THE NATURAL NUMBERS
What are natural numbers
The four operations
The powers
Expressions with natural numbers
The properties of operations
The properties of powers
The multiples and divisors of a number
The maximum common divisor is the common minimum
The numbering systems
THE INNER NUMBERS
What are integers
Addition and subtraction
Multiplication, division and power
The laws of monotony
RATIONAL NUMBERS AND REAL NUMBERS
From fractions to rational numbers
The comparison of rational numbers
The operations in Q
Powers with negative integer exponent
The rational numbers and the decimal numbers
The real numbers
Fractions and proportions
Percentages
The approximate calculation
The scientific notation and the order of magnitude
THE SETS AND THE LOGIC
What is a whole
The representations of a whole
The subsets
Operations with sets
The set of parts and the partition of a whole
The logical propositions
Logical connectives and expressions
Valid forms of reasoning
Logic and sets
The quantifiers
RELATIONS AND FUNCTIONS
Binary relationships
The relationships defined in a whole and their properties
Equivalence relations
Order relations
Functions
The numerical functions
The Cartesian plane and the graph of a function
The circular functions
THE MONOMES
What are monomials
Operations with monomials
Maximum common divisor and least common multiple of monomials
I POLINOMIES
What are polynomials
Operations with polynomials
The remarkable products
Polynomial functions
The division between polynomials
Ruffini's rule
The theorem of the rest
Ruffini's theorem
BREAKDOWN IN FACTORS
The decomposition into polynomial factors
The MCD and the mcm between polynomials
Algebraic fractions
The calculation with algebraic fractions
THE LINEAR EQUATIONS
Identities
The equations
The principles of equivalence
The whole numerical equations
Equations and problems
The fractional equations
Literal equations
LINEAR DISEQUATIONS
Numerical inequalities. Inequalities
Whole inequalities
The systems of inequalities
Equations with absolute values
Inequalities with absolute values The study of the sign of a product
Broken inequalities
Numbers and digital information
Problems and algorithms
Program with Python
INTRODUCTION TO STATISTICS
Statistical data
The graphic representation of the data
The indexes of central position
The indices of variability
plane geometry
Euclidean geometry
Geometrical objects and properties
The postulates of belonging and order
Fundamental entities
Segment operations and angles Figures and demonstrations
Lengths, widths, measures
THE TRIANGLES
First definitions on triangles
The first criterion of congruence
The second criterion of congruence
The properties of the isosceles triangle
The third criterion of congruence; Congruence criteria and isosceles and equilateral triangles
Inequalities in triangles
PERPENDICULAR AND PARALLEL
Perpendicular lines
The parallel lines
The properties of polygon angles
The criteria of congruence of the right-angled triangles
PARALLELOGRAMS AND TRAPEZIES
The parallelogram
The rectangle
The roar
The square
The trapeze
The correspondences in a bundle of parallel lines
LINEAR SYSTEMS
The systems of two equations in two unknowns
The replacement method
The determined, impossible, indeterminate systems
The comparison method
The reduction method
The matrices and determinants
Cramer's method
The systems of three equations in three unknowns
Literal and fract systems
Linear systems and problems
THE RADICALS
The real numbers
Square roots and cubic roots
The nth root
Simplification and comparison of radicals
OPERATIONS WITH RADICALS
Multiplication and division of roots
The transport of a factor outside or inside the root sign
The power and root of a radical
The addition and subtraction of radicals
The rationalization of the denominator of a fraction
THE CARTESIAN PLAN AND THE STRAIGHT
Points and segments
The distance between two points is the midpoint
The equation of a line passing through the origin
The general equation of the line
Straight lines and linear systems
Parallel lines and perpendicular lines
The bundles of lines
How to determine the equation of a line
The distance of a point from a line
The parts of the plane and the line
THE SECOND GRADE EQUATIONS AND THE PARABOLA
Second degree equations: definitions
The resolution of a second degree equation
The whole numerical equations
The quadratic function and the parabola
The relations between the roots and the coefficients
The rule of Descartes
The breakdown of a second degree trinomial
Second degree equations and problems
APPLICATIONS OF SECOND-GRADE EQUATIONS
The fractional and literal equations
Equations and problems
The second degree parametric equations
Equations higher than the second degree
SECOND DEGREE SYSTEMS AND UPPER DEGREE
The second degree systems
The graphic interpretation of second degree systems
Systems higher than the second degree
Problems with higher than second degree systems
THE SECOND DEGREE DISEQUATIONS AND UPPER DEGREE
Linear inequalities
