ch16

1.1 Rational Numbers

• Definition: A number that can be expressed in the form \(p/q\), where \(p\) and \(q\) are integers and \(q \neq 0\).
• Examples: \(1/2, -3/4, 5, 0, -7/1\)
• Properties of Rational Numbers:
• Closure Property:
• Addition: Rational numbers are closed under addition. \(a+b\) is a rational number.
• Subtraction: Rational numbers are closed under subtraction. \(a-b\) is a rational number.
• Multiplication: Rational numbers are closed under multiplication. \(a \times b\) is a rational number.
• Division: Rational numbers are NOT closed under division (division by zero is undefined).
• Commutativity:
• Addition: \(a+b = b+a\)
• Multiplication: \(a \times b = b \times a\)
• Subtraction/Division: NOT commutative.
• Associativity:
• Addition: \((a+b)+c = a+(b+c)\)
• Multiplication: \((a \times b) \times c = a \times (b \times c)\)
• Subtraction/Division: NOT associative.
• Distributivity: \(a \times (b+c) = a \times b + a \times c\) and \(a \times (b-c) = a \times b - a \times c\)
• Additive Identity: 0 (\(a+0 = 0+a = a\))
• Multiplicative Identity: 1 (\(a \times 1 = 1 \times a = a\))
• Additive Inverse (Negative): For a rational number \(a/b\), its additive inverse is \(-a/b\). (\(a/b + (-a/b) = 0\))
• Multiplicative Inverse (Reciprocal): For a rational number \(a/b\), its multiplicative inverse is \(b/a\) (if \(a \neq 0\)). (\(a/b \times b/a = 1\))
• Representation on Number Line:
• Divide the unit length between integers into \(q\) equal parts for \(p/q\).
• Rational Numbers Between Two Rational Numbers:
• Method 1: Mean Method. The mean of two rational numbers \(a\) and \(b\) is \((a+b)/2\), which lies between \(a\) and \(b\).
• Method 2: Equivalent Rational Numbers. Convert to equivalent rational numbers with a common denominator and then find numbers between their numerators.

2.1 Linear Equations

• Equation: A statement of equality between two algebraic expressions involving one or more variables.
• Linear Equation: An equation where the highest power of the variable is 1.
• Linear Equation in One Variable: An equation with only one variable and its highest power is 1.
• General Form: \(ax + b = 0\), where \(a \neq 0\).
• Solving Linear Equations:
• Balancing Method: Perform the same operation (add, subtract, multiply, divide) on both sides of the equation to isolate the variable.
• Transposing Method: Move terms from one side to the other by changing their sign.
• Addition becomes subtraction, multiplication becomes division, etc.
• Equations with Variables on Both Sides: Collect all variable terms on one side and constant terms on the other.
• Equations Reducible to Linear Form: Equations involving fractions can often be simplified by cross-multiplication or by multiplying both sides by the LCM of the denominators.
• Applications: Word problems involving age, money, perimeter, etc., can be translated into linear equations and solved.

3.1 Polygons

• Polygon: A simple closed curve made up of only line segments.
• Classification by number of sides:
• 3 sides: Triangle
• 4 sides: Quadrilateral
• 5 sides: Pentagon
• 6 sides: Hexagon
• 7 sides: Heptagon
• 8 sides: Octagon
• Diagonals: A line segment connecting two non-consecutive vertices of a polygon.
• Convex Polygon: All diagonals lie entirely inside the polygon.
• Concave Polygon: At least one diagonal lies partly or wholly outside the polygon.
• Regular Polygon: A polygon that is both equiangular (all angles equal) and equilateral (all sides equal).
• Examples: Square, equilateral triangle.
• Angle Sum Property:
• Sum of interior angles of a polygon: \((n-2) \times 180^{\circ}\), where \(n\) is the number of sides.
• Each interior angle of a regular polygon: \(\frac{(n-2) \times 180^{\circ}}{n}\)
• Sum of exterior angles of any polygon: \(360^{\circ}\)
• Each exterior angle of a regular polygon: \(\frac{360^{\circ}}{n}\)
• Relationship: Interior Angle + Exterior Angle = \(180^{\circ}\)

