Ensembles de nombres ℕ, ℤ, ℚ, 𝔻, ℝ
4. 📚 Exercices corrigés
Ensembles de nombres et calculs – Tronc Commun Sciences BIOF
🔒 Document protégé – Copie, clic droit et sélection désactivés
✅ Exercice 1 – Symboles ∈ ou ∉
Compléter avec ∈ ou ∉ :
| \(0 \ldots \mathbb{Z}^*\) | \(0 \notin \mathbb{Z}^*\) (car \(\mathbb{Z}^*\) exclut 0) |
| \(\sqrt{97} \ldots \mathbb{R}^-\) | \(\sqrt{97} \notin \mathbb{R}^-\) (car \(\sqrt{97} > 0\)) |
| \(\frac{1}{12} \ldots \mathbb{D}\) | \(\frac{1}{12} \in \mathbb{D}\) (car \(1/12 = 0,08333...\) décimal) |
| \(\frac{2}{3} \ldots \mathbb{Q}\) | \(\frac{2}{3} \in \mathbb{Q}\) (rationnel) |
| \(4,1 \ldots \mathbb{Z}\) | \(4,1 \notin \mathbb{Z}\) (n'est pas entier) |
| \(2 \ldots \mathbb{N}\) | \(2 \in \mathbb{N}\) |
| \(-301 \ldots \mathbb{Q}^+\) | \(-301 \notin \mathbb{Q}^+\) (négatif) |
| \(433 \ldots \mathbb{Z}^*\) | \(433 \in \mathbb{Z}^*\) |
| \(0 \ldots \mathbb{N}\) | \(0 \in \mathbb{N}\) |
| \(5,33 \ldots \mathbb{Q}\) | \(5,33 = \frac{533}{100} \in \mathbb{Q}\) |
| \(5,33 \ldots \mathbb{D}\) | \(5,33 \in \mathbb{D}\) (décimal) |
| \(\frac{17}{2} \ldots \mathbb{D}^+\) | \(\frac{17}{2} = 8,5 \in \mathbb{D}^+\) |
| \(\sqrt{7} \ldots \mathbb{R}^-\) | \(\sqrt{7} \notin \mathbb{R}^-\) (positif) |
| \(\frac{n(n+1)}{2} \ldots \mathbb{N}\) | \(\frac{n(n+1)}{2} \in \mathbb{N}\) (somme des n premiers entiers) |
| \(\sqrt{16} + 2\sqrt{9} \ldots \mathbb{Q}\) | \(4 + 2 \times 3 = 10 \in \mathbb{Q}\) |
✅ Exercice 2 – Symboles ⊂ ou ⊄
| \(\mathbb{R}^- \ldots \mathbb{R}^+\) | \(\mathbb{R}^- \not\subset \mathbb{R}^+\) |
| \(\{0,2,3\} \ldots \mathbb{Z}\) | \(\{0,2,3\} \subset \mathbb{Z}\) |
| \(\{-1\} \ldots \mathbb{Z}^+\) | \(\{-1\} \not\subset \mathbb{Z}^+\) |
| \(\{1,3\} \ldots \mathbb{Z}\) | \(\{1,3\} \subset \mathbb{Z}\) |
| \(\mathbb{N} \ldots \mathbb{Q}^-\) | \(\mathbb{N} \not\subset \mathbb{Q}^-\) (les naturels sont positifs) |
| \(\mathbb{N} \ldots \mathbb{R}\) | \(\mathbb{N} \subset \mathbb{R}\) |
| \(\mathbb{N} \ldots \mathbb{Z}\) | \(\mathbb{N} \subset \mathbb{Z}\) |
| \(\mathbb{N} \ldots \mathbb{D}^+\) | \(\mathbb{N} \subset \mathbb{D}^+\) |
| \(\mathbb{R}^- \ldots \mathbb{Z}\) | \(\mathbb{R}^- \not\subset \mathbb{Z}\) |
| \(\mathbb{N} \ldots \mathbb{Z}^+\) | \(\mathbb{N} \subset \mathbb{Z}^+\) |
| \(\mathbb{N}^* \ldots \mathbb{Z}^*\) | \(\mathbb{N}^* \subset \mathbb{Z}^*\) |
✅ Exercice 3 – Écriture en extension
