Mathematical demonstrations

Common algebraic calculation

• Properties of fractions\( : \frac{a}{b} = \frac{c}{d} \Longleftrightarrow ad = bc \)

• Properties of the powers of x (for natural exponents)\( : x^a x^b = x^{a+b}\)

Binomials

• Newton's binomial\( : (a + b)^n = \sum \binom{n}{p} a^{n-p}b^p \)

• Geometrical identity\( : a^n - b^n = (a-b) \sum a^{n-p-1}b^p \)

• Properties of the binomial\( : \binom{n}{p} = \binom{n -1}{p -1 } + \binom{n - 1}{p} \)

Analysis and enumerating

• Combinatorial analysis and enumerating formulas \( : \binom{n}{p} = \frac{n!}{p! (n-p)!} \)

Linear algebra

• Properties of matrix\( : (A \times B)_{i,j} = \sum a_{i,k} \times b_{k,j} \)

Geometry of the triangle

• Pythagorean theorem and its reciprocal\( : a \perp b \Longleftrightarrow a^2 + b^2 = c^2 \)

• Thales' theorem and its reciprocal\( : BC \parallel DE \Longleftrightarrow \frac{AB}{AD} = \frac{AC}{AE} = \frac{BC}{DE} \)

• Geometric laws of a triangle\( : S_{abc} = \sqrt{p(p-a)(p-b)(p-c)} \)

• Similarity of two triangles

Geometry of the circle

• Right-angled triangle inscribed in a circle

• Calculation of Pi \( (\pi)\) by geometrical method

• Potency of a point in relation to a circle\( : \overline{OA} \times \overline{OB} = \overline{OC} \times \overline{OD} \)

Space and vectors

• Calculation of surfaces and volumes by integration\( : S_{sphere} = 4\pi R^2 \)

• Properties of the scalar product\( : \vec{u}\cdot\vec{v} = ||\vec{u}|| \times ||\vec{v}|| \times \cos(\vec{u}, \vec{v})\)

• Properties of the vector product\( : || \vec{u} \land \vec{v} || = || \vec{u}|| \times || \vec{v}|| \times \sin(\vec{u}, \vec{v}) \)

• Analytical geometry in space\( : M \in \mathcal{P}(A, \vec{n}) \hspace{0.1em} \Longleftrightarrow \hspace{0.1em} ax + by + cz + d = 0 \)

Trigonometry

• Trigonometric operations formulas\( : \sin(2\alpha) = 2 \sin(\alpha) \cos(\alpha) \)

• Euler's trigonometric formulas\( : e^{ix} = \cos(x) + i \sin(x) \)

Complex numbers

• Properties of complex numbers\( : \arg(z^n) = n \cdot \arg(z) \)

• Moivre's formula\( : \Bigl[\cos(\theta) + i \hspace{0.2em} \sin(\theta)\Bigr]^n = \cos(n\theta) + i \hspace{0.2em} \sin(n\theta) \)

Usual functions

• Logarithmic and exponential functions\( : \ln(ab) = \ln(a) + \ln(b) \)

Polynomials and equations

• Solving quadratic equations\( : P_2(X) = aX^2 + bX + c = 0 \)

• Solving cubic equations\( : P_3(X) = aX^3 + bX^2 + cX + d = 0\)

• Lagrange interpolating polynomial\( : L(X) = y_0 L_0(X) + y_1 L_1(X) + \ ... \ + \hspace{0.2em} y_n L_n(X) \)

Differential calculus

• L'Hôpital's rule\( : \enspace \lim_{x \to \alpha} \enspace \frac{f(x)}{g(x)} = \lim_{x \to \alpha} \enspace \frac{f'(x)}{g'(x)} \)

• Derivativity\( : f'(x) = \lim_{h \to 0} \enspace \frac{f(x+h) - f(x)}{h} \)

• Derivatives of standard functions\( : (x^n)' = nx^{n - 1} \)

• Derivatives of trigonometric functions\( : \operatorname{Arcsin}(x)' = \frac{1}{\sqrt{1 - x^2}} \)

• Derivatives of operations on functions\( : (f \circ g)' = g'(f' \circ g) \)

Studies of functions

• Convexity\( : f \enspace convex \enspace on \enspace I \Longleftrightarrow f(x) \geqslant f'(a)(x - a) + f(a) \)

• Mean value theorem\( : \exists c \in \bigl ]a,b \bigr[, \ f'(c) = \frac{ f(b) - f(a)}{b-a} \)

• Newton's method: a way to approximate any value\( : a_{n + 1} = a_n - \frac{f(a_n)}{f'(a_n)} \)

