#!/usr/bin/env python3
# Pyromaths
# Un programme en Python qui permet de créer des fiches d'exercices types de
# mathématiques niveau collège ainsi que leur corrigé en LaTeX.
# Copyright (C) 2006 -- Jérôme Ortais (jerome.ortais@pyromaths.org)
#
# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program; if not, write to the Free Software
# Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
#
'''
Created on 19 déc. 2014
@author: jerome
'''
from pyromaths.outils.Arithmetique import carrerise
from pyromaths.outils.decimaux import decimaux
from pyromaths.outils import Priorites3
from pyromaths import classes
from math import sqrt
[docs]class SquareRoot:
'''
Définit la classe SquareRoot permettant de manipuler des racines carrées.
SquareRoot([a,b], [c, d], e) ou SquareRoot([a,b], [c, d], [e, None]) permet de définir a*sqrt(b)+c*sqrt(d)+e
Cette définition permet d'utiliser pyromaths.outils.Priorites3
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> repr(SquareRoot(-4,[-2,1],[3,45],[-1,7],8))
SquareRoot([[-4, None], [-2, 1], [3, 45], [-1, 7], [8, None]])
'''
[docs] def __init__(self, *radicandes):
'''
Constructor
'''
# print radicandes, len(radicandes), radicandes[0], len(radicandes[0])
if len(radicandes) == 1 and len(radicandes[0]) == 2 and not isinstance(radicandes[0][0], list) and not isinstance(radicandes[0][1], list):
# SquareRoot([3, 4]),
self.racines = [radicandes[0]]
else:
if len(radicandes) == 1 and (len(radicandes[0]) != 2 or isinstance(radicandes[0][0], list) or isinstance(radicandes[0][1], list)):
# SquareRoot([1, [3, 4]])
radicandes = radicandes[0]
self.racines = []
for arg in radicandes:
if isinstance(arg, list) and len(arg) == 2:
# On gère a*sqrt(b)
if isinstance(arg[1], (float, int))and arg[1] < 0:
raise ValueError(u'Le radicande doit être un nombre positif.')
else:
self.racines.append(arg)
elif isinstance(arg, (float, int)):
self.racines.append([arg, None])
else:
raise ValueError(u'Not Implemented : SquareRoot(%s)' % arg)
[docs] def __str__(self):
r"""Renvoie une version LaTeX d'un objet SquareRoot.
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> str(SquareRoot([[-4, None], [-2, 1], [3, 45], [-1, 7], [8, None]]))
-4-2\,\sqrt{1}+3\,\sqrt{45}-\sqrt{7}+8
:rtype: String
"""
def print_coef(coef):
"""Gère le format du coef
"""
if isinstance(coef, (float, int)):
if coef > 0: return "+" + decimaux(coef)
else: return decimaux(coef)
if isinstance(coef, classes.Fractions.Fraction):
if isinstance(coef.n, int) and isinstance(coef.d, int) and coef.n < 0 and coef.d > 0:
return "-" + str(Fraction(-coef.n, coef.d, coef.code))
return "+" + str(coef)
if isinstance(coef, str):
texte = "(" + "".join(Priorites3.texify([Priorites3.splitting(coef)])) + ")"
if texte[0] != "-": return "+" + texte
else: return texte
s = ""
for m in self.racines:
if m[1] == None:
# pas de racine ici
s = s + print_coef(m[0])
else:
# Racine carrée
if m[0] == 1:
s += r'+\sqrt{%s}' % m[1]
elif m[0] == -1:
s += r'-\sqrt{%s}' % m[1]
else:
s += print_coef(m[0]) + r'\,\sqrt{%s}' % m[1]
# supprime le + en début de séquence
s = s.lstrip("+")
if not s: s = "0"
return s
[docs] def __repr__(self):
"""Renvoie une chaîne de caractère représentant un :mod:`SquareRoot`
évaluable pour créer un :mod:`SquareRoot`.
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> repr(SquareRoot(-4, [-2, 1], [3, 45], [-1, 7], 8))
SquareRoot([[-4, None], [-2, 1], [3, 45], [-1, 7], [8, None]])
:rtype: String
"""
return "SquareRoot(%s)" % self.racines
[docs] def __len__(self):
"""*object*\ .\ **__len__**\ ()
Renvoie le nombre d'éléments de l'objet SquareRoot.
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> len(SquareRoot(3, [2, 2], [4, 5]))
3
:rtype: integer
"""
return len(self.racines)
[docs] def __getitem__(self, i):
"""*object*\ .\ **__getitem__**\ (*integer*)
Renvoie le i ème élément de l'objet SquareRoot.
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> SquareRoot(3, [2, 2], [4, 5])[2]
[4, 5]
:rtype: list
"""
return self.racines[i]
[docs] def __add__(self, other):
"""Renvoie la somme d'un objet SquareRoot et d'un nombre.
