Dimostrare

Skill — Dimostrare.

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Linked atoms

130 linked atoms.

01 Numeri e operazioni fondamentali

• The idea behind the proof of the rules
• Why it is enough to stop at the square root of N

04 Percentuali

• Increase and decrease do not cancel out

06 Sistemi di equazioni lineari

• Verify that (AB) inverse equals B inverse times A inverse

07 Teoria degli insiemi

• Bernoulli’s inequality
• Checking an equivalence with the truth table
• Divisibility of n cubed minus n by 6
• Equivalence of two implications
• Inequality 2 to the n greater than n squared
• Lexicographic order and order on pairs
• Logical equivalences and De Morgan’s laws
• Newton’s binomial by induction
• Sum of cubes by induction
• Sum of the first n naturals
• Sum of the first n squares
• The principle of induction
• Total order and the strict less-than

08 Fondamenti di geometria euclidea

• A scheme for proofs
• Applying the scheme
• Base angles of the isosceles triangle
• Bisector, median and altitude
• Congruent opposite angles and parallelogram
• Diagonals that bisect each other and parallelogram
• Euclid and the Elements
• Perpendicular bisector of a segment
• Problem — Congruence with the altitudes
• Proof by contradiction
• Proofs using parallelism
• Sum of the interior angles of a polygon
• Sum of the interior angles of a triangle
• The converse theorem
• The exterior angle as the sum of the non-adjacent interior angles
• The exterior angle is greater than the non-adjacent interior angles
• The five most frequent moves
• The logical steps of a proof
• The perpendicular bisector as a locus
• The rhombus and the bisecting diagonal
• The rhombus and the perpendicular diagonals
• The syllogism
• Theory, postulates and theorems
• Vertically opposite angles
• What a theorem is

09 Metodo di lavoro — come risolvere un problema

• Example — carrying out the angle-bisector proof
• Example — finding the path, the isosceles triangle
• Example — reviewing the angle-bisector proof
• Example — understanding a geometry problem
• Worked problem — the bisector of the isosceles triangle

10 Equazioni di secondo grado e fratte

• Checking a root of the cubic
• The case of the vanishing discriminant

14 Equivalenza e teorema di Pitagora

• Equidecomposability of a triangle and a parallelogram
• Pythagoras’ theorem

15 Similitudine

• Proof of Euclid’s theorems
• The Hermit Crab problem

16 Circonferenza

• Arithmetic mean and geometric mean
• Centroid - concurrency of the medians
• Circumcentre - concurrency of the perpendicular bisectors
• Incentre - concurrency of the bisectors
• Orthocentre - concurrency of the altitudes
• Proof of the central angle theorem
• Radius of the inscribed circle
• Tangents from an external point

17 Disequazioni irrazionali

• Squaring without splitting into cases

19 Luoghi geometrici nel piano cartesiano

• Formulae for focus, vertex and directrix
• Why the locus is a line

20 Circonferenza nel piano cartesiano

• From the definition to the equation
• The radical axis

21 Ellisse

• Proof of the canonical equation

22 Iperbole e funzione omografica

• Proof of the asymptotes
• Rectangular hyperbola xy=6

23 Funzioni, linguaggio e proprietà

• Effect of a tax on the equilibrium
• (x²+1)

26 Funzione logaritmica

• Proof of the change of base
• Proof of the product property

28 Circonferenza goniometrica e funzioni sin, cos, tan

• Fundamental identity with complementary arcs
• Identity with arctan
• Periodicity
• The fundamental identity

29 Formule di addizione, duplicazione, bisezione

• Example: linearising sin⁴α
• How the other formulae are derived
• Linearising sin²α cos²α
• Proof of the bisection formulae
• Proof of the cosine subtraction formula
• The duplication formulae
• Triple-angle formula for the cosine

30 Formule parametriche, prostaferesi, Werner

• Identity on the three angles of a triangle
• The parametric formulae
• Werner formulae: from product to sum

33 Trigonometria dei triangoli

• Heron’s formula as a corollary
• Proof of the area formula
• Proof of the law of cosines
• Proof of the sine rule
• Prove Heron’s formula

34 Combinatoria

• Proof of Tartaglia’s formula

35 Probabilità avanzata

• Formula of the binomial distribution
• Problem — Bertrand’s three answers
• Total probability theorem

36 Numeri complessi

• A branch of a hyperbola
• Centroid of the roots
• Generalised Vieta’s relations

37 Geometria analitica nello spazio

• Distance between two skew lines
• Three planes and a common point

38 Geometria sintetica nello spazio

• Euler’s formula for the pentagonal prism
• Theorem of the three perpendiculars

39 Successioni numeriche

• Sum of the first n terms of an AP
• Theorem of bounded monotone sequences
• Worked exercise — Binet’s formula for Fibonacci
• Worked exercise — Nested radical
• n^n

40 Limiti di funzioni

• n)
• Accumulation point

41 Continuità e discontinuità

• Function with no maximum on [0,1]
• Newton exact for linear functions
• Newton iteration for the reciprocal
• The zero theorem
• Zero of x³−3x+1 with bisection

42 Derivate

• Lagrange’s Theorem
• Rolle’s theorem

44 Integrale

• Cavalieri at work - prism and sphere
• Cavalieri’s principle
• Equivalent pyramids via Cavalieri
• Equivalent triangles (Cavalieri 2D)
• Hemisphere and cylinder via Cavalieri
• Improper integral with a root
• Volume of revolution via Cavalieri

45 Teoremi del calcolo

• Proof of Rolle’s theorem
• Proof of the fundamental theorem
• Proof of the integral mean value theorem

47 Distribuzioni di probabilità

• Mean of the exponential distribution
• Properties of entropy
• Study of the function -p log p