• Length of a curve over an interval\( : {\displaystyle L_{\bigl[a, b \bigr]}(f) \approx \int_a^b \sqrt{1 + \bigl[f'(t)\bigr]^2} \ dt} \)

Integral calculus

• Properties of integrals\( : {\displaystyle \int^x } \bigl(\lambda f + \mu g \bigr) \ dt = \lambda {\displaystyle \int^x } f \ dt + \mu {\displaystyle \int^x } g \ dt \)

• Link between integrals and antiderivatives\( : F(x) = F(a) + {\displaystyle \int_a^x } f(t) \ dt \)

• Standard antiderivatives and general integration methods\( : {\displaystyle \int_a^b } (f'g) \hspace{0.2em}dt = \Bigl[fg\Bigr]_{a}^b - {\displaystyle \int_a^b } (fg') \hspace{0.2em}dt \)

• Integration methods for rational fractions \( : {\displaystyle \int^x } \frac{1}{t^2 + pt + q} \ dt = \frac{1}{(x_1 - x_2)} \ \ln \left| \frac{x-x_1}{x-x_2} \right| \)

• Integration methods for rational fractions with square roots \( : {\displaystyle \int^x } \frac{dt}{\sqrt{a^2 + t^2}} = \ln \left|\sqrt{ a^2 + x^2 } + x\right| \)

• Antiderivatives of trigonometric functions\( : {\displaystyle \int^x } \tan(t) \ dt = - \ln|\cos(x)| \)

• Fourier series \( : {\displaystyle f(t) = a_0 + \sum_{n = 1}^{+ \infty} a_n \cos(n \omega t) + \sum_{n = 1}^{+ \infty} b_n \sin(n \omega t) } \)

Differential equations

• Solving 1 ^st order linear differential equations with continuous coefficient\( : y' + a(x)y = f(x) \)

• Solving 2 ^nd order linear differential equations with constant coefficients\( : y'' + ay' + by= f(x) \)

• Superposition principle

Asymptotic analysis

• Taylor formulas and common Taylor expansions\( : f(x) = \hspace{0.2em} f(a) + f'(a)(x - a) \hspace{0.2em} + ...\hspace{0.2em} + o\bigl((x - a)^n\bigr) \)

Sequences

• Properties of numerical sequences\( : {\displaystyle \lim_{n \to \infty} \bigl[ u_n \bigr] = l \Longrightarrow \lim_{n \to \infty} \left[ \frac{1}{n + 1} \sum_{k = 0}^n u_k \right] = l } \)

• The Fibonacci sequence and the golden ratio\( : F_n = \frac{\varphi^n}{\sqrt{5}} - \frac{(1 - \varphi)^n}{\sqrt{5}} \)

Sums

• Usual sums\( : {\displaystyle \sum_{k = 0}^n} k = \frac{n(n+1)}{2} \)

• Properties of sums\( : {\displaystyle \sum_{k=0}^n \bigl [a_{k+1} - a_k \bigr] = a_{n+1} - a_{0} } \)

Series

• Convergence criteria for series\( : {\displaystyle \int_{1}^{n + 1} f(t) \ dt \leqslant \sum_{k = 0}^n f(k) \leqslant \sum_{k = 1}^n \int_{0}^n f(t) \ dt } \)

• Properties of convergent series\( : {\displaystyle \sum_{k = 0}^{+ \infty} (\lambda \ a_k) = \lambda \sum_{k = 0}^{+ \infty} a_k } \)

Divisibility

• Euclid's algorithm\( : b \nmid a \Longleftrightarrow a = bq + R \)

• Properties of divisibility\( : ka \mid kb \hspace{0.2em} \Longrightarrow \hspace{0.2em} a \mid b \)

• Properties of the GCD of two natural numbers\( : a = bq + R \Longrightarrow GCD(a, b) = GCD(b, R) \)

Prime numbers

• Properties of prime numbers\( : p \nmid a \hspace{0.2em} \Longleftrightarrow \hspace{0.2em} a \wedge p = 1 \)

Modular arithmetic

• Properties of congruences\( : a \equiv b \hspace{0.2em} \bigl[ n \bigr] \Longleftrightarrow n \mid (a - b) \)

• Bezout's theorem ans its corollary\( : a \wedge b = 1 \Longleftrightarrow au + bv = 1 \)

• Gauss' theorem ans its corollary\( : (a \mid bc) \text{ and } (a \wedge b = 1) \hspace{0.2em} \Longrightarrow \hspace{0.2em} a \mid c \)

• Fermat's small theorem and its corollary\( : a^p \equiv a \hspace{0.2em} \bigl[p\bigr] \)