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History of Mathematics

• Leonhard Euler
  • The life of Leonhard Euler
  • The language that Euler left us
  • The Basel problem
  • The number $e$ and compound interest
  • Euler's formula and the "most beautiful equation"
  • The spring and differential equations
  • The Königsberg bridges and the birth of graphs
  • $V - A + F = 2$ and the soccer ball
  • Infinitely many prime numbers from a sum
  • Fermat's number that was not prime
  • Euler's theorem and cryptography
  • Other contributions and legacy
  • Image credits
• René Descartes
  • The life of René Descartes
  • Coordinates: geometry becomes algebra
  • Square root with ruler and compass
  • The rule of signs
  • The angles of polyhedra and Euler's formula
  • The kissing circles theorem
  • Why does the rainbow appear at 42°
  • Descartes and the idea of ​​function
  • A reading of the Cogito: listening and speaking
  • Descartes and the problem of cannons
  • The word "function"
  • *Physis*: deciphering the traits of nature
  • Infinity and division by zero
  • Other contributions and legacy
  • Image credits
• Carl Friedrich Gauss
  • The life of Carl Friedrich Gauss
  • The sum of 1 to 100
  • The 17 sided polygon
  • Clock arithmetic
  • The fundamental theorem of algebra
  • Counting prime numbers
  • Ceres and the least squares method
  • The normal curve, Gauss's bell
  • The Egregium Theorem and curvature
  • Systems of equations: the determinant and Gaussian elimination
  • Gaussian quadrature: integrate with two points
  • Other contributions and legacy
  • Image credits
• Isaac Newton
  • The life of Isaac Newton
  • The generalized binomial: beyond whole-number powers
  • Fluxions: how to measure an instantaneous change
  • Areas and integration: the inverse of fluxions
  • Newton's method: approximating roots with tangents
  • The laws of motion: forces and changes in velocity
  • Gravitation: from the apple to the Moon
  • Light and colours: the prism does not make the rainbow
  • Newton's legacy
  • Image credits
• Gottfried Wilhelm Leibniz
  • The life of Gottfried Wilhelm Leibniz
  • Leibniz's notation: a language for change
  • The product rule: two changes in one calculation
  • The ∫ sign: from sum to accumulation
  • Binary arithmetic: writing numbers with 0 and 1
  • Leibniz's machine: turning sums into motion
  • The Leibniz series for π: a simple, slow sum
  • Calculemus: the idea of calculating with reasoning
  • Leibniz's legacy
  • Image credits
• Pierre de Fermat
  • The life of Pierre de Fermat
  • Fermat and analytic geometry: a curve in an equation
  • Maxima and minima: the rectangle with the largest area
  • Fermat's little theorem: powers and remainders
  • Fermat numbers: a conjecture and a counterexample
  • Fermat and Pascal: fairly dividing an interrupted game
  • Fermat's principle: light takes the fastest path
  • Infinite descent: a smaller solution leads to a contradiction
  • Fermat's last theorem: from the margin to Wiles's proof
  • Fermat's legacy
  • Image credits
• Blaise Pascal
  • The life of Blaise Pascal
  • Pascal's triangle: build it by adding
  • Combinations: counting without listing
  • The Chevalier de Méré's dice and the fair split
  • The Pascaline: a sum made of gears
  • The mystic hexagram: six points on a conic
  • The cycloid: the toothache curve
  • Pascal's principle: multiplying forces with water
  • The weight of air: the Puy de Dôme experiment
  • Pascal's wager
  • Pascal's legacy
  • Image credits
• Joseph-Louis Lagrange
  • The life of Joseph-Louis Lagrange
  • The calculus of variations
  • The Analytical Mechanics
  • The Lagrange points
  • Four squares are enough
  • Shuffling the roots
  • The derivative and the mean value
  • Lagrange multipliers
  • Lagrange's legacy
  • Image credits
• Sophie Germain
  • The life of Sophie Germain
  • The letters to Gauss
  • The theorem on Fermat
  • Sophie Germain primes
  • The Sophie Germain identity
  • Vibrating plates
  • Mean curvature
  • Sophie Germain's legacy
  • Image credits
• James Bernoulli
  • The life of James Bernoulli
  • Interest and the number e
  • The harmonic series
  • The spiral and the lemniscate
  • The art of conjecturing
  • The law of large numbers
  • The Bernoulli numbers
  • The legacy of James Bernoulli
  • Image credits
• Albert Einstein
  • The life of Albert Einstein
  • The photoelectric effect
  • Brownian motion
  • Special relativity
  • E = mc²
  • General relativity
  • The legacy of Albert Einstein
  • Image credits
• Johann Bernoulli
  • The life of Johann Bernoulli
  • The catenary
  • L'Hôpital's rule
  • The brachistochrone
  • The exponential calculus
  • Virtual work
  • The legacy of Johann Bernoulli
  • Image credits
• Euclid of Alexandria
  • Euclid: the author without a face
  • The Elements
  • The equilateral triangle
  • Pythagoras's theorem
  • Geometric algebra
  • Euclid's algorithm
  • The primes never run out
  • The fifth postulate
  • The legacy of Euclid
  • Image credits
• James Clerk Maxwell
  • The life of James Clerk Maxwell
  • Colour
  • Saturn's rings
  • The speeds of molecules
  • Maxwell's equations
  • The displacement current
  • Light is a wave
  • The legacy of James Clerk Maxwell
  • Image credits
• John Nash
  • The life of John Nash
  • The game of Hex
  • The Nash equilibrium
  • Mixed strategies
  • The Nash bargaining solution
  • The embedding theorem
  • The differential equations
  • The legacy of John Nash
  • Image credits
• Niels Henrik Abel
  • The life of Niels Henrik Abel
  • The fifth-degree equation
  • The binomial series
  • Elliptic functions
  • The lemniscate
  • Abelian groups
  • The legacy of Niels Henrik Abel
  • Image credits
• Évariste Galois
  • The life of Évariste Galois
  • The group of an equation
  • What a group is
  • Solving by radicals
  • Finite fields
  • The impossible constructions
  • The legacy of Évariste Galois
  • Image credits
• John Napier
  • The life of John Napier
  • Multiplying by adding
  • Napier's two points
  • Briggs and base 10
  • Napier's bones
  • The slide rule
  • The legacy of John Napier
  • Image credits
• Alan Turing
  • The life of Alan Turing
  • The Turing machine
  • The halting problem
  • Computable numbers
  • The universal machine
  • The Enigma
  • Banburismus
  • The imitation game
  • Morphogenesis
  • The legacy of Alan Turing
  • Image credits

