Mahesh

Students can go through AP Board 8th Class Maths Notes Chapter 7 Frequency Distribution Tables and Graphs to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 7 Frequency Distribution Tables and Graphs

→ The Central Tendencies are 3 types. They are

1. Arithmetic Mean
2. Median
3. Mode

→ Information, available in the numerical form or verbal form or graphical form that helps in taking decisions or drawing conclusions is called Data.

→ Arithmetic mean of the ungrouped data = (short representation) where ∑x_i represents the sum of all x_is, where ‘i’ takes the values from 1 to n.

→ Arithmetic mean = Estimated mean + Average of deviations

→ Mean is used in the analysis of numerical data represented by unique value.

→ Median represents the middle value of the distribution arranged in order.

→ The median is used to analyse the numerical data, particularly useful when there are a few observations that are unlike mean, it is not affected by extreme values.

→ Mode is used to analyse both numerical and verbal data.

→ Mode is the most frequent observation of the given data. There may be more than one mode for the given data.

→ Representation of classified distinct observations of the data with frequencies is called ‘Frequency Distribution’ or ‘Distribution Table’.

→ Difference between upper and lower boundaries of a class is called length of the class denoted by ‘C’.

→ In a class the initial value and end value of each class is called the lower limit and upper limit respectively of that class.

→ The average of upper limit of a class and lower limit of successive class is called upper boundary of that class.

→ The average of the lower limit of a class and upper limit of preceding class is called the lower boundary of the class.
The progressive total of frequencies from the last class of the table to the lower boundary of particular class is called Greater than Cumulative Frequency (G.C.F).

→ The progressive total of frequencies from first class to the upper boundary of particular class is called Less than Cumulative Frequency (L.C.F.).

→ Histogram is a graphical representation of frequency distribution of exclusive class intervals. When the class intervals in a grouped frequency distribution are varying we need to construct rectangles in histogram on the basis of frequency density.
Frequency density = \(\frac{\text { Frequency of class }}{\text { Length of that class }}\) × Least class length in the data

→ Frequency polygon is a graphical representation of a frequency distribution (discrete/ continuous).

→ Infrequency polygon or frequency curve, class marks or mid values of the classes are taken on X-axis and the corresponding frequencies on the Y-axis.

→ Area of frequency polygon and histogram drawn for the same data are equal.

→ A graph representing the cumulative frequencies of a grouped frequency distribution against the corresponding lower/upper boundaries of respective class intervals is called Cumulative Frequency Curve or “Ogive Curve”.

Students can go through AP SSC 10th Class Maths Notes Chapter 13 Probability to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 13 Probability

→ Theory of probability has its origin date back to 16th century.

→ J. Cardan, an Italian physician and mathematician wrote the first book on probability named “The Book of Games of Chance”.

→ James Bernoulli (1654 – 1705), A.De Moivre (1667 – 1754) and Pierre Simon Laplace (1749 – 1827) made significant contribution to the theory of probability.

→ Experimental or empirical probability : The probability estimated on the basis of results of an actual experiment is called experimental probability of empirical probability.
Eg : An unbiased coin is tossed for 1000 times, head turned up for 455 times and tail turned up 545 times, then the probability or likelyhood of getting a head is = \(\frac{455}{1000}\) = 0.455.

Thus experimental probability = \(\frac{\text { No. of trials in which the event happened }}{\text { Total no. of trials }}\)

→ Classical or Theoretical probability: Classical probability of an event (E) is defined Number as

P(E) = \(\frac{\text { Number of outcomes favourable to E }}{\text { No. of all possible outcomes of the experiment }}\)

This definition was given by ‘Pierre Simon Laplace’.
Eg: The probability of getting a head when a coin is tossed is given by Number of outcomes favourable to this event getting a head = 1 Number of all possible outcomes of this experiment = 2 (Head, Tail)

∴ P(E) = \(\frac{\text { No. of favourable outcomes }}{\text { Total events }}\) = \(\frac{1}{2}\)

Note: If an experiment is conducted for many number of times, then the experimental probability may become closer and closer to theoretical probability.

→ The probability of a sure event is 1.

→ The probability of an impossible event is zero.

