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2 TABLE OF CONTENTS Topic.Page# 1. Numbers Ratio, Profit & Loss Angles Interest Algebra Quadratic Equations Logarithms Series Sphere Coordinate Geometry Trigonometry Matrices and Determinants Vectors Probability...29 Our aim is to give people math skillss in a very simple way Raymond J Page 2 of 29 Easy revision

3 About the Book This guide book is the formula book for Ordinary Level students who are eager to learn mathematics. It comprises formulas for the topics of mathematics from Form 1 up to Form 4. You can find more information about these formulas and their solved questions at Easy revision. About the Author J N Raymond works as independent tutor and web developer. He is also a student at University of Dar es Salaam taking BSc. in Meteorology. He recently opened an online platform for students to learn mathematics in easy ways for free at Contact him at ray@wesovu.com or raymondjohn001@gmail.com Phone: I dedicate this guide book to all Ordinary level students who love and live in mathematics all over the world J N Raymond 2018 All rights reserved. Page 3 of 29 Easy revision

4 1. NUMBERS Number: an arithmetic value, expressed by a word, symbol or figure representing a particular quantity and used in counting and making calculations. Numeral: a figure, symbol denoting a number. Digit: any of the numerals from 0 to 9, especially when forming part of a number. Natural numbers (N): counting numbers {1, 2, 3, 4, 5, 6 } Whole numbers (N 0 ): numbers without fraction and no negative which start from 0. Example {0, 1, 2, 3, 4 } Integers (Z): positive and negative counting numbers as well as zero. Example {. -4, -3, -2, -1, 0, 1, 2 } Real numbers (R): all numbers on a number line with decimal representation. They can be positive, negative or zero. Rational numbers (Q): numbers that can be expressed in form of where q is not equal to zero. All integers are rational but the converse is not true. Irrational numbers: numbers that are not rational (i.e. they cannot be expressed in form of p/q). Example 3 Complex number (C): includes real numbers, imaginary numbers and sums and differences of real and imaginary numbers. Examples 1, -i, 3 + 2i, 5 i (i is an imaginary unit). Page 4 of 29 Easy revision

5 Relationship between N, Z, Q, R and C N Z Q R C Composite number: number that can be factored into a product of smaller integers. Every integer greater than one is either prime or composite. Even number: an integer which is divisible by 2 (i.e. n = 2k where k is any integer). Example 2, -2, 8, -10, 0. Odd number: an integer which is not a multiple of 2 (i.e. n = 2k + 1). Example -3, -9, 5, and 35. Prime numbers: whole numbers that have no factors other than one and number itself. The number 1 is usually not considered to be a prime number. Prime twins: pairs of primes that have a difference of 2. Example 5, 7 (7 5 = 2) Goldbach s conjecture Every even number greater than 2 can be expressed as the sum of two prime numbers Significant figures: Is each of the digits of a number that are used to express it to a required degree of accuracy. Standard form (Standard index form/scientific form): is the way of expressing numbers that are too big or too small to be conveniently written in decimal form Page 5 of 29 Easy revision

6 2. RATIO, PROFIT AND LOSS Profit made = Selling price Buying price Loss made = Buying price Selling price Percentage profit = x 100 Percentage loss = x 100 Discount a reduction made from a regular or list price. Discount (allowed/received) expressed in fractions) = List price x Discount rate (discount rate Interior angle, β = ANGLES Sum of interior angles = (n 2)180 0 Interior angle + Exterior angle = (sum of angles of a straight line) Exterior angle = where n = number of sides of a regular polygon. Simple Interest, I = Compound Interest, A n = P INTEREST where P = Principal R = Rate T = Time (e.g annual, semi-annual) n = number of years Page 6 of 29 Easy revision n

7 5. ALGEBRA Factoring Formulas Given that a, b and c are real numbers and n is a natural number then: a 2 b 2 = (a + b)(a b) a 3 b 3 = (a b)(a 2 + ab + b 2 ) a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 4 b 4 = (a 2 b 2 )(a 2 + b 2 ) = (a b)(a + b)(a 2 + b 2 ) a 5 b 5 = (a b)(a 4 + a 3 b + a 2 b 2 + ab 3 + b 4 ) a 5 + b 5 = (a + b)(a 4 - a 3 b + a 2 b 2 - ab 3 + b 4 ) If n is even, then a n b n = (a b)(a n-1 + a n-2 b + a n-3 b ab n-2 + b n-1 ) a n + b n = (a + b)(a n-1 - a n-2 b + a n-3 b ab n-2 + b n-1 ) If n is odd, then a n + b n = (a + b)(a n-1 - a n-2 b + a n-3 b ab n-2 + b n-1 ) Product Formulas Given that a, b and c are real numbers and n, k are whole numbers, then: (a b) 2 = a 2 2ab + b 2 (a + b) 2 = a 2 + 2ab + b 2 (a b) 3 = a 3 3a 2 b + 3ab 2 b 3 (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (a b) 4 = a 4 4a 3 b + 6a 2 b 2 4ab 3 + b 4 (a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4 (a + b + c) 2 = a 2 + b 2 + c 2 + 2ab + 2ac + 2bc Page 7 of 29 Easy revision