3.2 Quadrilaterals

• Quadrilateral: A four-sided polygon.
• Types of Quadrilaterals:
• Trapezium (Trapezoid): A quadrilateral with at least one pair of parallel sides.
• Kite: A quadrilateral where two distinct pairs of equal adjacent sides.
• Diagonals are perpendicular.
• One diagonal bisects the other.
• One pair of opposite angles are equal.
• Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
• Properties:
• Opposite sides are equal.
• Opposite angles are equal.
• Diagonals bisect each other.
• Adjacent angles are supplementary (sum to \(180^{\circ}\)).
• Rhombus: A parallelogram with all four sides equal.
• Properties: All properties of a parallelogram + diagonals are perpendicular bisectors of each other.
• Rectangle: A parallelogram with all four angles equal to \(90^{\circ}\).
• Properties: All properties of a parallelogram + diagonals are equal.
• Square: A rectangle with all four sides equal (or a rhombus with all four angles equal to \(90^{\circ}\)).
• Properties: All properties of parallelogram, rhombus, and rectangle.

3.3 Special Parallelograms Summary

4.1 Constructing Quadrilaterals

• To construct a unique quadrilateral, at least five measurements are needed.
• Cases for Construction:

1. Four sides and one diagonal are given:

• Construct a triangle using SSS criterion with the diagonal and two sides.
• Use the remaining two sides and the other endpoint of the diagonal to complete the quadrilateral.

1. Two diagonals and three sides are given:

• Construct a triangle using SSS criterion.
• Use the remaining diagonal and side to locate the fourth vertex.

1. Two adjacent sides and three angles are given:

• Draw one side.
• Construct the two angles at its endpoints.
• Mark the length of the adjacent side.
• Construct the third angle to find the intersection point.

1. Three sides and two included angles are given:

• Draw the middle side.
• Construct the two included angles at its endpoints.
• Mark the lengths of the other two sides along the angle arms.
• Join the endpoints.

1. When other special properties are known: (e.g., square, rhombus, rectangle, parallelogram)

• Use the specific properties (e.g., all sides equal, diagonals bisect each other, angles are \(90^{\circ}\)) to deduce missing measurements and then follow one of the above methods.

5.1 Data

• Data: A collection of numerical facts gathered to give some information.
• Raw Data: Data collected in its original form.
• Frequency: The number of times a particular observation occurs.
• Frequency Distribution: A table showing the frequency of different observations.
• Grouped Frequency Distribution: Data grouped into class intervals.
• Class Interval: The range of data values (e.g., 0-10, 10-20).
• Lower Class Limit: The smallest value in a class interval.
• Upper Class Limit: The largest value in a class interval.
• Class Size/Width: Difference between the upper and lower class limits.

5.2 Graphical Representation of Data

• Pictograph: Uses pictures to represent data.
• Bar Graph: Uses bars of uniform width, with heights proportional to the frequency of the data.
• Bars are separated.
• Histogram: A bar graph for continuous class intervals.
• Bars are adjacent (no gaps).
• Height of bar represents frequency.
• Width of bar represents class size.
• Circle Graph (Pie Chart): Shows the relationship between a whole and its parts.
• The whole circle represents \(360^{\circ}\) or 100%.
• Each sector's angle is proportional to the component it represents: \(\text{Angle of sector} = \frac{\text{Value of component}}{\text{Total value}} \times 360^{\circ}\)

5.3 Chance and Probability

• Experiment: An operation that can produce some well-defined outcomes.
• Random Experiment: An experiment where the outcome cannot be predicted with certainty.
• Outcome: A possible result of a random experiment.
• Event: A collection of one or more outcomes of an experiment.
• Probability of an Event (P(E)):
• \(P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}\)
• Range of Probability: \(0 \le P(E) \le 1\)
• \(P(E) = 0\) for an impossible event.
• \(P(E) = 1\) for a sure event.