| \(\{-1\} \cap \mathbb{Z}^+\) | \(\emptyset\) (aucun élément commun) |
| \(\{1,3\} \cap \mathbb{Z}\) | \(\{1,3\}\) |
| \(\mathbb{N} \cap \mathbb{Q}^-\) | \(\{0\}\) |
| \(\mathbb{N} \cap \mathbb{R}\) | \(\mathbb{N}\) |
| \(\mathbb{N} \cap \mathbb{Z}\) | \(\mathbb{N}\) |
| \(\mathbb{N} \cap \mathbb{R}^*\) | \(\mathbb{N}^*\) |
| \(\mathbb{D} \cap \mathbb{Q}\) | \(\mathbb{D}\) |
| \(\mathbb{Z}^- \cap \mathbb{Z}^*\) | \(\mathbb{Z}^- \setminus \{0\}\) |
| \(\mathbb{R} \cap \mathbb{R}^+\) | \(\mathbb{R}^+\) |
| \(\{0,2,3\} \cap \mathbb{Z}\) | \(\{0,2,3\}\) |
| \(\mathbb{N} \cap \mathbb{D}^+\) | \(\mathbb{N}\) |
| \(\mathbb{R} \cap \mathbb{Z}\) | \(\mathbb{Z}\) |
| \(\mathbb{N} \cap \mathbb{Z}^+\) | \(\mathbb{N}\) |
| \(\mathbb{N}^* \cap \mathbb{Z}^*\) | \(\mathbb{N}^*\) |
| \(\{-1\} \cup \{1,3,4\}\) | \(\{-1,1,3,4\}\) |
| \(\{1,3\} \cup \mathbb{Z}\) | \(\mathbb{Z}\) |
| \(\mathbb{N} \cup \mathbb{Q}^*\) | \(\mathbb{Q}\) |
| \(\mathbb{N} \cup \mathbb{R}\) | \(\mathbb{R}\) |
| \(\mathbb{N} \cup \mathbb{R}^+\) | \(\mathbb{R}^+\) |
| \(\{0,2,3\} \cup \mathbb{Z}\) | \(\mathbb{Z}\) |
✅ Exercice 4 – Calculs (fractions et racines)
\( A = \left( \frac{3}{4} - \frac{5}{3} \right) \times \frac{2 - \frac{4}{7}}{3} \times \frac{1}{\frac{4}{3} - \frac{1}{2}} \)
\( \frac{3}{4} - \frac{5}{3} = \frac{9 - 20}{12} = -\frac{11}{12} \)
\( 2 - \frac{4}{7} = \frac{14 - 4}{7} = \frac{10}{7} \)
\( \frac{4}{3} - \frac{1}{2} = \frac{8 - 3}{6} = \frac{5}{6} \)
\( A = -\frac{11}{12} \times \frac{10/7}{3} \times \frac{1}{5/6} = -\frac{11}{12} \times \frac{10}{21} \times \frac{6}{5} \)
\( A = -\frac{11 \times 10 \times 6}{12 \times 21 \times 5} = -\frac{660}{1260} = -\frac{11}{21} \)
\( B = \frac{10101}{10101} \)
\( B = 1 \)
\( C = \) (expression complexe)
Après simplification, \( C = \frac{3(\sqrt{2} - \sqrt{3}) + \frac{1}{\sqrt{6}(1/\sqrt{2} - 1/\sqrt{3})}}{\sqrt{2} + \sqrt{3} - \frac{1}{\sqrt{6}(1/\sqrt{2} + 1/\sqrt{3})}} \)
\( C = 1 \)
\( D = 2 + \frac{1}{4 + \frac{1}{\sqrt{5+2}}} \)
\( \sqrt{5+2} = \sqrt{7} \approx 2,64575 \)
\( D = 2 + \frac{1}{4 + \frac{1}{\sqrt{7}}} = 2 + \frac{1}{\frac{4\sqrt{7} + 1}{\sqrt{7}}} = 2 + \frac{\sqrt{7}}{4\sqrt{7} + 1} \)
✅ Exercice 5 – Calculs avec racines
\( A = 3\left(1 + \frac{1}{3} - \frac{3}{2}\right) + \left(2 - \frac{1}{3}\right)\left(2 - \frac{3}{2}\right) \)
\( 1 + \frac{1}{3} - \frac{3}{2} = \frac{6 + 2 - 9}{6} = -\frac{1}{6} \)
\( 3 \times (-\frac{1}{6}) = -\frac{1}{2} \)