Mathematics for Children

• Natural numbers
  • Elementary I (1st to 5th year)
    • Introduction
    • The number zero
    • Sum
    • Subtraction
    • Multiplication
    • Division
    • Even and odd numbers
    • Introduction to Geometry
    • Quadrilateral
  • Elementary II (6th to 9th year)
    • Prime numbers
    • Greatest Common Divisor
    • Least Common Multiple
    • Geometry with sequences of numbers
    • Triangles
  • High School
    • Introduction to Arithmetic Progression
    • Arithmetic Progression
  • Solved exercises (all levels)
    • Natural Number Subtraction Exercises
    • Exercises for Adding Natural Numbers
• Whole Numbers
  • Elementary I (1st to 5th year)
    • Above and below zero
    • Measure
  • Elementary II (6th to 9th year)
    • Introduction to Integers
    • Straight and operations
    • Signal rule
    • Potentiation of integers
    • Angles and Geometry
      • Angles in basic planes
      • Property of Angles in Planes
  • High School (1st to 3rd grade)
    • Pascal's triangle
    • Function Basics
      • Function Discovery
      • Cartesian Plane
      • Classification in mathematics
      • Graphs for integers
• Rational Numbers
  • Elementary I (1st to 5th year)
    • Equal parts and fractions
    • Decimals in everyday life
  • Elementary II (6th to 9th year)
    • Introduction to Rational Numbers
    • Operations with Rational Numbers
    • Percentage Basics
    • Proportions
    • Rule of 3
    • Fees
    • Quantity Analysis
    • Proportions and Geometry
    • Algebraic Expressions
    • First Degree Equations
  • High School (1st to 3rd grade)
    • Power with Negative Exponent
    • Nth Root
    • Models and restrictions
• Irrational Numbers
  • Elementary I (1st to 5th year)
    • Measure and discover
  • Elementary II (6th to 9th year)
    • Operation with Radicals
      • Introduction to Irrational Numbers
      • Introduction to Radiciation
    • Approaches on the straight
    • Geometry - Irrational Numbers
      • Circumferences
      • Pi Number
      • Circle Area
      • Angles and Circumferences
      • Triangles and Circles
      • Pythagorean Suits
  • High School (1st to 3rd grade)
    • Arguments and operations
    • Logarithm
      • Introduction to Logarithm Table
      • Logarithm Properties
    • Rectangle Triangles
    • Basic Finance
      • Increases and Discounts
      • Simple Interest
      • Compound interest and Euler's number
• Real Numbers
  • Elementary I (1st to 5th year)
    • Measures between integers
    • Everyday data
  • Elementary II (6th to 9th year)
    • Introduction to Real Numbers
    • Rational, irrational and approximations
    • Inequalities of real numbers
    • 2° degree equation
    • Bhaskara
    • Notions of Probability
      • Introduction to Probability
      • Fundamentals
      • Equiprobable Events
  • High School (1st to 3rd grade)
    • Notions of Sets
      • Introduction to Sets
      • Equivalence between Sets
      • Venn-Euler Diagrams
      • Diagram Operations
    • Intervals and infinity
      • Notion of Infinity
      • Ranges in real numbers
      • Interval Operations
    • Functions
      • Introduction to Function Theory
      • Understanding Relationships
      • Correspondence law
      • Function Concept
      • First Degree Functions
      • Domain and Image
    • Statistics Notions
      • Introduction to Statistics
      • Basic Concepts of Statistics
      • Descriptive Statistics
        • Position Measurements
        • Dispersion Measures
• Plane Geometry
  • Elementary I (1st to 5th year)