6.1 Squares

• Square Number (Perfect Square): A natural number is called a perfect square if it is the product of two identical integers (i.e., \(n \times n = n^2\)).
• Examples: \(1, 4, 9, 16, 25, \dots\)
• Properties of Square Numbers:
• Numbers ending in 2, 3, 7, or 8 are never perfect squares.
• Numbers ending in an odd number of zeros are never perfect squares.
• The unit digit of a square number is the unit digit of the square of its unit digit.
• If a number ends in 1 or 9, its square ends in 1.
• If a number ends in 2 or 8, its square ends in 4.
• If a number ends in 3 or 7, its square ends in 9.
• If a number ends in 4 or 6, its square ends in 6.
• If a number ends in 0 or 5, its square ends in 0 or 5 respectively.
• The square of an even number is even. The square of an odd number is odd.
• A perfect square can be expressed as the sum of consecutive odd numbers starting from 1.
• \(1 = 1^2\)
• \(1+3 = 4 = 2^2\)
• \(1+3+5 = 9 = 3^2\)
• Pythagorean Triplets: Three natural numbers \(m, n, p\) form a Pythagorean triplet if \(m^2 + n^2 = p^2\).
• For any natural number \(m > 1\), \((2m, m^2-1, m^2+1)\) forms a Pythagorean triplet.

6.2 Square Roots

• Square Root: The square root of a number \(x\) is the number \(y\) such that \(y^2 = x\).
• Represented by the symbol \(\sqrt{}\).
• Every positive number has two square roots: a positive and a negative one (e.g., \(\sqrt{25} = \pm 5\)). Usually, we refer to the positive square root.
• Methods to find Square Root:
• Repeated Subtraction Method: Subtract consecutive odd numbers starting from 1 until 0 is obtained. The number of subtractions is the square root.
• Example: \(\sqrt{9}\)
• \(9-1=8\)
• \(8-3=5\)
• \(5-5=0\)
• 3 subtractions, so \(\sqrt{9}=3\).
• Prime Factorisation Method:

1. Find the prime factors of the number.
2. Group the factors in pairs.
3. Take one factor from each pair and multiply them.

• Long Division Method: A general method for finding square roots of large numbers or non-perfect squares.
• Estimating Square Roots: Find two perfect squares between which the given number lies.

7.1 Cubes

• Cube Number (Perfect Cube): A natural number is called a perfect cube if it is the product of three identical integers (i.e., \(n \times n \times n = n^3\)).
• Examples: \(1, 8, 27, 64, 125, \dots\)
• Properties of Cube Numbers:
• Cubes of even numbers are even. Cubes of odd numbers are odd.
• Numbers ending in 0, 1, 4, 5, 6, 9 have their cubes ending in the same digit.
• Numbers ending in 2 have their cubes ending in 8.
• Numbers ending in 8 have their cubes ending in 2.
• Numbers ending in 3 have their cubes ending in 7.
• Numbers ending in 7 have their cubes ending in 3.
• A perfect cube can end in any digit from 0 to 9.

7.2 Cube Roots

• Cube Root: The cube root of a number \(x\) is the number \(y\) such that \(y^3 = x\).
• Represented by the symbol \(\sqrt[3]{}\).
• Every number has exactly one real cube root.
• Methods to find Cube Root:
• Prime Factorisation Method:

1. Find the prime factors of the number.
2. Group the factors in triplets.
3. Take one factor from each triplet and multiply them.

• Estimating Cube Roots: Similar to square roots, find two perfect cubes between which the given number lies. Also, use the unit digit property to guess the unit digit of the cube root.

8.1 Ratios and Percentages

• Ratio: Comparison of two quantities of the same unit. \(a:b\) or \(a/b\).
• Percentage: A way of comparing quantities. 'Per cent' means 'per hundred'.
• To convert a fraction/decimal to percentage, multiply by 100.
• To convert a percentage to fraction/decimal, divide by 100.
• Finding Increase/Decrease Percent:
• \(\text{Percentage Change} = \frac{\text{Amount of Change}}{\text{Original Amount}} \times 100\%\)

