teaching:progappchim:math_nombres [Didier Villers, UMONS

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====== Mathématiques et nombres ======
<WRAP center round tip 60%>
Quelques programmes et algorithmes reliés aux mathématiques et aux nombres.
</WRAP>


  * [[wp>fr:Théorie_des_nombres|Théorie des nombres]]
  * [[wp>fr:Nombre_remarquable|Nombre_remarquable]]
  * ...

===== Calculs en précision arbitraire =====
<note tip>
Les calculs suivants renvoient des nombres avec des décimales bien particulières :

  * 1/9² = 0.0123456790123456790123456790123456790123456790123457...
  * 1/99² = 0.0001020304050607080910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485868788899091929394959697990001020304050607080910111...

Quelle est l'explication de ces particularités. Comment manipuler de tels nombres et les construire ?

L'idée est de s'intéresser au développement en [[wp>fr:Série_de_Taylor|série de Taylor]] de 1/x² autour de a=1, ou de manière équivalente à la série de Maclaurin de $1/(1-x)^2$
</note>

Taylor, pour f(x) = 1/x² :

$$f(x)=\sum _{n=0}^{\infty}{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}}=1-2(x-1)+3(x-1)^2−4(x-1)^3+5(x-1)^4+...$$


Maclaurin :

$$1/(1-x)^2 \approx 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7x^6 + 8x^7 + 9x^8 + ...$$

Ce développement va introduire des décimales particulières si x = 0.1, 0.01 ou 0.001,...

==== Programme ====

<code python>
#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
'Nombres magiques' 100/81 (et 10000/9801) dont les valeurs montrent des décimales
consécutives

Librairie de multiprécision : http://mpmath.org/doc/current/basics.html

"""

from mpmath import mp # multiprecision library
n = 41
mp.dps = n
#mp.pretty = True
print('standard float value : ', 1/0.81)

print('Multiprécision :')
sumt = mp.mpf(0)
for i in range(40, 0, -1):
    t = mp.mpf(i) * (mp.mpf(1)/mp.mpf(10))**mp.mpf(i-1)
    
    sumt += t
    print(mp.nstr(t, n, strip_zeros=False, min_fixed=-n)[:n],
          mp.nstr(sumt, n, strip_zeros=False, min_fixed=-n)[:n])

print(100/mpf(9**2))                                                     

n = 51
mp.dps = n
#mp.pretty = True
print('standard float value : ', 1/0.9801)

print('Multiprécision :')
sumt = mp.mpf(0)
for i in range(25, 0, -1):
    t = mp.mpf(i) * (mp.mpf(1)/mp.mpf(100))**mp.mpf(i-1)
    
    sumt += t
    print(mp.nstr(t, n, strip_zeros=False, min_fixed=-n)[:n],
          mp.nstr(sumt, n, strip_zeros=False, min_fixed=-n)[:n])

print(10000/mpf(99**2))                                                     
</code>



==== Résultats ====
<blockquote>
standard float value :  1.2345679012345678

Multiprécision :

0.000000000000000000000000000000000000040 0.000000000000000000000000000000000000040
0.000000000000000000000000000000000000390 0.000000000000000000000000000000000000430
0.000000000000000000000000000000000003800 0.000000000000000000000000000000000004230
0.000000000000000000000000000000000037000 0.000000000000000000000000000000000041230
0.000000000000000000000000000000000360000 0.000000000000000000000000000000000401230
0.000000000000000000000000000000003500000 0.000000000000000000000000000000003901230
0.000000000000000000000000000000034000000 0.000000000000000000000000000000037901230
0.000000000000000000000000000000330000000 0.000000000000000000000000000000367901230
0.000000000000000000000000000003200000000 0.000000000000000000000000000003567901230
0.000000000000000000000000000031000000000 0.000000000000000000000000000034567901230
0.000000000000000000000000000300000000000 0.000000000000000000000000000334567901230
0.000000000000000000000000002900000000000 0.000000000000000000000000003234567901230
0.000000000000000000000000028000000000000 0.000000000000000000000000031234567901230
0.000000000000000000000000270000000000000 0.000000000000000000000000301234567901230
0.000000000000000000000002600000000000000 0.000000000000000000000002901234567901230
0.000000000000000000000025000000000000000 0.000000000000000000000027901234567901230
0.000000000000000000000240000000000000000 0.000000000000000000000267901234567901230
0.000000000000000000002300000000000000000 0.000000000000000000002567901234567901230
0.000000000000000000022000000000000000000 0.000000000000000000024567901234567901230
0.000000000000000000210000000000000000000 0.000000000000000000234567901234567901230
0.000000000000000002000000000000000000000 0.000000000000000002234567901234567901230
0.000000000000000019000000000000000000000 0.000000000000000021234567901234567901230
0.000000000000000180000000000000000000000 0.000000000000000201234567901234567901230
0.000000000000001700000000000000000000000 0.000000000000001901234567901234567901230
0.000000000000016000000000000000000000000 0.000000000000017901234567901234567901230
0.000000000000150000000000000000000000000 0.000000000000167901234567901234567901230
0.000000000001400000000000000000000000000 0.000000000001567901234567901234567901230
0.000000000013000000000000000000000000000 0.000000000014567901234567901234567901230
0.000000000120000000000000000000000000000 0.000000000134567901234567901234567901230
0.000000001100000000000000000000000000000 0.000000001234567901234567901234567901230
0.000000010000000000000000000000000000000 0.000000011234567901234567901234567901230
0.000000090000000000000000000000000000000 0.000000101234567901234567901234567901230
0.000000800000000000000000000000000000000 0.000000901234567901234567901234567901230
0.000007000000000000000000000000000000000 0.000007901234567901234567901234567901230
0.000060000000000000000000000000000000000 0.000067901234567901234567901234567901230
0.000500000000000000000000000000000000000 0.000567901234567901234567901234567901230
0.004000000000000000000000000000000000000 0.004567901234567901234567901234567901230
0.030000000000000000000000000000000000000 0.034567901234567901234567901234567901230
0.200000000000000000000000000000000000000 0.234567901234567901234567901234567901230
1.000000000000000000000000000000000000000 1.234567901234567901234567901234567901230