8 Binomial Expansion Formula (a + b) n = n C 0 a n + n C 1 a n-1 b + n C 2 a n-2 b n C n-1 ab n-1 + n C n b n where n C k =!!( )! are the binomial coefficients. 6. QUADRATIC EQUATIONS Standard form of Quadratic equation ax 2 + bx + c = 0 where a 0 ax 2 + bx = 0, x 1 = 0, x 2 = - ax 2 + c = 0, x 1 = x 2 = ± Quadratic general formula x = ± where b 2 4ac = D (Discriminant) For perfect square, Discriminant, D = 0 ( b 2 4ac = 0) Viete s Formula If given x 2 + px + q = 0, then x + x = p x x = q 7. LOGARITHMS Given that: X, Y, a, c, k are positive real numbers and n is a natural number then: y = log a x if and only if x = a y, a > 0, a 1. Page 8 of 29 Easy revision

9 log a 1 = 0 log a a = 1 Product rule of Logarithms log a (XY) = log a X + log a Y Quotient rule of Logarithms log a = log a X log a Y Power rule of Logarithms log a (X n ) = nlog a X log a X = log a X Other rules of logarithms log a X = = log c X log a c provided that c > 0, c 1 log a c = X = a Natural logarithm log e X = ln X where e = = log X = ln X = log X ln X where ln 10 is a constant value equals to Page 9 of 29 Easy revision

10 8. SERIES Arithmetic Series (Arithmetic Progression) Given that: A 1 is an initial term (first term) A n is the general term (nth term) d is the difference between successive terms (common difference) S n is the sum of the first n terms A n = A n-1 + d = A n-2 + 2d = = A 1 + (n 1)d (formula for general term of Arithmetic Progression) A 1 + A n = A 2 + A n-1 = = A i + A n+1-i d = A 2 A 1 = A 3 A 2 = A n A n-1 Arithmetic Mean A i = Sum of the nth terms of AP (Arithmetic Progression) S n = x n = (2A 1 + (n 1)d Geometric Series (Geometric Progression) Given that: G 1 is an initial term (first term) G n is the general term (nth term) r is the common ratio n is the number of terms in the series S n is the sum of the first n terms S is the sum to infinity G n = G 1 r n 1 = G n-1 x r (formula for general term/ nth term) G 1 G n = G 2 G n-1 = = G i G n+1-i Geometric Mean G i = G x G Page 10 of 29 Easy revision

11 Sum of the nth term S n = G 1 Sum to infinity S n = lim S = for r < 1, the sum m S converges as n. 9. SPHERE Circumference of Small Circle, C C = 2πRcos θ where R is the radius of the e earth and is the latitude. Nautical Mile 1 0 = 60 nm (where nm stands for Nautical Miles) Distance between two points on the longitudes, D (in Kilometers) same latitude but different D = Where: R = Radius of the earth, α = Difference in longitudes and θ = latitude Page 11 of 29 Easy revision

12 Distance between two points on the same longitude but different latitudes, D (in Kilometers) D = where α = Difference in latitudes Knot A knot is the velocity of ship in nautical miles per hour. 1Knot = 1nm/hr 1Knot = 1.852km/hr Surface area of a Sphere, A = 4πR 2 Volume of a Sphere, V = πr 3 where R is a given radius. Page 12 of 29 Easy revision

13 Slope/Gradient (m) m = = 10. COORDINATE GEOMETRY Equation of a straight line y = mx + c where: m = Slope/Gradient c = y-intercept General equation of a straight line is: Ax + By + C = 0 Distance between two points, d Page 13 of 29 Easy revision

14 Mid-point of a line segment M(x 0,y 0 ) Page 14 of 29 Easy revision

15 Page 15 of 29 Easy revision

16 11. TRIGONOMETRY sin α = cos α = tan α = cot α = sec α = cosec α = Signs of Trigonometric Functions Page 16 of 29 Easy revision

17 Trigonometric Functions of Common Angles Trigonometry Identities Page 17 of 29 Easy revision

18 Reduction Formulas Page 18 of 29 Easy revision

19 Relations between Trigonometric Functions Page 19 of 29 Easy revision

20 Compound Angles Page 20 of 29 Easy revision

21 Double Angles Page 21 of 29 Easy revision

22 12. MATRICES AND DETERMINANTS Matrices Definition: An m x n matrix A is a rectangular array of elements (numbers or functions) with m rows and n columns. A m x n = [a ij ] = Order of matrix is written as m x n which represents m rows and n columns. Example of Common Matrices 2 x 2 Matrix A 2 X 2 = [a 22 ] = 3 x 3 Matrix A 3 X 3 = [a 33 ] = Square matrix is a matrix of order n x n (that means m=n). A square matrix [a ij ] is symmetrical if a ij = a ji A square matrix [a ij ] is skew-symmetrical if a ij = -a ji Diagonal matrix is a square matrix with all elements zero except those on the leading diagonal. 0 A = is a 2 x 2 diagonal matrix 0 Unit matrix is a diagonal matrix in which the elements on the leading diagonal are all unity. The unit matrix is denoted by I. I = 1 0 is a 2 x 2 unit matrix. 0 1 I = is a 3 x 3 unit matrix. Page 22 of 29 Easy revision