8.2 Profit, Loss, Discount

• Cost Price (CP): The price at which an article is purchased.
• Selling Price (SP): The price at which an article is sold.
• Profit: \(SP > CP\), Profit \(= SP - CP\)
• Loss: \(CP > SP\), Loss \(= CP - SP\)
• Profit/Loss Percent: Always calculated on CP.
• \(\text{Profit Percent} = \frac{\text{Profit}}{CP} \times 100\%\)
• \(\text{Loss Percent} = \frac{\text{Loss}}{CP} \times 100\%\)
• Discount: Reduction given on the Marked Price (MP) or List Price.
• \(\text{Discount} = MP - SP\)
• \(\text{Discount Percent} = \frac{\text{Discount}}{MP} \times 100\%\)
• Sales Tax (ST)/Value Added Tax (VAT)/Goods and Services Tax (GST): Added to the selling price of the article.
• \(\text{Bill Amount} = \text{Cost of item} + \text{Tax}\)

8.3 Simple and Compound Interest

• Principal (P): The money borrowed or deposited.
• Interest (I): The extra money paid for using borrowed money.
• Rate (R): The interest rate per annum.
• Time (T): The period for which the money is borrowed/deposited.
• Amount (A): Principal + Interest.
• Simple Interest (SI): Interest calculated only on the original principal amount.
• \(SI = \frac{P \times R \times T}{100}\)
• \(A = P + SI\)
• Compound Interest (CI): Interest calculated on the principal amount and also on the accumulated interest of previous periods.
• \(A = P(1 + \frac{R}{100})^n\), where \(n\) is the number of compounding periods.
• \(CI = A - P\)
• Compounding Period:
• Annually: \(R\) and \(n\) remain as given.
• Half-yearly (Semi-annually): Rate becomes \(R/2\), time becomes \(2n\).
• Quarterly: Rate becomes \(R/4\), time becomes \(4n\).

9.1 Algebraic Expressions

• Expression: A combination of variables and constants connected by some mathematical operations (+, -, \(\times\), /).
• Terms: Parts of an expression separated by + or - signs.
• Factors: Products that form a term.
• Coefficients: The numerical factor of a term.
• Monomial: An expression with one term.
• Binomial: An expression with two terms.
• Trinomial: An expression with three terms.
• Polynomial: An expression with one or more terms.
• Like Terms: Terms having the same algebraic factors (same variables with same powers).
• Unlike Terms: Terms having different algebraic factors.
• Addition and Subtraction: Only like terms can be added or subtracted.
• Multiplication:
• Monomial by Monomial: Multiply coefficients and add powers of same variables.
• Monomial by Polynomial: Distribute the monomial to each term of the polynomial.
• Polynomial by Polynomial: Multiply each term of the first polynomial by each term of the second polynomial.

9.2 Algebraic Identities

• Identity: An equality that is true for all values of its variables.
• Standard Identities:

1. \((a+b)^2 = a^2 + 2ab + b^2\)
2. \((a-b)^2 = a^2 - 2ab + b^2\)
3. \((a+b)(a-b) = a^2 - b^2\)
4. \((x+a)(x+b) = x^2 + (a+b)x + ab\)

• Using Identities: Identities provide a shortcut for multiplying algebraic expressions and for factorising them.

10.1 2D and 3D Shapes

• 2D Shapes (Plane Figures): Have only length and breadth (e.g., square, circle, triangle).
• 3D Shapes (Solid Figures): Have length, breadth, and height/depth (e.g., cube, cuboid, cylinder, cone, sphere).
• Faces: The flat surfaces of a 3D shape.
• Edges: The line segments where two faces meet.
• Vertices: The corners where three or more edges meet.

10.2 Views of 3D Shapes

• Different Views: A 3D object can look different from different positions.
• Front View: What you see when looking directly at the front.
• Side View: What you see when looking from the side.
• Top View: What you see when looking from above.
• Mapping Space Around Us: Using maps to locate places. Maps use symbols and scales.

10.3 Euler's Formula

• For any polyhedron (a 3D solid whose faces are polygons), the following relationship holds:
• \(F + V - E = 2\)
• Where \(F\) = number of faces, \(V\) = number of vertices, \(E\) = number of edges.
• This formula is true for all convex polyhedra.