→ The probability of an event E is a number P(E) such that 0 ≤ P(E) ≤ 1.

→ An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.

→ For any event E, P(E) + P(\(\overline{\mathrm{E}}\)) = 1, where E and \(\overline{\mathrm{E}}\) are complementary events.

→ Playing cards and their probability : A deck of playing cards consists of 52 cards which are divided into four suits of 13 cards each.
They are:

→ The cards in each suit are:

Eg : When a card is drawn at random from a deck of cards then

• Getting a black or red card – equally likely exhaustive events.
• Getting an ace or king – mutually exclusive.
  
• Getting an ace or a hearts – not mutually exclusive since the hearts contain an ace.
  

→ When a coin is tossed, the outcomes are H, T (Head, Tail).

→ When a dice is thrown the outcomes are 1, 2, 3, 4, 5 and 6.

→ When two dice are thrown, the outcomes are

→ If a coin is tossed n-times or n – coins are tossed simultaneously, then the number of total outcomes = 2ⁿ.

→ If a dice is thrown for n – times or n – dice are thrown simultaneously then the number of total outcomes = 6ⁿ.

Students can go through AP Board 8th Class Maths Notes Chapter 6 Square Roots and Cube Roots to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 6 Square Roots and Cube Roots

→ The product of two same numbers is called its square.
Ex: 1) x × x = x²
2) 6 × 6 = 6² = 36

→ The digits in the units place of a square number are 0, 1, 4, 5, 6, 9.

→ If the digits 2,3, 7 or 8 are in the units place of an umber then it is not a perfect square.

→ If there are ‘n’ digits in a number then the no.of digits in its square = 2n or (2n -1).

→ Sum of ‘n ‘ consecutive odd numbers = n²

→ The square of any odd number say ‘n’ can be expressed as the sum of two consecutive numbers as

→ If a, b, c are any three positive integers and a² + b² = c² then a, b, c are called Pythagorean triplets. Ex: (3, 4, 5) (5, 12, 13).

→ If a square number is expressed, as the product of two equal factors, then one of the factors is called the square root of that square number. Thus, the square root of 169 is 13. It can be expressed as √169 = 13 (symbol used for square root is √). Thus it is the inverse operation of squaring.

→ If the same number is multiplied itself by 3 times then it is called a cube of a number. Ex: cube of x = x × x × x = x³

→ If a cube number is expressed, as the product of 3 equal factors, then one of he factors is called the cube root of that number.
Symbol for cube root is \(\sqrt[3]{ }\)
Ex: \(\sqrt[3]{64}=\left(4^{3}\right)^{1 / 3}\) = 4

→

Students can go through AP Board 8th Class Maths Notes Chapter 5 Comparing Quantities Using Proportion to understand and remember the concepts easily.

AP State Board Syllabus 8th Class Maths Notes Chapter 5 Comparing Quantities Using Proportion

→ Two simple ratios are expressed like a single ratio as the ratio of product of antecedents to product of consequents and we call it compound ratio of the given two simple ratios.
a : b and c : d are any two ratios, then their compound ratio is \(\frac{a}{b}\) × \(\frac{c}{d}\) = \(\frac{ac}{bd}\) i.e. ac : bd.

→ A percentage(%) compares a number to 100. The word percent means “per every hundred” or “out of every hundred”. 100% = \(\frac{100}{100}\) it is also a fraction with denominator 100.

→ Discount is a decrease percent of marked price. Price reduction is called rebate or discount. It is calculated on marked price or list price.

→ Profit or loss is always calculated on cost price. Profit is an example of increase percent of cost price and loss is an example of decrease percent of cost price.

→ VAT will be charged on the selling price of an item and will be included in the bill.
VAT is an increase percent on selling price.

→ Simple interest is an increase percent on the principal.

→ Simple interest (I) = \(\frac{P \times T \times R}{100}\)
where P = Principal, T = Time inyears, R = Rate of interest.

→ Amount = Principal + Interest = P + \(\frac{P \times T \times R}{100}\) = P\(\left(1+\frac{T \times R}{100}\right)\)

→ Compound interest allows you to earn interest on interest.

