D.S de MATHÉMATIQUES

\(\frac{36}{9} = 4\). Le plus petit ensemble est \(\mathbb{N}\) (entiers naturels), défini par \(\mathbb{N} = \{0, 1, 2, 3, ...\}\).

\(-\frac{7}{2} = -3,5\). Le plus petit ensemble est \(\mathbb{D}\) (nombres décimaux), défini par \(\mathbb{D} = \{\frac{a}{10^p} \mid a \in \mathbb{Z}, p \in \mathbb{N}\}\).

\(-\frac{35}{5} = -7\). Le plus petit ensemble est \(\mathbb{Z}\) (entiers relatifs), défini par \(\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}\).

\(\frac{5}{6}\). Le plus petit ensemble est \(\mathbb{Q}\) (nombres rationnels), défini par \(\mathbb{Q} = \{\frac{a}{b} \mid a \in \mathbb{Z}, b \in \mathbb{N}^*\}.\) Ce nombre n'a pas une partie décimale finie.

\(A = (a-b)-(c+d) = (-1-3)-(-1-2) = -4 - (-3) = -4+3 = \boldsymbol{-1}\)

\(B = -(b-a) \times c - c \times d = -(3-(-1)) \times (-1) - (-1) \times (-2) = -(4)(-1) - 2 = 4-2 = \boldsymbol{2}\)

\(C = (-a-b) \times (-c+d) = (-(-1)-3) \times (-(-1)+(-2)) = (1-3) \times (1-2) = (-2) \times (-1) = \boldsymbol{2}\)

\(D = a - b \times d - d \times a = -1 - 3 \times (-2) - (-2) \times (-1) = -1 - (-6) - 2 = -1+6-2 = \boldsymbol{3}\)

\(E = -3(x-1)(2-4x)-x(1-x) = -3(-4x^2+6x-2) -x+x^2 = 12x^2-18x+6-x+x^2 = \boldsymbol{13x^2-19x+6}\)

\(F = -(p+1)p - (1-p)(2p-1) = -(p^2+p) - (2p-1-2p^2+p) = -p^2-p - (-2p^2+3p-1) = -p^2-p+2p^2-3p+1 = \boldsymbol{p^2-4p+1}\)

\(G = -5x+10x^3 = \boldsymbol{5x(2x^2-1)}\)

\(H = ab-ab^2 = \boldsymbol{ab(1-b)}\)

\(J = 7 - 7 \times \frac{-5}{14} = 7 - \frac{-35}{14} = 7 + \frac{5}{2} = \frac{14}{2} + \frac{5}{2} = \boldsymbol{\frac{19}{2}}\)

\(K = \frac{35}{4} / \frac{49}{8} = \frac{35}{4} \times \frac{8}{49} = \frac{35 \times 8}{4 \times 49} = \frac{(5 \times 7) \times (2 \times 4)}{4 \times (7 \times 7)} = \frac{5 \times 2}{7} = \boldsymbol{\frac{10}{7}}\)

Soit \(L = 0,444...\).
\(10L = 4,444...\)
\(10L - L = 9L = 4\)
\(L = \boldsymbol{\frac{4}{9}}\)

Soit \(M = 1,7272...\). On pose \(x = 0,7272...\).
\(100x = 72,7272...\)
\(100x - x = 99x = 72 \implies x = \frac{72}{99} = \frac{8}{11}\).
\(M = 1 + x = 1 + \frac{8}{11} = \boldsymbol{\frac{19}{11}}\).