\( 2 - \frac{1}{3} = \frac{5}{3} \), \( 2 - \frac{3}{2} = \frac{1}{2} \)
\( \frac{5}{3} \times \frac{1}{2} = \frac{5}{6} \)
\( A = -\frac{1}{2} + \frac{5}{6} = \frac{-3 + 5}{6} = \frac{2}{6} = \frac{1}{3} \)
\( C = \sqrt{96} + 2\sqrt{6} - 2\sqrt{24} - 3\sqrt{54} \)
\( \sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6} \)
\( 2\sqrt{24} = 2\sqrt{4 \times 6} = 4\sqrt{6} \)
\( 3\sqrt{54} = 3\sqrt{9 \times 6} = 9\sqrt{6} \)
\( C = 4\sqrt{6} + 2\sqrt{6} - 4\sqrt{6} - 9\sqrt{6} = (4+2-4-9)\sqrt{6} = -7\sqrt{6} \)
\( D = \frac{1}{3}\sqrt{363} + \sqrt{108} - \sqrt{300} + \frac{2}{\sqrt{12}} - 2\sqrt{\frac{75}{36}} \)
\( \sqrt{363} = \sqrt{121 \times 3} = 11\sqrt{3} \) → \( \frac{1}{3} \times 11\sqrt{3} = \frac{11}{3}\sqrt{3} \)
\( \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3} \)
\( \sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3} \)
\( \frac{2}{\sqrt{12}} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \)
\( 2\sqrt{\frac{75}{36}} = 2 \times \frac{\sqrt{75}}{6} = \frac{\sqrt{75}}{3} = \frac{5\sqrt{3}}{3} \)
\( D = \frac{11}{3}\sqrt{3} + 6\sqrt{3} - 10\sqrt{3} + \frac{\sqrt{3}}{3} - \frac{5\sqrt{3}}{3} \)
\( D = \left(\frac{11}{3} + 6 - 10 + \frac{1}{3} - \frac{5}{3}\right)\sqrt{3} = \left(\frac{11 + 1 - 5}{3} - 4\right)\sqrt{3} = \left(\frac{7}{3} - 4\right)\sqrt{3} = -\frac{5}{3}\sqrt{3} \)
✅ Exercice 6 – Calculs supplémentaires
\( E = \frac{\sqrt{2} - \frac{1}{\sqrt{2}}}{\sqrt{2} + \frac{1}{\sqrt{8}}} \)
\( \sqrt{2} - \frac{1}{\sqrt{2}} = \frac{2 - 1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \)
\( \sqrt{2} + \frac{1}{\sqrt{8}} = \sqrt{2} + \frac{1}{2\sqrt{2}} = \frac{4 + 1}{2\sqrt{2}} = \frac{5}{2\sqrt{2}} \)
\( E = \frac{1/\sqrt{2}}{5/(2\sqrt{2})} = \frac{1}{\sqrt{2}} \times \frac{2\sqrt{2}}{5} = \frac{2}{5} \)
\( F = 6 - \frac{\frac{5}{3} + \frac{3}{2}}{\frac{3}{2} - \frac{5}{4}} \)
\( \frac{5}{3} + \frac{3}{2} = \frac{10 + 9}{6} = \frac{19}{6} \)
\( \frac{3}{2} - \frac{5}{4} = \frac{6 - 5}{4} = \frac{1}{4} \)
\( \frac{19/6}{1/4} = \frac{19}{6} \times 4 = \frac{76}{6} = \frac{38}{3} \)
\( F = 6 - \frac{38}{3} = \frac{18 - 38}{3} = -\frac{20}{3} \)
\( G = \frac{\frac{3}{1} + \frac{1}{3}}{\frac{1}{4} + \frac{2}{2}} \)
\( \frac{3}{1} + \frac{1}{3} = 3 + \frac{1}{3} = \frac{10}{3} \)
\( \frac{1}{4} + \frac{2}{2} = \frac{1}{4} + 1 = \frac{5}{4} \)
\( G = \frac{10/3}{5/4} = \frac{10}{3} \times \frac{4}{5} = \frac{40}{15} = \frac{8}{3} \)