Smartbook of maths rules

INTRODUCTION

SET THEORY

Sets

• Representing a set


• Subsets and the power set


• Intersection, union, and difference of sets


• Complement of a set


• Cartesian product


• Problems involving sets

• The set N and operations in N


• Powers and their properties


• Prime factorisation


• Highest Common Factor (HCF)


• Lowest Common Multiple (LCM)


• Problems involving HCF and LCM


• The set Z and operations in Z


• Properties of powers in Z


• The set Q and operations in Q


• Converting decimals to fractions


• Powers with negative integer exponents


• Scientific notation

• Monomials: definitions


• Operations with monomials


• HCF and LCM of monomials

• Polynomials: definitions


• Operations with polynomials


• Special products


• Polynomial division


• Remainder theorem and Ruffini’s rule


• Polynomial division using Ruffini’s rule

• Common factor extraction (full)


• Common factor extraction (partial)


• Factorisation using special products


• Factorisation of the perfect square trinomial


• Factorisation using Ruffini’s rule


• Factorisation of the sum or difference of cubes

• Definitions and conditions of existence


• Simplifying algebraic fractions


• HCF and LCM of polynomials


• Reducing fractions to the same denominator


• Operations with algebraic fractions

• Linear equations with one unknown


• Principles of equivalence


• Solving whole linear equations


• Fractional linear equations and solution conditions


• Solving fractional linear equations


• Higher degree equations reducible to linear form


• Literal linear equations


• Solving fractional literal linear equations


• Word problems involving linear equations

• What is a linear inequality


• Principles of equivalence


• Solving whole linear inequalities


• Solving fractional linear inequalities


• Systems of linear inequalities


• Higher-degree inequalities reducible to linear

• Points on the Cartesian plane


• Equation of a straight line


• Graphing a line from its equation

• Systems of linear equations in two unknowns


• Graphical interpretation of a linear system


• Solving by substitution


• Solving by comparison


• Solving by elimination


• Solving using Cramer’s rule


• Solving systems not in standard form


• Solving fractional linear systems


• Word problems solvable using systems

• Irrational and real numbers


• Roots


• Conditions for the existence of a root


• Invariant properties and simplification of roots


• Reducing roots to the same index


• Taking factors out of the root


• Like radicals and addition of radicals


• Multiplying and dividing radicals


• Raising a radical to a power and taking roots


• Rationalising denominators


• Expressions involving roots


• Equations and inequalities with irrational coefficients


• Powers with rational exponents

• Whole quadratic equations


• Solving incomplete quadratic equations


• Solving complete quadratic equations using the formula


• Solving fractional quadratic equations


• Word problems involving quadratic equations


• Literal quadratic equations


• Relationships between solutions and coefficients: parametric equations

• Solving higher-degree equations by factorisation


• Binomial equations


• Trinomial equations


• Reciprocal cubic equations

• Graphical interpretation of second-degree systems


• Solving by substitution


• Symmetrical systems


• Word problems solved using second-degree systems

• Solving whole second-degree inequalities


• Finding solutions to second-degree inequalities


• Solving systems of second-degree inequalities


• Solving fractional second-degree inequalities

• Basic geometric elements


• Polygons


• Triangles


• Properties of triangles


• Quadrilaterals


• Geometric transformations


• Congruence


• Triangle congruence criteria


• Pythagoras’ theorem


• Euclid’s first and second theorems


• Thales’ theorem


• Triangle similarity criteria


• Circumference and circle


• Relative positions: line and circle


• Inscribed polygons


• Circumscribed polygons


• Regular polygons

• Introduction to descriptive statistics and frequency


• Graphical representation of data


• Averages and measures of central tendency


• Measures of variability

• Events and probability


• Complementary events


• Compatible and incompatible events


• Intersection of events


• Union of events


• Dependent and independent events


• Tree diagrams for probability