From 4909359eac45d691891f20b372fc91e80615b1ba Mon Sep 17 00:00:00 2001 From: Florian Mathieu Date: Mon, 3 Jul 2023 11:54:24 +0200 Subject: [PATCH] maj chapitre 3 --- representation_base/.DS_Store | Bin 10244 -> 10244 bytes representation_base/chapitre_3/README.md | 61 +++++++++++++++---- representation_base/chapitre_3/assets/bo.png | Bin 0 -> 439908 bytes 3 files changed, 48 insertions(+), 13 deletions(-) create mode 100644 representation_base/chapitre_3/assets/bo.png diff --git a/representation_base/.DS_Store b/representation_base/.DS_Store index 4a5dcbbafda2dc09168b37b48932273084572082..41b318a06205f328b82a8beefe81c401a000ff53 100644 GIT binary patch delta 25 bcmZn(XbIRbLv->7;pokuMVInJ8G=jznXL;Y delta 25 bcmZn(XbIRbLv*r(c=YDaqD%Rq3_&IUh}j9v diff --git a/representation_base/chapitre_3/README.md b/representation_base/chapitre_3/README.md index 663b3d4..8474667 100644 --- a/representation_base/chapitre_3/README.md +++ b/representation_base/chapitre_3/README.md @@ -1,12 +1,12 @@ # Représentation des décimaux : les flottants -## 1. Attendus +## Le Programme -| Contenus | Capacités attendues | -| :--: | :-- | -| Représentation approximative des nombres réels : notion de nombre flottant | Calculer sur quelques exemples la représentation de nombres réels : 0.1, 0.25 ou 1/3. | +![bo.png](assets/bo.png) -## 2. Contexte +------- + +## Contexte Nous avons appris à encoder des nombres entiers naturels, et relatifs, et nous avons vu que les limites physiques des machines imposaient des limites sur l'étendue des valeurs. Par exemple, sur un octet on dispose de $`2^8 = 256`$ valeurs distinctes qui permettent d'encoder: @@ -15,7 +15,7 @@ Nous avons appris à encoder des nombres entiers naturels, et relatifs, et nous Maintenant que nous allons tenter de coder les réels, les limites de notre machine vont encore entraîner des limites sur l'étendue des valeurs, mais également sur la __précision des valeurs__. -### 2.1. À Faire +### À Faire 1. effectuez le calcul $`0.2 + 0.1`$ en Python : @@ -39,7 +39,9 @@ Maintenant que nous allons tenter de coder les réels, les limites de notre mach Ainsi, un calcul avec des nombres à virgule ne peut-être qu'__approximatif__. Cependant plus on augmente la taille du registre du processeur et plus nous pourrons représenter de valeurs, et plus nos calculs gagneront en précision. -## 3. Codage de la virgule +--------------------- + +## Codage de la virgule Il existe deux façons de coder les nombres réels, en virgule fixe ou virgule flottante. @@ -57,16 +59,16 @@ C’est extrêmement simple. Cette manière de faire s’appelle virgule fixe, c L’inconvénient de cette méthode est que, pour un nombre avec peu de chiffres après la virgule, on perd un espace de stockage significatif. Si le nombre en question est `0110 1000`, on perd trois bits “inutilement”. -#### 3.1.1. À Faire +#### À Faire Représenter les valeurs suivants sur 8 bits, en virgule fixe : 1. 7,75 2. 0,1 -### 3.2. Virgule flottante +### Virgule flottante -#### 3.2.1. Notation scientifique +#### Notation scientifique Cette écriture se base sur la notation scientifique des nombres : $`\pm a \times 10^n`$ où $`1 \le a \lt 10`$ et $`n \in \mathbb{Z}^*`$ @@ -81,14 +83,14 @@ Le terme __exposant__ correspond à la puissance de 10, et le terme __mantisse__ - la mantisse (ou significande) est « 1,23 » ; - l'exposant est « -2 ». -##### 3.2.1.1. À Faire +##### À Faire Exprimez les nombres suivants en notation scientifique : - La distance $`d`$ entre la Lune et la Terre est de 384 400km - Le poids $`p`$ du moustique Tigre est d'environ 0, 000 001 07kg. -#### 3.2.2. Notation scientifique binaire +#### Notation scientifique binaire Un nombre binaire à virgule de quatre chiffres $`n_1n_0,n_{-1}n_{-2}`$ correspond au nombre décimal $`n_1 \times 2^1 + n_0 \times 2^0 + n_{-1} \times 2^{-1} + n_{-2} \times 2^{-2}`$. @@ -101,7 +103,34 @@ Par exemple : _N.B : Dans le cas de la notation scientifique binaire, le nombre avant la virgule doit être compris entre $`1_2`$ inclus et $`10_2`$ exclus (c'est-à-dire 2 exclus), c'est-à-dire que sa partie entière est nécessairement 1._ -#### 3.2.3. Principe du codage en virgule flottante + + +#### 3.2.3 Passage de l'écriture décimale à l'écriture binaire pour un nombre flottant + +Si vous avez bien suivi, vous savez qu'un nombre est composé de deux parties : + +- partie entière, à gauche de la virgule +- partie décimale (ou fractionnaire), à droite de la virgule + +On traite ces deux parties indépendamment l'une de l'autre : + +- La partie entière va s'écrire comme nous l'avons appris précédemment. + +- Pour la partie décimale, on va procéder par multiplications par 2 successivement. + - Après chaque multiplication, le résultat est reporté sans sa partie entière + - On poursuit les multiplications jusqu'à obtenir 1 comme résultat. + - On prend la partie entière de chaque résultat, de haut en bas, pour obtenir l'écriture binaire du nombre. + +Exemple : + +- $` 0,6875 \times 2 = **1**,375`$ +- $`0,375 \times 2 = **0**,75 `$ +- $`0,75 \times 2 = **1**,5`$ +- $`0,5 \times 2 = **1** `$ + +0,6875 s'écrit donc 0,10112 + +#### 3.2.4. Principe du codage en virgule flottante Un nombre flottant est formé de trois éléments : la mantisse, l'exposant et le signe. @@ -132,6 +161,10 @@ Supposons un nombre flottant codé sur un octet utilisant 1 bit de signe, 3 bits Le code `1 101 1011` sur un octet utilisant 1 bit de signe, 3 bits pour l'exposant et 4 bits pour la mantisse représente donc: $`-1,6875 \times 2^2 = -6,75`$ + + +---------------- + ## 4. La norme IEEE 754 > L'__IEEE 754__ est une norme pour la représentation des nombres à virgule flottante en binaire. Elle est la norme la plus employée actuellement pour le calcul des nombres à virgule flottante dans le domaine informatique. @@ -201,6 +234,8 @@ $`2^{-23}`$ . On ne peut pas trouver un flottant compris entre les deux. Les nombres flottants sont une représentation approximative des nombres réels dans un ordinateur. En particulier, il n’est pas possible de représenter de manière exacte en machine tous les nombres réels. La manipulation de nombres réels par un langage informatique est donc à prendre avec précaution car elle peut engendrer des résultats surprenants, en particulier il ne faut jamais tester l’égalité entre deux flottants. +---------- + ## 5. Exercices ### 5.1. 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