2017-02-13-204055.ipynb

Kernel: SageMath (stable)

Prise en main du logiciel Sage

Plot

plot(floor(x), (x, 1, 10), exclude=[1..10])

plot(x**2,xmin=-5,xmax=5,ymin=-5,ymax=5,color='purple',legend_label='x**2')

plot([sin(x), tan(x), 1-x^2], legend_label='automatic')    # AUTOMATIC S'APPLIQUE A CE QUI EST ENTRE CROCHETS !

Tester une propriété

is_even(3)

False

is_prime(71)

True

is_prime_power(1872)

False

Symboles, expressions et variables

diff(x**3,x)

3*x^2

integrate(x**3,x)

1/4*x^4

integrate(exp(x),x)

e^x

var('y')
integrate(exp(x),y)

y*e^x

integral(ln(x),x,0,1)

-1

integral(exp(-x^2),x,-oo,+oo)

sqrt(pi)

factor(4*x+4*y+2*x**2*y+9*x**2+2*x**3+9*x*y)

(2*x + 1)*(x + y)*(x + 4)

numerical_approx(pi,digits=200)

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303820

var('n')
sum(1/n**2,n,1,+oo)

1/6*pi^2

sum(1/factorial(n),n,0,+oo)

e

Quelques objets

Z7 = IntegerModRing(7)

b = Z7(10)

b

3

ZZ

Integer Ring

QQ

Rational Field

a = Z7(2)

a

2

type(a)

<type 'sage.rings.finite_rings.integer_mod.IntegerMod_int'>

a**-1

4

a*4

1

Z9 = IntegerModRing(9)

b = Z9(2)

a == b

False

10//3

3

s = [2^i for i in [1..10]] ; s

[2, 4, 8, 16, 32, 64, 128, 256, 512, 1024]

s+s

[2,
 4,
 8,
 16,
 32,
 64,
 128,
 256,
 512,
 1024,
 2,
 4,
 8,
 16,
 32,
 64,
 128,
 256,
 512,
 1024]

4*s

[2,
 4,
 8,
 16,
 32,
 64,
 128,
 256,
 512,
 1024,
 2,
 4,
 8,
 16,
 32,
 64,
 128,
 256,
 512,
 1024,
 2,
 4,
 8,
 16,
 32,
 64,
 128,
 256,
 512,
 1024,
 2,
 4,
 8,
 16,
 32,
 64,
 128,
 256,
 512,
 1024]

s.append(pi)

s

[2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, pi]

L = []
f = x*exp(x)
for k in [1..6]:
    f = diff(f)
    L += [factor(f)]
print(L)

[(x + 1)*e^x, (x + 2)*e^x, (x + 3)*e^x, (x + 4)*e^x, (x + 5)*e^x, (x + 6)*e^x]

parent(5)

Integer Ring

parent (5/2)

Rational Field

QQ

Rational Field

parent(x)

Symbolic Ring

Un peu de programmation

def factorielle(n):
    res = 1
    for k in [2..n]:
        res = res*k
    return res

for k in [1..10]:
    print(factorielle(k))

1
2
6
24
120
720
5040
40320
362880
3628800

def factorielle2(n):
    if n==0 or n==1 : return 1
    return n*factorielle2(n-1)

for k in [1..10]:
    print(factorielle2(k))

1
2
6
24
120
720
5040
40320
362880
3628800

for k in [1..10]:
    print(factorial(k))

1
2
6
24
120
720
5040
40320
362880
3628800

next_prime(11)

13

def liste_premiers(N):
    a = 2
    L = []
    while a <= N :
        L += [a]
        a = next_prime(a)
    return L

liste_premiers(100)

[2,
 3,
 5,
 7,
 11,
 13,
 17,
 19,
 23,
 29,
 31,
 37,
 41,
 43,
 47,
 53,
 59,
 61,
 67,
 71,
 73,
 79,
 83,
 89,
 97]

L = []
for k in [2..1000]:
    if is_prime(k) and is_prime(k+2) : L += [(k,k+2)]
L

[(3, 5),
 (5, 7),
 (11, 13),
 (17, 19),
 (29, 31),
 (41, 43),
 (59, 61),
 (71, 73),
 (101, 103),
 (107, 109),
 (137, 139),
 (149, 151),
 (179, 181),
 (191, 193),
 (197, 199),
 (227, 229),
 (239, 241),
 (269, 271),
 (281, 283),
 (311, 313),
 (347, 349),
 (419, 421),
 (431, 433),
 (461, 463),
 (521, 523),
 (569, 571),
 (599, 601),
 (617, 619),
 (641, 643),
 (659, 661),
 (809, 811),
 (821, 823),
 (827, 829),
 (857, 859),
 (881, 883)]

def fib(n):
    if n in [0,1] : return 1
    else : return fib(n-1)+fib(n-2)

@cached_function
def fibb(n):
    if n in [0,1] : return 1
    else : return fibb(n-2)+fibb(n-1)

fib(30)

1346269

fibb(100)

573147844013817084101

def fib2(n):
    a, b = 1,1
    for i in [1..n-1]:
        a, b = b+a,a
    return a

for k in [0..10]:
    print(k,fib2(k))

(0, 1)
(1, 1)
(2, 2)
(3, 3)
(4, 5)
(5, 8)
(6, 13)
(7, 21)
(8, 34)
(9, 55)
(10, 89)

X = [(k,fibb(k+1)/fibb(k)) for k in [1..30]]
point(X)

N(lim(fibb(k+1)/fibb(k),k=+oo))

1.61803398874965

def syracuse(u0):
    X = [(0,u0)]
    k = 0
    uk = u0
    while uk !=1:
        if uk%2 == 0 : uk = uk/2
        else : uk = 3*uk+1
        k += 1
        X += [(k,uk)]
    return X

point(syracuse(127))