    • Flat shapes
    • Lines, parallel and perpendicular
    • Angles: turns and corners
    • Polygons
    • Symmetry
    • Perimeter and area in the mesh
    • Enlargement and reduction
  • Elementary II (6th to 9th year)
    • 1. Primitive notions and propositions
      • A little history
      • I. Primitive notions
      • II. Primitive propositions
      • III. Competing straights
      • Exercises
    • 2. straight segment
      • I. The notion of "being between"
      • II. Segment and ray
      • III. Positions of two segments
      • IV. Segment congruence
      • V. Midpoint
      • VI. Measurement of a segment
      • VII. Distance between two points
      • Segments and straight lines in everyday life
      • Exercises
    • 3. Angles
      • I. Introduction: convexity and half-planes
      • II. Angle: definitions
      • III. Congruence and comparison
      • IV. Right, acute and obtuse angles — measurement
      • Exercises
    • 4. Triangles
      • I. Concept, elements and classification
      • II. Congruence of triangles
      • III. Inequalities in triangles
      • Exercises
      • Reading: Euclid and deductive geometry
    • 5. Parallelism
      • I. Parallel lines
      • II. Existence and uniqueness of parallel
      • III. Consequences in the triangle
      • Exercises
      • Reading: The fifth postulate
    • 6. Perpendicularity
      • I. Definitions — right angle
      • II. Existence and uniqueness of the perpendicular
      • III. Projections and distance
      • Exercises
    • Constructions with ruler and compass
    • 7. Notable quadrilaterals
      • I. Quadrilateral — definition and elements
      • II. Notable quadrilaterals — definitions
      • III. Properties of trapezoids
      • IV. Properties of parallelograms
      • V. Rectangle, rhombus and square
      • VI. Consequences — medium bases
      • Exercises
    • 9. Polygons
      • I. Definitions and elements
      • II. Diagonals, interior angles and exterior angles
      • Exercises
    • Areas of flat figures
    • 10. Circumference and circle
      • I. Definitions and elements
      • II. Relative positions of straight line and circle
      • III. Relative positions of two circles
      • IV. Tangent segments — circumscribable quadrilaterals
      • Exercises
    • Circumference length
    • Pythagoras Theorem
  • High School
    • 8. Notable points of the triangle
      • I. centroid — medians
      • II. incenter — angle bisectors internal
      • III. circumcenter — perpendicular bisectors
      • IV. orthocenter — heights
      • Exercises
      • Reading: Papus: the epilogue of Greek geometry
    • 11. Angles on the circumference
      • I. Congruence, addition and inequality of arcs
      • II. Central angle
      • III. Inscribed angle
      • IV. Segment angle (semi-inscribed)
      • V. Eccentric angles
      • Exercises
    • 12. Thales' theorem
      • I. Thales' Theorem
      • II. angle bisectors Theorems
      • Exercises
      • Reading: Legendre: towards a rigorous and didactic geometry
    • 13. Similarity of triangles and power of point
      • I. Similarity of triangles
      • II. Cases (criteria) of similarity
      • III. Point power
      • Exercises
    • 14. Right triangles
      • I. Metric relationships
      • II. Applications of the Pythagorean theorem
      • Exercises
    • 15. Any triangles
      • I. Sines theorem
      • II. Metric relations and cosine theorem
      • III. Calculation of notable lines
      • Exercises
    • Regular Polygons
    • Circumference ratio in regular polygons