→ Amount at the end of ‘n’ years using compound interest is A = P \(\left(1+\frac{R}{100}\right)^{n}\)

→ The time period after which interest is added to principal is called conversion period.
When interest is compounded halfyearly, there are two conversion periods in a year, each after 6 months. In such a case, ha If year rate will be half of the annual rate.

→ Note: 1.615 : 1 is called as golden ratio.
In ancient Greece, artists and architects believed there was a particular rectangular shape that looked very pleasing to the eye. For rectangles of this shape, the ratio of long side to the short side is roughly 1.615 : 1. This ratio is very close to what is known as golden ratio.

Students can go through AP SSC 10th Class Maths Notes Chapter 12 Applications of Trigonometry to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 12 Applications of Trigonometry

→ If a person is looking at an object then the imaginary line joining the object and the eye of the observer is called the line of sight or ray of view.

→ An imaginary line parallel to earth surface and passing through the point of observation is called the horizontal.

→ If the line of sight is above the horizontal then the angle between them is called “angle of elevation”.

→ If the line of sight is below the horizontal then the angle between them is called the angle of depression.

→ Useful hints to solve the problems:

1. Draw a neat diagram of a right triangle or a combination of right triangles if necessary.
2. Represent the data given on the triangle.
3. Find the relation between known values and unknown values.
4. Choose appropriate trigonometric ratio and solve for the unknown.

→ The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.

→ To use this application of trigonometry, we should know the following terms.

→ The terms are Horizontal line, Line of Sight, Angle of Elevation and Angle of Depression.

→ Horizontal line: A line which is parallel to earth from observation point to object is called “horizontal line”.

→ Line of Sight (or) Ray of Vision: The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer.

→ Angle of Elevation: The line of sight is above the horizontal line then angle between the line of sight and the horizontal line is called “angle of elevation”.

Note:

1. If the angle of observer moves towards the perpendicular line (pole/tree/ building), then angle of elevation increases and if the observer moves away from the perpendicular line (pole/tree/building), then angle of elevation decreases.
2. If height of tower is doubled and the distance between the observer and foot of the tower is also doubled, then the angle of elevation remains same.
3. If the angle of elevation of sun above a tower decreases, then the length of shadow of a tower increases.

→ Angle of Depression: The line of sight is below the horizontal line then angle between the line of sight and the horizontal line is called angle of depression.

Note:

1. The angle of elevation and depression are always acute angles.
2. The angle of elevation of a point P as seen from a point ‘O’ is always equal to the angle of depression of ‘O’ as seen from P.

→ Points to be kept in mind:
I. Trigonometric ratios in a right triangle:

II. Trigonometric ratios of some specific angles:

→ Solving Procedure:
When we want to solve the problems of height and distances, we should consider the following :

1. All the objects such as tower, trees, buildings, ships, mountains, etc. shall be considered as linear for mathematical convenience.
2. The angle of elevation or angle of depression is considered with reference to the horizontal line.
3. The height of the observer is neglected, if it is not given in the problem.
4. To find heights and distances, we need to draw figures and with the help of these figures we can solve the problems.

Students can go through AP Board 9th Class Maths Notes Chapter 4 Lines and Angles to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 4 Lines and Angles

→ A ray is a part of line. It begins at a point and goes on endlessly in a specified direction.

→ A part of a line with two end points is called a line segment.

→ Points on the same line are called collinear points.

→ The angle is formed by rotating a ray from an initial position to a terminal position.

→ One complete rotation makes an angle 360°.

→ Angles are named according to their measure.
Obtuse angle 90° < x < 180°.

→ Straight angle y = 180°

→ Reflex angle 180° < z < 360°

→ If two lines have no common points, they are called parallel lines. In the figure l // m.

→ If two lines have a common point then they are called intersecting lines.

→ Three or more lines meet at a point are called concurrent lines.

→ Two angles are said to be supplementary angles if their sum is 180°.
E.g.: (100°, 80°), (110°, 70°), (120°, 60°), (179°, 1°), (90°, 90°) etc.

→ The supplementary angle to x° is given by (180° – x°).

→ Two angles are said to be complementary if their sum is 90°.