The sign of entire second degree inequalities
The resolution of entire second degree inequalities
Full inequalities of degree higher than the second
Broken inequalities
The systems of inequalities
Problems with inequalities
APPLICATIONS OF DISEQUATIONS
Parametric equations
The irrational equations
Irrational inequalities
Equations with absolute values
Inequalities with absolute values
The graphs of functions with absolute values
INTRODUCTION TO PROBABILITY
The events and the sample space
The classical definition of probability
Operations with events
Theorems related to the calculation of probabilities
Other probability definitions
THE CIRCUMFERENCE
The geometric places
The circumference and the circle
Theorems on the strings
Circumferences and lines
The reciprocal positions between two circles
The angles on the circumference
INSCRIBED AND CIRCUSCRIBED POLIGONES
The inscribed polygons
The circumscribed polygons
The triangles and the remarkable points
The quadrilaterals inscribed and circumscribed
The regular polygons
EQUIVALENT SURFACES AND AREAS
The equivalence of surfaces
Equivalence of parallelograms
Triangles and equivalence
The equivalence between a circumscribed polygon and a triangle
The construction of equivalent polygons
Measurement of polygon areas
THE THEOREMS OF EUCLIDE AND PYTHAGORAS
Euclid's first theorem
The Pythagorean theorem
Applications of the Pythagorean theorem
Euclid's second theorem
PROPORTIONALITY
The geometric sizes
The commensurable and incommensurable magnitudes
THE SIMILITUDE
The simile and the triangles
The triangularity similarity criteria
The similarity and the theorems of Euclid
The simile and the polygons
The similarity and the circumference
The golden section and its applications
The length of the circumference and the area of the circle
GEOMETRIC TRANSFORMATIONS
Geometric transformations and isometries
The translation
The rotation
The central symmetry
Axial symmetry
A non-isometric transformation: homothetia
EQUATIONS AND DISEQUATIONS
Inequalities and equivalence principles
First degree inequalities
Second degree inequalities
Full second degree inequalities
Inequalities of a grade higher than the second
Broken inequalities
Systems of inequalities
Equations and inequalities with absolute values
Irrational equations and inequalities
FUNCTIONS
Functions and their characteristics
Injective, surjective and bijective functions
Inverse function
Properties of the functions
Compound functions
Geometric and graphic transformations
SUCCESSIONS AND PROGRESSIONS
Numerical sequences
Induction principle
Arithmetic progressions
Geometric progressions
CARTESIAN AND STRAIGHT PLAN
Coordinates in the plan
Length of a segment
Midpoint of a segment, center of gravity of a triangle
Parallel lines and perpendicular lines
Distance of a point from a line
Geometric and straight places
Bundles of lines
Problems with straight lines
DISH
Parable and its equation
Parabola with axis parallel to the x axis Parabola and functions
Parabola and geometric transformations
Lines and parabolas
Determine the equation of a parabola
Search for the equation of a parabola
Bundles of parabolas
CIRCUMFERENCE
Circumference and its equation
Lines and circumferences
Determine the equation of a circle
Position of two circles
Bundles of circumferences
ELLIPSE
Ellipse and its equation
Ellipses and lines
Determine the equation of an ellipse
Ellipse and geometric transformations
HYPERBOLE
Hyperbole and its equation
Hyperbolas and lines
Determine the equation of a hyperbola
Translated hyperbola
Equilateral hyperbole
CONICHE
Definition of a conic by eccentricity
Second degree inequalities in two unknowns
Conics and geometric problems
EXPONENTIAL
Powers with real exponent
Exponential function
Exponential equations
Exponential inequalities
logarithms
Definition of logarithm
Logarithm properties
Logarithmic function
Logarithmic equations
Logarithmic equations
Logarithmic inequalities
Logarithmic inequalities
Logarithms and exponential equations and inequalities
Domain and sign of functions with exponentials and logarithms
Logarithmic equations and inequalities that can be solved only graphically
Logarithmic and semilogarithmic coordinates
UNIVARIAT STATISTICS
Statistics
Position indexes and variability
Gaussian distribution
Statistical reports
Effectiveness, efficiency, quality
Indicators of effectiveness, efficiency, quality
Reports and indicators
BIVARIATE STATISTICS
Introduction to bivariate statistics
Regression
Correlation
POLAR COORDINATES IN THE PLAN
Polar coordinates
Curve equations
Uniform circular motion
APPROXIMATE CALCULATION
The approximations
The propagation of errors
CARRIERS
Vectors in the plan
Linearly dependent and independent vectors
Scalar product and vector product
Cartesian representation of vectors
MATRICES AND DETERMINANTS
Matrices
Square matrices
Operations with matrices
Determinants
Properties of determinants
Rank
Reverse matrix
Some applications of matrices and determinants
LINEAR EQUATION SYSTEMS
What are linear systems