Mathematical programming

• Project Euler
  • Level Problems 1

Pure Mathematics

• Elements of Mathematics
  • Mathematical Logic
  • Truth Table
  • Quantifier
  • Axioms and Postulates
  • Theorems
  • Demo
• Set Theory
  • Primitive Notions
  • Introduction to Set Theory
  • Operations with Sets
  • Set Family
  • Cartesian Product of Sets
• Relations
  • Introduction to Relationships
  • Relationship Properties
  • Equivalence Relations
  • Set Partition
• Natural Numbers
  • Peano axioms

OBMEP Exercises

• 2018
  • OBMEP 2018
  • OBMEP 2018 - Second Phase
• 2019
  • OBMEP 2019 - Second Phase
• 2021
  • OBMEP 2021 - 1 Phase
  • OBMEP 2021 - 2 Phase
• 2022
  • OBMEP 2022 - 2 Phase - N1
  • OBMEP 2022 - 1 Phase
• 2023
  • 1st phase – Level 1
  • 1st phase – Level 2
  • 1st phase – Level 3
  • 2nd phase – Level 1
  • 2nd phase – Level 2
  • 2nd phase – Level 3

Entrance Exam Exercises

• ENEM
  • ENEM 2025
    • Questions 136 to 150
    • Questions 151 to 165
    • Questions 166 to 180
  • ENEM 2024
    • Questions 136 to 150
    • Questions 151 to 165
    • Questions 166 to 180
  • ENEM 2023
    • Questions 136 to 150
    • Questions 151 to 165
    • Questions 166 to 180
  • ENEM 2022
    • Questions 136 to 150
    • Questions 151 to 165
  • ENEM 2021
    • Questions 136 to 150
    • Questions 151 to 165
    • Questions 166 to 180
• Fuvest
  • FUVEST 2026
    • 1st phase
    • 2nd phase
  • FUVEST 2025
    • 1st phase
    • 2nd phase
  • FUVEST 2024
    • 1st phase
    • 2nd phase
  • FUVEST 2023
    • 1st phase
    • 2nd phase
  • FUVEST 2022
    • 1st phase
    • 2nd phase
  • FUVEST 2021 – 2nd phase

The Semiotics of Mathematical Notation

• 1. From word to symbol
  • Radix quadrati: mathematics has a script
  • Zero and place value
  • Euclid: the figure and the proof
  • Sine, leg and hypotenuse
  • From words to letters
  • +, −, × and =: the history of the signs
  • Napier and the logarithm
  • Leibniz against the dots
  • From exhaustion to the integral
  • Euler and functions
  • Cauchy and the limit
  • Galois and Cayley
  • Hamilton, Gibbs and Heaviside
  • Differential equations
  • Boole, Frege and Peano
  • References and credits
• 2. The mathematical sign
  • What is a sign?
  • Peirce: icon, index and symbol
  • The object and its representations
  • The grammar of formulas
  • When place is the algorithm
  • The equals sign is a balance
  • Square root: the side of the square
  • Formula, table and graph
  • How to read a notation
  • References and credits
• 3. Variables, unknowns and functions
  • One letter, five roles
  • The unknown
  • Letters that generalise
  • The variable
  • Parameters
  • Naming a function
  • Dummy letters
  • When the letter misleads
  • Guide and glossary
  • References and credits
• 4. Equality, equivalence and proof
  • One sign, many equals
  • Frege and the morning star
  • Equation, identity, definition
  • Equivalence
  • ⇒ and ⇔
  • The chain of equalities
  • The words of a proof
  • Examples do not prove
  • Guide and glossary
  • References and credits

mainAngGeoen

• Introduction to Angles
• Angle Comparison
• Angle Measures

mainCirc

• Stewart Ratio

mainCircen

• Introduction to Circumference
• Position relative to the Circumference
• Circumscript segments
• Arc Properties
• Inscribed angle
• Segment angle
• Point power
• Theorem for any triangles

mainGeoPlanaen

• Primitive Notions
• Straight lines
• Region in Plans

mainIRRAen

• Radicals Operations

mainPolien

• Introduction to Polygons
• Polygon Properties

mainQuadGeoen

• Introduction to Quadrilaterals
• Trapezium
• Parallelogram
• Rectangle, Rhombus and Square

mainTriAngGeoen

• Introduction to Triangles
• Triangle Congruence
• Parallelism
• Perpendicularity
• Triangle Similarity
• Thales' Theorem
• Notable points of the triangle
• Right triangle relations