→ The complementary angle to x° is (90° – x°).
E.g.: (89°, 1°), (70°, 20°), (60°, 30°) etc.

→ Two angles are said to be form a pair of adjacent angles if they have a common arm and lie on the either sides of the common arm.

∠1 and ∠2 are a pair of adjacent angles with OB as their common arm.

→ A pair of adjacent angles are said to be a linear pair of angles if their sum is 180°. ∠1 + ∠2 = 180°

→ When a pair of lines meet at a point, they form four angles. The two pairs of angles which have no common arm are called vertically opposite angles.
In the figure (∠1, ∠3) and (∠2, ∠4) are the pairs of vertically opposite angles.

→ When two lines intersect, the pairs of vertically opposite angles thus formed are equal a = c and b = d. (from the figure)

→ When a pair of lines intersected by a transversal, there forms eight angles.

→ When a pair of parallel lines intersected by a transversal the pairs of a) alternate interior angles b) corresponding angles c) alternate exterior angles are equal and the interior / exterior angles on the same side of the transversal are supplementary.

→ Lines which are parallel to same line are parallel to each other.

→ The sum of the interior angles of a triangle is 180°.

→ If one side of a triangle is produced, then the exterior angle thus formed is equal to the sum of the two interior opposite angles.

Students can go through AP SSC 10th Class Maths Notes Chapter 11 Trigonometry to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 11 Trigonometry

→ In our daily life, we can measure the heights, distances and slopes by using some mathematical techniques.

→ The mathematical techniques which come under a branch of mathematics is called ‘trigonometry’.

→ “Trigonometry” is the study of relationships between the sides and angles of a triangle.

→ Early astronomers used to find out the distances of the stars and planets from the Earth. Even today, most of the technologically advanced methods used in engineering and physical sciences are based on trigonometrical concepts.

→ Naming the sides in a right triangle:
Let’s take a right triangle ABC as shown in the figure.

Consider ∠CAB as A where angle ‘A’ is acute.
Since AC is the longest side, it is called “Hypotenuse”.

→ Now observe the position of side BC with respect to angle A. It is opposite to angle ‘A’ and we can call it as “Opposite side of angle A”.

→ And the remaining side AB can be called as “Adjacent side of angle A”.

→ Trigonometric Ratios:
The ratios of the sides of a right angled triangle with respect to its acute angles, are called Trigonometric ratios.

→ Consider a right angled triangle ABC having right angle at B as shown in the given figure.

Then, trigonometric ratios of the angle A in right angled triangle ABC are defined as follows:

→ There are three more ratios defined in trigonometry which are considered as multiplicative inverse of the above three ratios.

→ Multiplicative inverse of “sine A” is “cosecant A”.
i.e., cosec A = \(\frac{1}{\sin A}\) = \(\frac{\text { Hypotenuse }}{\text { Opposite side of the angle } A}\)

→ Multiplicative inverse of “cosine A” is “secant A”.
i.e., sec A = \(\frac{1}{\cos A}\) = \(\frac{\text { Hypotenuse }}{\text { Adjacent side of the angle } A}\)

→ Multiplicative inverse of “tangent A” is “cot A”.
i.e., cot A = \(\frac{1}{\tan A}\) = \(\frac{\text { Adjacent side of the angle } A}{\text { Opposite side of the angle } A}\)

Note:
i) The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangles, if the angle remains the same.
ii) Each trigonometric ratio is a real number and has no unit.
iii) “sin θ” is one symbol and sin, cos, tan etc., cannot be separated from θ.
iv) If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ratios of angle can be easily determined.
v)

→ Trigonometric ratios of some specific angles:
The values of various trigonometric ratios of 0°, 30°, 45°, 60° and 90°.

Note:
i) The values of “sin θ” and “cos θ” always lie between ‘0’ and ‘1’.
ii) In the case of tan θ, the values increase from 0 to ∞ (not determinate).
iii) In the case of cot θ, the values decrease from ∞ to 0.
iv) In the case of cosec θ, the values decrease from ∞ to 1.
v) In the case of sec θ, the values increase from 1 to ∞.