11.1 Area of Plane Figures

• Area: The amount of surface enclosed by a closed figure.
• Trapezium: \(\text{Area} = \frac{1}{2} \times (a+b) \times h\), where \(a\) and \(b\) are lengths of parallel sides, \(h\) is height.
• Rhombus: \(\text{Area} = \frac{1}{2} \times d_1 \times d_2\), where \(d_1\) and \(d_2\) are lengths of diagonals.
• General Quadrilateral: Can be divided into two triangles by a diagonal. \(\text{Area} = \frac{1}{2} \times d \times (h_1 + h_2)\), where \(d\) is the diagonal and \(h_1, h_2\) are perpendiculars from other vertices to the diagonal.
• Polygon: Can be divided into triangles and trapeziums.

11.2 Surface Area of 3D Shapes

• Surface Area: The sum of the areas of all the faces of a 3D object.
• Cuboid:
• Total Surface Area (TSA) = \(2(lb + bh + hl)\)
• Lateral Surface Area (LSA) = \(2h(l+b)\) (Area of 4 walls)
• Cube:
• TSA = \(6a^2\)
• LSA = \(4a^2\)
• Cylinder:
• Curved Surface Area (CSA) = \(2\pi rh\)
• TSA = \(2\pi r(r+h)\)

11.3 Volume of 3D Shapes

• Volume: The amount of space occupied by a 3D object.
• Cuboid: \(V = l \times b \times h\)
• Cube: \(V = a^3\)
• Cylinder: \(V = \pi r^2 h\)
• Capacity: The volume of a substance that a container can hold.
• \(1 m^3 = 1000 L\)
• \(1 cm^3 = 1 mL\)

12.1 Exponents

• Base and Exponent: In \(a^n\), \(a\) is the base and \(n\) is the exponent (or power).
• Negative Exponents: For any non-zero rational number \(a\) and positive integer \(n\), \(a^{-n} = \frac{1}{a^n}\).
• \((a/b)^{-n} = (b/a)^n\)
• Laws of Exponents: For any non-zero rational numbers \(a, b\) and integers \(m, n\):

1. \(a^m \times a^n = a^{m+n}\)
2. \(a^m \div a^n = a^{m-n}\)
3. \((a^m)^n = a^{mn}\)
4. \(a^m \times b^m = (ab)^m\)
5. \(a^m \div b^m = (a/b)^m\)
6. \(a^0 = 1\) (where \(a \neq 0\))

• Standard Form (Scientific Notation): A number is in standard form if it is expressed as \(k \times 10^n\), where \(1 \le k < 10\) and \(n\) is an integer.
• Example: \(3,450,000 = 3.45 \times 10^6\)
• Example: \(0.000000021 = 2.1 \times 10^{-8}\)

13.1 Direct Proportion

• Direct Proportion: Two quantities \(x\) and \(y\) are in direct proportion if an increase in \(x\) leads to a proportional increase in \(y\), and a decrease in \(x\) leads to a proportional decrease in \(y\).
• Mathematically, \(x \propto y\) or \(x/y = k\) (constant).
• If \(x_1, y_1\) and \(x_2, y_2\) are two sets of values, then \(\frac{x_1}{y_1} = \frac{x_2}{y_2}\).
• Example: More articles, more cost. More work, more time.

13.2 Inverse Proportion

• Inverse Proportion: Two quantities \(x\) and \(y\) are in inverse proportion if an increase in \(x\) leads to a proportional decrease in \(y\), and vice versa.
• Mathematically, \(x \propto 1/y\) or \(xy = k\) (constant).
• If \(x_1, y_1\) and \(x_2, y_2\) are two sets of values, then \(x_1 y_1 = x_2 y_2\).
• Example: More workers, less time to complete a job. More speed, less time to cover a distance.