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> repr(SquareRoot([3,45],3)+SquareRoot([2,45]))
SquareRoot([[3, None], [5, 45]])
:rtype: SquareRoot
"""
if not isinstance(other, SquareRoot):
other = SquareRoot([other, None])
self.racines.extend(other.racines)
return SquareRoot(self.racines).simplifie()
[docs] def __radd__(self, other):
"""
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> repr(2+SquareRoot([3,45],3))
SquareRoot([[5, None], [3, 45]])
:rtype: SquareRoot
"""
other = SquareRoot([other, None])
other.racines.extend(self.racines)
return SquareRoot(other.racines).simplifie()
[docs] def __neg__(self):
"""*object*\ .\ **__neg__**\ ()
``p.__neg__()`` est équivalent à ``-p`` est équivalent à ``p = -p``
Renvoie l'opposé d'un objet SquareRoot.
:rtype: SquareRoot
"""
if self.EstDecomposable() or self.EstReductible(): return '-%r' % self.simplifie()
r = list(self.racines)
for i in range(len(r)):
r[i][0] = -r[i][0]
return SquareRoot(r)
[docs] def __abs__(self):
""" Renvoie la valeur absolue d'un objet SquareRoot
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> repr(abs(SquareRoot([5, 5], [-2, 7])))
SquareRoot([[5, 5], [-2, 7]])
>>> repr(abs(SquareRoot([-5, 5], [2, 7])))
SquareRoot([[5, 5], [-2, 7]])
:rtype: SquareRoot
"""
t = 0
for e in self.racines:
if e[1] == None: t += e[0]
else: t += e[0] * sqrt(e[1])
if t > 0: return self
else: return -self
def evaluate(self, other):
t, t2 = 0, 0
for e in self.racines:
if e[1] == None:
t += e[0]
else:
t += e[0] * sqrt(e[1])
if isinstance(other, SquareRoot):
for e in other.racines:
if e[1] == None:
t2 += e[0]
else:
t2 += e[0] * sqrt(e[1])
else:
t2=other
return t, t2
[docs] def __lt__(self, other):
t, t2 = self.evaluate(other)
if t<t2: return True
else: return False
[docs] def __le__(self, other):
t, t2 = self.evaluate(other)
if t<=t2: return True
else: return False
[docs] def __eq__(self, other):
t, t2 = self.evaluate(other)
if t==t2: return True
else: return False
[docs] def __ne__(self, other):
t, t2 = self.evaluate(other)
if t!=t2: return True
else: return False
[docs] def __gt__(self, other):
t, t2 = self.evaluate(other)
if t>t2: return True
else: return False
[docs] def __ge__(self, other):
t, t2 = self.evaluate(other)
if t>=t2: return True
else: return False
[docs] def __mul__(self, other):
"""Multiplie un objet SquareRoot par un nombre.
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> repr(SquareRoot([3,45],3)*SquareRoot([2,45],-1))
SquareRoot([['6*45', None], [-3, 45], [6, 45], [-3, None]])
"""
if not isinstance(other, SquareRoot):
other = SquareRoot([other, None])
reduction = False
if self.EstReductible():
self = self.simplifie()
reduction = True
if other.EstReductible():
other = other.simplifie()
reduction = True
if reduction: return '%r*%r' % (self, other)
lprod = []
for e in self.racines:
for f in other.racines:
if e[1] == None or f[1] == None:
lprod.append([e[0] * f[0], e[1] or f[1]])
elif e[1] == f[1]:
lprod.append(['%r*%r' % (e[0] * f[0], e[1]), None])
elif carrerise(e[1]) == 1 or carrerise(f[1]) == 1:
if carrerise(e[1]) == 1: e[0], e[1] = e[0] * int(sqrt(e[1])), 1
if carrerise(f[1]) == 1: f[0], f[1] = f[0] * int(sqrt(f[1])), 1
lprod.append([e[0] * f[0], e[1] * f[1]])
else:
lprod.append([e[0] * f[0], e[1] * f[1]])
return SquareRoot(lprod)
[docs] def __rmul__(self, other):
"""
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> repr(5*SquareRoot([3,45],3))
SquareRoot([[15, 45], [15, None]])
:rtype: SquareRoot
"""
return SquareRoot([other, None]) * self
[docs] def __floordiv__(self, other):
"""
Division entière par un entier
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> repr(SquareRoot([10, 8], [15, 5])//5)
SquareRoot([[2, 8], [3, 5]])
:rtype: SquareRoot
"""
if isinstance(other, int):
r = list(self.racines)
for i in range(len(r)):
r[i][0] = r[i][0] // other
return SquareRoot(r)
else:
raise NotImplemented
[docs] def EstDecomposable(self):
"""
Renvoie True si une des racines est de la forme sqrt{a**2*b} avec a != 1