→ Trigonometric ratios of complementary angles:
Two angles are said to be complementary, if their sum is equal to 90°.
In a right angled triangle, if ∠B = 90°, then ∠A + ∠C = 90° i.e., ∠A and ∠C form a pair of complementary angles.
If ‘θ’ is an acute angle, then we can prove that
sin (90 – θ) = cos θ
cos (90 – θ) = sin θ
tan (90 – θ) = cot θ
cot (90 – θ) = tan θ
sec (90 – θ) = cosec θ
cosec (90 – θ) = sec θ

→ Trigonometric Identity: An identity equation having trigonometric ratios of an angle is called trigonometric identity. It is true for all the values of the angles involved in it.
We have three major trigonometric identities. They are
i) sin² A + cos² A = 1
ii) sec² A – tan² A = 1
iii) cosec² A – cot² A = 1
Note: sin² θ = (sin θ)² but sin θ² ≠ (sin θ)²

Students can go through AP Board 9th Class Maths Notes Chapter 3 The Elements of Geometry to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 3 The Elements of Geometry

→ Geometry is structured on its building blocks namely point, line and plane.

→ In geometry there are undefined terms like point, plane and line.

→ Angles, circles and triangles are the examples for defined terms.

→ No better entrance exists than Euclid’s time honoured ‘Elements’.

→ In ‘The Elements’, Euclid developed a new system of thought which laid the foundation for the advancement of the geometry.

→ Some of the Euclid’s axioms are:

• Things which are equal to same things are equal to one another.
• If equals are added to equals, the wholes are equal.
• If equals are subtracted from equals, the remainders are also equal.
• Things which coincide with one another are equal to one another.
• The whole is greater than part.
• Things which are double of the same things are equal to one another.
• Things which are halves of the same things are equal to one another.

→ Euclid’s postulates are
Postulate – 1 : To draw a straight line from any point to any point.
Postulate – 2 : A terminated line can be produced indefinitely.
Postulate – 3 : To describe a circle with any centre and radius.
Postulate – 4 : That all right angles are equal to one another.
Postulate – 5 : If a straight line falling on two straight lines makes the interior angles on the same side of it taken together is less than two right angles, then the two straight lines, if produced infinitely, meet on that side on which the sum of the angles is less than two right angles.

→ Equivalent versions of Euclid’s fifth postulate:

• Through a point not on a given line, exactly one parallel line may be drawn to the given line – John Play Fair (1748 – 1819).
• The sum of angles of any triangle is a constant and is equal to two right angles (Legendre).
• There exists a pair of lines everywhere equidistant from one another (Posidominus).
• If a straight line intersects any one of two parallel lines, then it will intersect the other also (Proclus).
• The statements that were proved to be true are called propositions or theorems.
• The statements neither proved nor disproved are called conjectures.
• There are non-Euclidian geometries.

Students can go through AP SSC 10th Class Maths Notes Chapter 10 Mensuration to understand and remember the concepts easily.

AP State Syllabus SSC 10th Class Maths Notes Chapter 10 Mensuration

→ A solid is a geometrical shape with three dimensions namely length, breadth and height.
Eg:

→ A solid has two types of area namely,
a) Lateral Surface Area (L.S.A.)
b) Total Surface Area (T.S.A,)

→ In general, L.S.A. of a solid is the product of its base perimeter and height.
Eg : L.S.A. of a cuboid = 2h(l + b)
L.S.A. of a cylinder = 2πrh

→ The T.S.A. of a solid is the sum of L.S.A. and the areas of its top and base.
Eg : T.S.A. of a cylinder = 2πrh + 2πr²
= 2πr(r + h)

→ In general, the volume of a solid is the product of its base area and height.
V = A. h
Eg: Volume of a cube = a² . a = a³
Volume of a cylinder = πr² . h = πr²h

→ The volume of solid formed by joining two basic solids is the sum of volumes of the constituents.

→ Surface area of the combination of solids: In calculating the surface area of the solid which is a combination of two or more solids, we can’t add the surface areas of all its constituents, because some part of the surface area disappears in the process of joining them.