1.2345679012345679012345679012345679012346

----

standard float value :  1.0203040506070808

Multiprécision :

0.0000000000000000000000000000000000000000000000250 0.0000000000000000000000000000000000000000000000250
0.0000000000000000000000000000000000000000000024000 0.0000000000000000000000000000000000000000000024250
0.0000000000000000000000000000000000000000002300000 0.0000000000000000000000000000000000000000002324250
0.0000000000000000000000000000000000000000220000000 0.0000000000000000000000000000000000000000222324250
0.0000000000000000000000000000000000000021000000000 0.0000000000000000000000000000000000000021222324250
0.0000000000000000000000000000000000002000000000000 0.0000000000000000000000000000000000002021222324250
0.0000000000000000000000000000000000190000000000000 0.0000000000000000000000000000000000192021222324250
0.0000000000000000000000000000000018000000000000000 0.0000000000000000000000000000000018192021222324250
0.0000000000000000000000000000001700000000000000000 0.0000000000000000000000000000001718192021222324250
0.0000000000000000000000000000160000000000000000000 0.0000000000000000000000000000161718192021222324250
0.0000000000000000000000000015000000000000000000000 0.0000000000000000000000000015161718192021222324250
0.0000000000000000000000001400000000000000000000000 0.0000000000000000000000001415161718192021222324250
0.0000000000000000000000130000000000000000000000000 0.0000000000000000000000131415161718192021222324250
0.0000000000000000000012000000000000000000000000000 0.0000000000000000000012131415161718192021222324250
0.0000000000000000001100000000000000000000000000000 0.0000000000000000001112131415161718192021222324250
0.0000000000000000100000000000000000000000000000000 0.0000000000000000101112131415161718192021222324250
0.0000000000000009000000000000000000000000000000000 0.0000000000000009101112131415161718192021222324250
0.0000000000000800000000000000000000000000000000000 0.0000000000000809101112131415161718192021222324250
0.0000000000070000000000000000000000000000000000000 0.0000000000070809101112131415161718192021222324250
0.0000000006000000000000000000000000000000000000000 0.0000000006070809101112131415161718192021222324250
0.0000000500000000000000000000000000000000000000000 0.0000000506070809101112131415161718192021222324250
0.0000040000000000000000000000000000000000000000000 0.0000040506070809101112131415161718192021222324250
0.0003000000000000000000000000000000000000000000000 0.0003040506070809101112131415161718192021222324250
0.0200000000000000000000000000000000000000000000000 0.0203040506070809101112131415161718192021222324250
1.0000000000000000000000000000000000000000000000000 1.0203040506070809101112131415161718192021222324250

1.02030405060708091011121314151617181920212223242526
</blockquote>


==== Références ====
  * [[https://mrob.com/]] → 1/99**2
    * [[https://mrob.com/pub/math/numbers-14.html]]
    * [[https://mrob.com/pub/seq/digits.html]]
  * [[https://fr.wikipedia.org/wiki/S%C3%A9rie_de_Taylor]]
  * [[https://socratic.org/questions/what-is-the-taylor-series-expansion-of-f-x-1-x-2-at-a-1]]

  * Multiprécision en Python : [[http://mpmath.org/]]
    * [[http://mpmath.org/doc/current/|Documentation]]