Inverse matrix method
Cramer's rule
Reduction method
Rouché-Capelli theorem
Homogeneous linear systems of n equations in n unknowns
CONICAL SECTIONS: THE SYNTHETIC VIEWPOINT
Dandelin theorems
The parabolic segment
SPEED OF VARIATION OF A GREATNESS
Average speed and instantaneous variation
GONIOMETRIC FUNCTIONS
Angle measurement
Sine and cosine functions
Tangent function
Secant and cosecant functions
Cotangent function
Goniometric functions of particular angles
Associated corners
Inverse goniometric functions
Goniometric functions and geometric transformations
GONIOMETRIC FORMULAS
Addition and subtraction formulas
Duplication formulas
Bisection formulas
Parametric formulas
Prosthesis and Werner formulas
EQUATIONS AND GONIOMETRIC DISEQUATIONS
Elementary goniometric equations
Linear equations in sine and cosine
Homogeneous second degree equations in the sine and cosine
Goniometric equation systems
Goniometric inequalities
TRIGONOMETRY
Applications of theorems on rectangular triangles
Any triangles
Applications to trigonometry
Cosine theorem
COMPLEX NUMBERS
Algebraic form of complex numbers
Work with imaginary numbers
Operations with complex numbers in algebraic form
Algebraic representation of complex numbers
Trigonometric form of a complex number
Operations between complex numbers in trigonometric form
Nth roots of the unit
VECTORS, MATRICES, DETERMINANTS
Vectors in the plan
Vectors in the Cartesian plane
Matrices
Operations with matrices
Determinants
Reverse matrix
Matrices and analytical geometry
GEOMETRIC TRANSFORMATIONS
Translation
Rotation
Central symmetry
Axial symmetry
isometries
dilation
simile
Affinity
EUCLIDEA GEOMETRY IN THE SPACE
Points, lines, planes in space
Perpendicularity and parallelism
Distances and angles in space
Geometric transformations
Polyhedra
Rotating solids
Solids areas
Extension and equivalence of solids
Solid volumes
ANALYTICAL GEOMETRY IN THE SPACE
Coordinates in space
Vectors in space
Plan and its equation
Straight line and its equation
Mutual position of a line and a plane
Some notable surfaces
COMBINATORIAL CALCULATION
What is combinatorial calculation
Provisions
Permutations
Combinations
Newton's binomial
CHANCE
Events
Classical conception of probability
Logical sum of events
Conditional probability
Logical product of events
Bayes theorem
Statistical probability concept
Subjective conception of probability
Axiomatic setting of probability
POLAR COORDINATES IN THE PLAN
Polar coordinates
Curve equations
Uniform circular motion
TRANSMISSION NUMBERS
Rational numbers and irrational numbers
Algebraic numbers and transcendent numbers
FUNCTIONS AND THEIR PROPERTIES
Real functions of real variable
Domain of a function
Properties of the functions
Inverse function
Composed function
FUNCTION LIMITS
Sets of real numbers
finite limits for x that tends to a finite value
infinite limits for x that tends to a finite value
finite limits for x that tends to an infinite value
infinite limits for x that tends to an infinite value
First theorems about limits
CALCULATION OF LIMITS AND CONTINUITY OF FUNCTIONS
Limit operations
Indeterminate forms
Notable limits
Calculation of limits
Infinitesimals, infinities and their comparison
Continuous functions
Theorems on continuous functions
Points of discontinuity of a function
Asymptotes
Search for asymptotes
Probable graph of a function
SUCCESSIONS AND SERIES
Numerical sequences
Progressions
Some properties of the sequences
Limit of a succession
Calculation of the limit of a succession
Induction principle
What is a series
Converging, divergent, indeterminate series
DERIVATIVES
Derivative of a function
Fundamental derivatives,
Operations with derivatives
Derivative of a composite function
Derivative of function power function
DIFFERENTIAL CALCULATION THEOREMS
Rolle's theorem
Lagrange theorem
Consequences of the Lagrange theorem
Cauchy theorem
De Hospìtal's theorem
MAXIMUM, MINIMUM AND FLEXES
definitions
Maxima, minima, horizontal inflections and first derivative
Flexes and second derivative
Maximum, minimum, inflections and subsequent derivatives
Optimization problems
FUNCTION STUDY
Study of a function
Graphs of a function and its derivative
Applications of the study of a function
Approximate resolution of an equation
INDEFINITE INTEGRALS
Immediate indefinite integrals
Integration by substitution
Integration by parts
Integration of rational functions used
INTEGRALS · DEFINED
Fundamental theorem of integral calculus
Area calculation
Volume calculation
Volume of a rotating solid
Improper integrals
Applications of integrals to physics
Numerical integration
DIFFERENTIAL EQUATIONS
What is a differential equation
First order differential equations
Second order differential equations
Differential equations and physics
PROBABILITY DISTRIBUTIONS
Discrete random variables and probability distributions
Values characterizing a discrete random variable
Frequently used probability distributions
Random games
Standardized random variables
Continuous random variables