→ Surface areas and volume of different solid shapes:

→ Some solid figures and their combination shapes:

Students can go through AP Board 9th Class Maths Notes Chapter 2 Polynomials and Factorisation to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 2 Polynomials and Factorisation

→ An algebraic expression ¡n which the variables involved have only non-negative integral powers is called a polynomial.
E.g.: 5x³ – 2x + 8

→ Polynomials can be classified as monomials. binomials, trinomials and polynomials based on the number of terms it contains.

→ A polynomial with a single term is a monomial.
E.g.: 2x, -5x², \(\frac{6}{7}\)x³ etc.

→ A polynomial with two terms is a binomial.
E.g.: 2x + 5y; -3x² + 5x etc.

→ A polynomial with three terms is a trinomial.
E.g.: 3x² + 5x – 8; 3x + 2y – 5z etc.

→ In general a polynomial may contain two or more than two terms.
E.g.: 5x + 8x² – 3x³ + 7

→ Degree of a polynomial ¡s the heighest degree of its variable terms.
E.g.: Degree of 3x² + 4xy³ + y² is 4.

→ Degree of a constant term is considered as zero.
E.g.: Degree of 4 is zero.

→ The general form of a polynomial is a₀xⁿ + a₁x^n-1 + a₂x^n-2 …….. + a_n-1x + a_n where a₀, a₁, a₂,…… a_n-1, a_n are constants and a₀ ≠ 0. Its degree is ‘n’.

→ Polynomials are again classified based on their degrees.

→ The zero of a polynomial p(x) is the value of x at which p(x) becomes zero (i.e.) p(x) = 0.
E.g.: Zero of 3x – 5 is x = \(\frac{5}{3}\)

→ To find the zero of a polynomial we equate the polynomial to zero and solve for the value of the variable.
E.g.: To find zero of 7x + 8.
7x + 8 = 0
7x = – 8
x = \(\frac{-8}{7}\)

→ Let p(x) be any polynomial of degree greater than or equal to one and let ‘a’ be any real number. If p(x) is divided by the linear polynomial (x – a), then the remainder is p(a). This is called the Remainder theorem.
E.g.: If p(x) = 4x³ + 3x + 8 then the remainder when it is divided by x – 1 is p(1).
i.e., p(1) = 4 + 3 + 8 = 15.

→ If p(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then
(i) (x – a) is a factor of p(x), if p(a) = 0
(ii) and its converse “if (x – a) is a factor of a polynomial p(x) then p(a) = 0. This is called Factor theorem”.
E.g.: Let p(x) = x² – 5x + 6 and p(2) = 2² – 5(2)+ 6 = 0 and hence (x – 2) is a factor of p(x) conversely; p(x) = x² + 7x + 10 and (x + 2) is a factor, then p(-2) = 0.

→ Algebraic identities

• (x + y)² = x² + 2xy + y²
• (x – y)² = x² – 2xy + y²
• (x + y) (x-y) = x² – y²
• (x + a) (x + b) = x² + (a + b) x + ab
• (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
• (x + y)³ = x³ + y³ + 3xy (x + y)
• (x – y)³ = x³ – y³ – 3xy (x – y)
• (x + y + z) (x² + y² + z² – xy – yz – zx) = x³ + y³ + z³ – 3xyz

Students can go through AP Board 9th Class Maths Notes Chapter 1 Real Numbers to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 1 Real Numbers

→ Numbers of the form \(\frac{p}{q}\) where p and q are integers and q ≠ 0 are called rational numbers, represented by ‘Q’.

→ There are infinitely many rational numbers between any two integers.
E.g.: 3 < \(\frac{19}{6}\), \(\frac{20}{6}\), \(\frac{21}{6}\), \(\frac{22}{6}\), \(\frac{23}{6}\), ……. < 4

→ There are infinitely many rational numbers between any two rational numbers.
E.g.: \(\frac{3}{4}\) < \(\frac{29}{8}\) < \(\frac{71}{16}\) < \(\frac{81}{14}\) ……. < \(\frac{13}{2}\)

→ To find the decimal representation of a rational number we divide the numerator of a rational number by its denominator.
E.g.: The decimal representation of \(\frac{5}{6}\) is

∴ \(\frac{5}{6}\) = 0.833 …. = 0.8 \(\overline{3}\)