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> SquareRoot([5, 8], [1, 7]).EstDecomposable()
True
>>> SquareRoot([5, 7], [1, 7]).EstDecomposable()
False
:rtype: Boolean
"""
for e in self.racines:
if e[1] != None and (carrerise(e[1]) != e[1] or e[1] == 1):
return True
return False
[docs] def EstReductible(self):
"""
Renvoie True si la somme de racines est réductible
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> SquareRoot([5, 8], [1, 45]).EstReductible()
False
>>> SquareRoot([5, 8], [1, 8]).EstReductible()
True
:rtype: Boolean
"""
lradicandes = []
rationnel = False
for e in self.racines:
if e[1] != None:
if e[1] in lradicandes:
return True
else:
lradicandes.append(e[1])
elif e[1] == None:
if rationnel: return True
else: rationnel = True
return False
[docs] def Decompose(self):
"""
Décompose une unique racine carrée de la forme a*sqrt(b^2*c) en a*sqrt(b^2)*sqrt(c)
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> SquareRoot([5, 8]).Decompose()
SquareRoot([[5, 4]])*SquareRoot([[1, 2]])
:rtype: string
"""
racine = self.racines[0]
if racine[1] == None: return repr(racine[0])
if isinstance(racine[1], int):
complement = carrerise(racine[1])
if complement == 1:
if racine[0] == 1:
return int(sqrt(racine[1]))
if racine[0] == -1:
return -int(sqrt(racine[1]))
if racine[1] == 1:
return str(racine[0])
return '%r*%r' % (racine[0], int(sqrt(racine[1])))
if complement == racine[1]:
return repr(self)
return '%r*%r' % (SquareRoot([racine[0], racine[1] // complement]), SquareRoot([1, complement]))
raise ValueError(u'Not Implemented : SquareRoot(%s)' % racine)
[docs] def simplifie(self):
"""
Additionne les nombres rationnels et décompose les racines carrées.
>>> from pyromaths.classes.SquareRoot import SquareRoot
>>> repr(SquareRoot([[3, 9]]).simplifie())
'3*3'
>>> repr(SquareRoot(-2, [-2, 1], [3, 1], [-1, 7], 8).simplifie())
SquareRoot([[6, None], [1, 1], [-1, 7]])
>>> repr(SquareRoot(-2, [-2, 1], [3, 45], [-1, 7], 8).simplifie())
SquareRoot([[6, None], [-2, 1], [3, 45], [-1, 7]])
>>> repr(SquareRoot([-2,1],[3,45]).simplifie())
'-2+SquareRoot([[3, 9]])*SquareRoot([[1, 5]])'
:rtype: SquareRoot ou String
"""
decomposable = self.EstDecomposable()
reductible = self.EstReductible()
if not decomposable and not reductible:
return self
if reductible:
racines = []
# racines = [[sum([x[0] for i, x in enumerate(self.racines) if x[1] == None]), None]]
# TODO: cas de fractions
lsomme = [x[0] for i, x in enumerate(self.racines) if x[1] == None]
s = ''
for ls in lsomme:
if isinstance(ls, str): s += '+%s' % ls
else: s += '+%r' % ls
s.lstrip('+')
racines.append([eval(s), None])
for pos in reversed([i for i, x in enumerate(self.racines) if x[1] == None]):
self.racines.__delitem__(pos)
if racines[0][0] == 0: racines = []
while len(self.racines) > 0:
# racines.append([sum([x[0] for i, x in enumerate(self.racines) if x[1] == self.racines[0][1]]), self.racines[0][1]])
# TODO: cas de fractions
lsomme = [x[0] for i, x in enumerate(self.racines) if x[1] == self.racines[0][1]]
s = ''
for ls in lsomme:
if isinstance(ls, str): s += '+%s' % ls
else: s += '+%r' % ls
s.lstrip('+')
racines.append([eval(s), self.racines[0][1]])
if racines[-1][0] == 0: racines.pop(-1)
for pos in reversed([i for i, x in enumerate(self.racines) if x[1] == self.racines[0][1]]):
self.racines.__delitem__(pos)
if racines:
if len(racines) == 1 and racines[0][1] == None:
return racines[0][0]
else:
return SquareRoot(racines)
else: return 0
# Décomposable
s = ''
if len(self.racines) == 1 and isinstance(SquareRoot(self.racines).Decompose(), int):
return SquareRoot(self.racines).Decompose()
for e in self.racines:
decomposee = SquareRoot(e).Decompose()
if isinstance(decomposee, int): decomposee = str(decomposee)
if decomposee[0]in'+-':
s += decomposee
else:
s += '+' + decomposee
s = s.lstrip('+')
return s
def __float__(self):
value=0
for item in self.racines:
if item[1]!=None:
if type(item[0])==str:
value = value + float(eval(item[0])) * sqrt(item[1])
else:
value = value + item[0]*sqrt(item[1])
else:
value = value + item[0]
return value
