
1 dozen = 12 units
Cost of 12 bananas = Rs 20
$\therefore \mathrm{Cost} \mathrm{of} 1 \mathrm{banana}=\mathrm{Rs}\frac{20}{12}$ 
$\therefore \mathrm{C}\mathrm{ost} \mathrm{of} 9 \mathrm{bananas}= Rs\frac{20}{12}\times 9=\mathrm{Rs} 15$
This method is known as unitary method.Chapter 13: Symmetry
 
❖ Symmetrical Figures
A figure is said to be symmetrical, if it is in an evenly-balanced proportion.
❖ Line of Symmetry
• A figure has line symmetry, if a line can be drawn dividing the figure into two identical parts. The line is called the line of symmetry.
Example:

 
In this figure, the line l divides the above figure into two identical parts. This line l is known as the line of symmetry. So, this figure has a line of symmetry.
• A figure may have no line of symmetry, only one line of symmetry, or multiple lines of symmetry.
❖ Types of Line of Symmetry
• If a vertical line divides the figure into two identical parts, then we say that the figure has a vertical line of symmetry.
• If a horizontal line divides a figure into two identical parts, then we say that the figure has a horizontal line of symmetry.
❖ The line of symmetry is closely related to mirror reflection. When we deal with mirror reflection, we have to take into account that the object and image are symmetrical with reference to the mirror line. There is no change in the length and angle of the object and the corresponding length and angle of the image, with respect to the mirror line; only the left-and-right alignment changes.
❖ The concept of symmetry is widely used in the field of technology, architecture, geometrical reasoning, designing, etc.Chapter 14: Practical Geometry
 
❖ We use mathematical instruments such as ruler, compass, divider, set squares, and protractor to construct different shapes in geometry.
❖ Using these instruments, we can construct:
• a circle if its radius is known
• a line segment if its length is known
• a copy of a line segment
• a perpendicular to a line segment through a point on it
• a perpendicular to a line segment through a point not on it
• a perpendicular bisector of a line segment
• an angle of a given measure using protractor
• a copy of an angle
• the bisector of a given angle
• some angles of special measures such as 30°, 45°, 60°, 90°, 120°, 135°, etc
 
❖ Construction of a perpendicular line to a line segment through a point on it.
Example: Draw any line $\overleftrightarrow{\mathrm{PQ}}$. Mark a point M on it. Through M, draw $\overleftrightarrow{\mathrm{AB}}$ such that $\overleftrightarrow{\mathrm{AB}}$ is perpendicular to $\overleftrightarrow{\mathrm{PQ}}$.
Solution:
A perpendicular $\overleftrightarrow{\mathrm{AB}}$ through point M on $\overleftrightarrow{\mathrm{AB}}$ can be constructed as
i. Draw a line $\overleftrightarrow{\mathrm{PQ}}$ and mark a point M on it
ii. With M as the centre and a convenient radius, construct an arc intersecting $\overleftrightarrow{\mathrm{PQ}}$ at two points i.e., X and Y. With X and Y as centres and radius greater than MX, construct two arcs that cut each other at N.
iii. Draw a line through points M and N and name this line as $\overleftrightarrow{\mathrm{AB}}$. Therefore, $\overleftrightarrow{\mathrm{AB}}\perp \overleftrightarrow{\mathrm{PQ}}.$

 
❖ Construction of a perpendicular to a line segment through a point not on it.
Example: Draw a line $\overleftrightarrow{\mathrm{AB}}$. Take a point M not on it. Through M, draw a perpendicular to $\overleftrightarrow{\mathrm{AB}}$.
Solution:
A perpendicular from point M (not on $\overleftrightarrow{\mathrm{AB}}$) to $\overleftrightarrow{\mathrm{AB}}$ can be drawn as
i. Draw line $\overleftrightarrow{\mathrm{PQ}}$. Mark a point M outside it.
ii. With M as the centre, draw an arc that intersects $\overleftrightarrow{\mathrm{PQ}}$ at two points i.e., X and Y.
iii. Using the same radius and with X and Y as centres, construct two arcs such that they intersect at N on the other side.
iv. Join $\overleftrightarrow{\mathrm{MN}}$ to get $\overleftrightarrow{\mathrm{MN}}\perp \overleftrightarrow{\mathrm{AB}}$ .

 
❖ Construction of a perpendicular bisector of a line segment.
Example: Draw the perpendicular bisector of a line segment $\overline{\mathrm{PQ}}$ where $\overline{\mathrm{PQ}}$ = 9.4 cm.
Solution:
The perpendicular bisector of line segment $\overline{\mathrm{PQ}}$ where $\overline{\mathrm{PQ}}$ = 9.4 cm can be drawn as
i. Draw a line segment $\overline{\mathrm{PQ}}$ whose length is 9.4 cm.
ii. With P as the centre and radius more than half of $\overline{\mathrm{PQ}}$, draw a circle using compass.
iii. With the same radius and Q as the centre, draw two arcs that cut the previous circle at points A and B. Join AB to get the perpendicular bisector of $\overline{\mathrm{PQ}}$.

 
❖ Construction of an angle of a given measure without using protractor.
Example: Draw ∠PQR = 55° without measuring. Further, draw an angle ∠ABC such that ∠ABC = PQR.
Solution:
∠ABC = ∠PQR such that ∠PQR = 55° can be constructed as
i. Construct ∠PQR = 55°
ii. Draw a line l and mark a point B on it.
iii. Place the compass at Q and draw an arc to cut the rays $\overrightarrow{\mathrm{QP}}$ and $\overline{\mathrm{QR}}$ at points X and Y respectively.
iv. Use the same compass setting to draw an arc with B as the centre, cutting l at C.
v. Set your compass to length XY.
vi. Place the compass pointer at C and draw the arc (with the same setting) that cuts the arc drawn earlier at A.
vii. Join AB.

 Now, ∠ABC = ∠PQR, where ∠PQR = 55°
 
❖ Construction of an angle of 60°.
Example: Draw an ∠A of measure 60°. Draw a ray $\overrightarrow{\mathrm{AD}}$ that bisects ∠A into two equal parts i.e., draw the bisector of ∠A.
Solution:
The bisector of ∠A, where ∠A = 60° can be drawn as
i. Draw ∠A such  that ∠A = 60°
ii. With A as the centre, draw an arc that cuts both the rays of ∠A at B and C.
iii. With B and C as centres and radius more than $\frac{1}{2}$ BC, draw two arcs that intersect each other at D.
iv. Join AD. AD is the bisector of ∠A.

 
❖ Constructing different angles by using ruler and compass
Example: Construct the following angles using ruler and compass.
(a) 60°
(b) 120°
(c) 30°
Solution:
Using ruler and compass, angles 60° and 120° can be constructed as
i. Draw a line l and mark a point O on it.
ii. Place the pointer of the compass at O and draw an arc of convenient radius that cuts l at D.
iii. With the same radius, draw an arc with centre P that cuts the previous arc at Q. Similarly, with the same radius, draw an arc with centre Q that cuts the arc at R.
iv. Join OQ and OR to get ∠QOP = 60° and ∠ROP = 120°.

Now, 30° is nothing but half of angle 60°. Therefore, 30°angle can be obtained by drawing the bisector of ∠QOP. Thus, ∠SOP = 30°.

 
❖ Similarly, we can draw other angles of measures 45°, 90°, 135°, and 150° using the above method.