→ Every rational number can be expressed as a terminating decimal or as a non-terminating repeating decimal. Conversely every terminating decimal or non¬terminating recurring decimal can be expressed as a rational number.
E.g.: 1.6 \(\overline{2}\) = \(\frac{161}{99}\)

→ A rational number whose denominator consists of only 2’s or 5’s or a combination of 2’s and 5’s can be expressed as a terminating decimal.
E.g. : \(\frac{13}{32}\) can be expressed as a terminating decimal (∵ 32 = 2 × 2 × 2 × 2 × 2)
\(\frac{7}{125}\) can be expressed as a terminating decimal (∵ 125 = 5 × 5 × 5)
\(\frac{24}{40}\) can be expressed as a terminating decimal (∵ 40 = 2 × 2 × 2 × 5)

→ Numbers which can’t written in the form \(\frac{p}{q}\) where p and q are integers and q ≠ 0, are called irrational numbers.
E.g.: √2, √3, √5,….. etc.
The decimal form of an irrational number is neither terminating nor recurring decimal.

→ Irrational numbers can be represented on a number line using Pythagoras theorem.
E.g.: Represent √2 on a number line.

→ If ‘n’ is a natural number which is not a perfect square, then √n is always an irrational number.
E.g.: 2, 3, 5, 7, 8, …… etc., are not perfect squares.
∴ √2, √3, √5, √7 and √8 are irrational numbers.

→ We often write π as \(\frac{22}{7}\) there by π seems to be a rational number; but π is not a rational number.

→ The collection of rational numbers together with irrational numbers is called set of Real numbers.
R = Q ∪ S

→ If a and b are two positive rational numbers such that ab is not a perfect square, then , √ab is an irrational number between ‘a’ and b’.
E.g.: Consider any two rational numbers 7 and 4.
7 × 4 = 28 is not a perfect square; then √28 lies between 4 and 7.
i.e., 4 < √28 < 1

→ If ‘a’ is a rational number and ‘b’ is arty irrational number then a + b, a – b, a.b or \(\frac{a}{b}\) is an irrational number.
E.g.: Consider 7 and √5 then 7 + √5, 7 – √5, 7√5 and \(\frac{7}{\sqrt{5}}\)= are all irrational numbers.

→ If the product of any two irrational numbers is a rational number, then they are said to be the rationalising factor of each other.
E.g.: Consider any two irrational number 7√3 and 5√3.
7√3 × 5√3 = 7 × 5 × 3 = 105 a rational number.
Also 7√3 × √3 = 21 – a rational number.
5√3 × √3 = 15 – a rational number.
So the rationalising factor of an irrational number is not unique.

→ The general form of rationalising factor (R.F.) of (a ± √b} is (a ∓ √b). They are called conjugates to each other.

→ Laws of exponents:

i) a^m × aⁿ = a^m+n
e.g.: 5⁴ . 5^-3 = 5^4+(-3) = 5¹ = 5

ii) (a^m)ⁿ = a^mn
e.g.: (4³)² = 4^3×2 = 4⁶

iii)
 = a^m-n if (m > n)
= 1 if m = n
= \(\frac{1}{a^{n-m}}\) if (m < n)

iv) a^m . b^m = (ab)^m
e.g.:(-5)³ . (2)³ = (-5 × 2)³ =(-10)³

v) \(\frac{1}{a^{n}}\) = a^-n
e.g.: (6)^-3 = \(\frac{1}{6^{3}}\) = \(\frac{1}{216}\)

vi) a⁰ = 1
e.g.: \(\left(\frac{-3}{4}\right)^{0}\) = 1
Where a, b are rationals and m, n are integers.

→ Let a, b be any two rational numbers such that a = bⁿ then b = \(\sqrt[n]{a}\) = \((\mathrm{a})^{1 / \mathrm{n}}\)
Here ‘b’ is called n^th root of a.
e.g.: 4² = 16 then \(16^{1 / 2}\) or \(\sqrt[2]{16}\)
3⁴ = 81 then 3 = \(\sqrt[4]{81}\) or \((81)^{1 / 4}\)

→ Let ‘a’ be a positive number and n > 1 then \(\sqrt[n]{a}\) i.e., n^th root of a is called a surd.