14.1 Factors of Algebraic Expressions

• Factorisation: The process of writing an algebraic expression as a product of two or more expressions.
• Irreducible Factors: Factors that cannot be expressed further as a product of factors (other than 1 and the expression itself).
• Methods of Factorisation:

1. Common Factors: Taking out the common factor from all terms.

• Example: \(3xy + 6x = 3x(y+2)\)

1. Regrouping Terms: Grouping terms such that common factors can be taken out from each group, leading to a common binomial factor.

• Example: \(ax + by + bx + ay = (ax+bx) + (ay+by) = x(a+b) + y(a+b) = (a+b)(x+y)\)

1. Using Identities: Applying the standard algebraic identities in reverse.

• \(a^2 + 2ab + b^2 = (a+b)^2\)
• \(a^2 - 2ab + b^2 = (a-b)^2\)
• \(a^2 - b^2 = (a+b)(a-b)\)
• \(x^2 + (a+b)x + ab = (x+a)(x+b)\)

1. Factors of the form \((x+a)(x+b)\): For expressions like \(x^2 + Px + Q\), find two numbers \(a\) and \(b\) such that \(a+b = P\) and \(ab = Q\).

14.2 Division of Algebraic Expressions

• Division of Monomial by Monomial: Divide coefficients and subtract powers of same variables.
• Example: \(12x^3y^2 \div 4xy = 3x^2y\)
• Division of Polynomial by Monomial: Divide each term of the polynomial by the monomial.
• Example: \((6x^2y + 9xy^2) \div 3xy = \frac{6x^2y}{3xy} + \frac{9xy^2}{3xy} = 2x + 3y\)
• Division of Polynomial by Polynomial: Use long division method or factorisation.
• If \(P(x) = Q(x) \times R(x)\), then \(P(x) \div Q(x) = R(x)\).

15.1 Graphs

• Bar Graph: Represents data using bars of uniform width. Used for discrete data.
• Pie Graph (Circle Graph): Shows the relationship of parts to a whole. Each sector's angle is proportional to the value it represents.
• Histogram: Bar graph for continuous data (class intervals). No gaps between bars.
• Line Graph: Displays data that changes continuously over periods of time. Points are plotted and connected by line segments.
• Linear Graph: A line graph where all the points lie on a single straight line.

15.2 Cartesian System

• Axes: Two perpendicular lines, the horizontal x-axis and the vertical y-axis, intersecting at the origin.
• Origin: The point of intersection of the x-axis and y-axis, denoted by \((0,0)\).
• Coordinates: A pair of numbers \((x,y)\) that uniquely locate a point on a graph.
• \(x\)-coordinate (abscissa): Distance from the y-axis.
• \(y\)-coordinate (ordinate): Distance from the x-axis.
• Plotting Points: To plot \((x,y)\), move \(x\) units horizontally from the origin and then \(y\) units vertically.
• Applications: Distance-time graphs, principal-interest graphs, etc.

16.1 Generalised Form of Numbers

• Generalised Form: Writing a number as the sum of the products of its digits and powers of 10.
• Two-digit number \(ab\): \(10a + b\)
• Three-digit number \(abc\): \(100a + 10b + c\)
• Reversing Digits:
• Two-digit number: If \(N = 10a+b\), reversed number is \(10b+a\).
• Sum: \(N + (10b+a) = 11a + 11b = 11(a+b)\) (divisible by 11)
• Difference: \(N - (10b+a) = 9a - 9b = 9(a-b)\) (divisible by 9)
• Three-digit number: If \(N = 100a+10b+c\), reversed number is \(100c+10b+a\).
• Difference: \(N - (100c+10b+a) = 99a - 99c = 99(a-c)\) (divisible by 99)

16.2 Divisibility Rules

• Divisibility by 2: A number is divisible by 2 if its unit digit is 0, 2, 4, 6, or 8.
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 5: A number is divisible by 5 if its unit digit is 0 or 5.
• Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
• Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
• Divisibility by 10: A number is divisible by 10 if its unit digit is 0.

16.3 Letters for Digits

• Puzzles where letters stand for digits. Each letter represents a unique digit (0-9).
• Rules:
• Each letter must stand for exactly one digit.
• The first digit of a number cannot be zero.
• Solve by using logical reasoning and trial-and-error, applying divisibility rules.