23 A null matrix is one whose elements are all zero. B = 0 0 is a 2 x 1 zero or null matrix. Operations with Matrices Equal Matrices Two matrices A and B are equal if, and only if, they are both of the same order m x n and corresponding elements are equal. Example: A 2x2 = and B 2x2 = , hence A = B Addition and Subtraction Two matrices A and B can be added or subtracted if and only if, they have the same order m x n Example: A = and B =, hence A + B = B + A (Addition of matrix is commutative) and A B = Difference of a matrix. A + B = A B = Scalar Multiplication If k is a scalar, and A is a 2 x 2 matrix, then ka = Page 23 of 29 Easy revision

24 Multiplication of Two Matrices Two matrices can be multiplied together only when the number of columns in the first matrix is equal to the number of rows in the second matrix. If A 2x2 = and B 2x1 = then AB = C is compatible since number of columns of matrix A is 2 and number of rows of matrix B is 2, hence C is 2 x 1 matrix given as: C = AB = + + Simply, if A is an m x p matrix and B is a p x n matrix, then AB is an m x n matrix. That is, A mxp = C mxn =[c ij ] = AB = and B pxn = then where c ij = a i1 b 1j + a i2 b 2j + + a ip b pj = (i = 1, 2,, m ; j = 1, 2,, n) NB: Multiplication of matrices requires much more care than their addition, since the algebraic properties of matrix multiplication differ from those satisfied by the real numbers. Part of the problem is due to the fact that AB is defined only when the number of columns of A is the same as the number of rows of B. Page 24 of 29 Easy revision

25 Properties of Matrix Multiplication o Associativity. (AB)C = A(BC) o Not generally commutative. That is, often AB BA o Distributed over addition and subtraction. C(A + B) = CA + CB o AB = 0 does not necessarily imply that either A = 0 or B = 0 Determinant of a Matrix Determinant is denoted det, for instance determinant of matrix A is written as det A Second Order Determinant If A is 2 x 2 matrix, then det A = = a 1 b 2 a 2 b 1 Transpose of a Matrix If the rows and columns of a matrix are interchanged, then the new matrix is called the transpose of the original matrix. If A is the original matrix, its transpose is denoted A T or If the matrix product AB is defined, then (AB) T = B T A T Adjoint of Matrix If A is a square n x n matrix, its adjoint, denoted by adj A, is the transpose of the matrix of cofactors C ij of A: adj A = [C ij ] T. Inverse of a Matrix If A is a square n x n matrix with a nonsingular determinant det A, then its inverse A -1 is given by A -1 = If the matrix product AB is defined, then (AB) -1 = B -1 A -1. Page 25 of 29 Easy revision

26 Cramer s rule If given a system of linear equations: a x + b y = d a x + b y = d then x =, and y = where D = = a 1 b 2 a 2 b 1 D x = = d 1 b 2 d 2 b 1 D y = = a 1 d 2 a 2 d 1 Note: If D 0, then the system has a single solution. If D = 0 and D x 0 (or D y 0), then the system has no solution. If D = D x = D y = 0, then the system has infinitely many solutions. 13. VECTORS Definition: Vectors are quantities with both magnitude and direction. For instance force, momentum and velocity. It can be denoted as, a, b,, or Unit vectors = (1,0,0) x direction. = (0,1,0) y direction. = (0,0,1) z direction. For two dimension vectors, k = 0 They are called Unit vectors because have magnitude of 1. i = j = k = 1 Vector Addition w = u + v, is called Resultant of the two vectors. Page 26 of 29 Easy revision

27 Vector Subtraction w = u - v Magnitude of a Vector It is a distance between two points, denoted = = ( ) + ( y 2 ). If AB = r, then BA = -r Unit Vector Is a vector with magnitude of 1. It is given as d by ) where A(x 1, y 1 ) and B = (x 2, Page 27 of 29 Easy revision

28 = Directional Cosines X = cos Y = cos Z = cos Page 28 of 29 Easy revision

29 Let Events: A, B Probability: P 14. PROBABILITY Probability of an Event P(A) =, Where m is the number of possible outcomes, n is the total number of possible outcomes. Range of Probability Values 0 P(A) 1 Certain Event P(A) = 1 Impossible Event P(A) = 0 Complement P(A) = 1 P(A) Independent Events P(A/B) = P(A) P(B/A) = P(B) Addition Rule for Independent Events P(A B) = P(A) + P(B) Multiplication Rule for Independent Events P(A B) = P(A) P(B) Conditional Probability ( ) P(A/B) = ( ) Page 29 of 29 Easy revision

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