Mathematical demonstrations

$$ (translation \ is \ up \ to \ date !) $$

This site is devoted to mathematics, and especially to demonstrations. It is designed to present certain concepts for middle/high school level (it may be a bit more), and to encourage students learning.

Furthermore, it gives support for my private lessons students, and anyone who wants it.

Josselin DOUINEAU, maths teacher.

List of themes

Geometry of the triangle

Geometry of the circle

Trigonometry

Complex numbers

Geometry in space

Calculation of surfaces and volumes by integration$: S_{sphere} = 4\pi R^2 $
Properties of the scalar product$: \vec{u}.\vec{v} = ||\vec{u}|| \times ||\vec{v}|| \times cos(\vec{u}, \vec{v})$
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 $

Polynomials

Limits and continuity

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)} $

Calculus

Integrals/antiderivatives

Differential equations

Asymptotic analysis

Taylor-Young's formula with integral remainder$: f_{n,a}(x) = \hspace{0.2em} f(a) + f'(a)(x - a) \hspace{0.2em} + ...\hspace{0.2em} + \hspace{0.2em} {\displaystyle \int^x } \frac{f^{(n+1)}(t)}{n!}(x-t)^n \hspace{0.2em} dt $

Miscellaneous

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

Pages structure

Title

1) Formula 1

$$ \forall (a, c) \in \hspace{0.05em} \mathbb{R}^2, \enspace (b, d) \in \hspace{0.05em} \bigl[\mathbb{R}^* \bigr]^2, $$

$$ \frac{a}{b} = \frac{c}{d} \Longleftrightarrow ad = bc $$

2) Formula 2

$$ \forall (a, c) \in \hspace{0.05em} \mathbb{R}^2, \enspace (b, d) \in \hspace{0.05em} \bigl[\mathbb{R}^* \bigr]^2, \enspace \ \Bigl \{ (b+d) \Bigr \} \ \neq 0, $$

$$ \frac{a}{b} = \frac{c}{d} = \frac{a+c}{b+d}$$

3) Récapitulatif

Demonstrations

1) Formula 1

We demonstrate the following formula ...etc.

$$ \frac{a}{c} = \frac{b}{d} $$

$$ \frac{ad}{c} = \frac{bd}{d} $$

$$ \frac{adc}{c} = bc$$

$$...etc.$$

$$ \forall (a, c) \in \hspace{0.05em} \mathbb{R}^2, \enspace (b, d) \in \hspace{0.05em} \bigl[\mathbb{R}^* \bigr]^2, $$

$$ \frac{a}{b} = \frac{c}{d} \Longleftrightarrow ad = bc $$

2) Formula 2

We demonstrate this formula by...etc.

$$ \frac{a}{c} = \frac{b}{d} $$

$$ ... $$

$$ ...etc. $$

$$ \forall (a, c) \in \hspace{0.05em} \mathbb{R}^2, \enspace (b, d) \in \hspace{0.05em} \bigl[\mathbb{R}^* \bigr]^2, \enspace \ \Bigl \{ (b+d) \Bigr \} \ \neq 0, $$

$$ \frac{a}{b} = \frac{c}{d} = \frac{a+c}{b+d}$$

3) Recap table

$$ condition $$	$$ function $$
$$ \forall (a, c) \in \hspace{0.05em} \mathbb{R}^2, \enspace (b, d) \in \hspace{0.05em} \bigl[\mathbb{R}^* \bigr]^2, $$	$$ \frac{a}{b} = \frac{c}{d} \Longleftrightarrow ad = bc $$
$$ \forall (a, c) \in \hspace{0.05em} \mathbb{R}^2, \enspace (b, d) \in \hspace{0.05em} \bigl[\mathbb{R}^* \bigr]^2, \enspace \ \Bigl \{ (b+d) \Bigr \} \ \neq 0, $$	$$ \frac{a}{b} = \frac{c}{d} = \frac{a+c}{b+d} $$

Examples

1) Example 1

Here is an example of application of this theorem:

$$ \frac{20 [g]}{6 [L]} = \frac{X [g]}{1 [L]} $$

Let us apply the theorem to find the unknown...etc.

Mathematical demonstrations

Elementary calculation \( (a + b)^n = \sum \binom{n}{p} a^{n-p}b^p \)

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

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 binom\(: \binom{n}{p} = \binom{n -1}{p -1 } + \binom{n - 1}{p} \)

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

Geometry \( a \perp b \Longleftrightarrow a^2 + b^2 = c^2 \)

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

Al-Kashi's theorem\(: a^2 = b^2 + c^2 - 2bc.cos(\alpha) \)

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

Law of sines\(: \frac{sin(\alpha)}{a} = \frac{sin(\beta)}{b} = \frac{sin(\gamma)}{c} \)

Similarity of two triangles

Right-angled triangle inscribed in a circle

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

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

Trigonometric duplication and addition formulas\(: sin(2\alpha), \ sin(\alpha + \beta)... \)

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

Properties of complex numbers\(: |z|, \ arg(z), \ \bar{z} \)

Moivre's formula\(: \Bigl[cos(\theta) + isin(\theta)\Bigr]^n = cos(n\theta) + isin(n\theta) \)

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

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

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

Analysis \( F(x) = F(a) + {\displaystyle \int_a^x } f(t) \ dt \)

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

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

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\(: f'(x) = lim_{\Delta_x \to 0} \ \frac{\Delta_{f(x)}}{\Delta_x} = \frac{df}{dx} \)

Derivatives of standard functions\(: (x^2)' , \ \big[ln(x)\big]' , \ \left(\frac{1}{x}\right)'... \)

Derivatives of trigonometric functions\(: sin(x)', \ cos(x)' , \ tan(x)'... \)

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

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

Mean value theorem\(: \exists c \in \hspace{0.05em} ]a, b[, \ f'(c) = \frac{ f(b) - f(a)}{b-a} \)

Newton's method: a way to approximate any value

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\(: {\displaystyle \int^x } tan(t) \ dt = - ln|cos(x)| \)

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

Taylor-Young's formula with integral remainder\(: f_{n,a}(x) = \hspace{0.2em} f(a) + f'(a)(x - a) \hspace{0.2em} + ...\hspace{0.2em} + \hspace{0.2em} {\displaystyle \int^x } \frac{f^{(n+1)}(t)}{n!}(x-t)^n \hspace{0.2em} dt \)

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

Sequences and series \( \sum \bigl [a_{k+1} - a_k \bigr] = a_{n+1} - a_{0} \)

Telescoping of terms in recurring numerical series\(: \sum \bigl [a_{k+1} - a_k \bigr] = a_{n+1} - a_{0} \)

Usual sums\(: \sum k, \ \sum k^2, \ \sum k^3, \ \sum (2k+1)... \)

Arithmetic \( b \nmid a \Longrightarrow a = bq + R \)

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

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

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

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

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

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

Gauss' theorem ans its corollary\(: a / bc \enspace et \enspace a \wedge b = 1 \hspace{0.2em} \Longrightarrow \hspace{0.2em} a/c \)

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

Glossary of symbols

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\(: y'' + ay' + by= f(x) \)