Algebarske strukture
Monoid: M=(S,∗), pri čemu su S skup i ∗ binarna operacija
Grupa: G=(S,∗)
Abelova ili komutativna grupa
Primjeri
Prsten: R=(S,+,⋅), pri čemu su + i ⋅ binarne operacije
Komutativni prsten
Prsten s jedinicom
Prsten s dijeljenjem
Polje — komutativni prsten s dijeljenjem
Primjeri
Vektorski prostor V nad poljem S: (V,S,+,⋅), pri čemu su
Primjeri
Slobodni modul: (V,S,+,⋅), ali je, za razliku od vektorskog prostora, S prsten s jedinicom
Algebra (nad poljem): vektorski prostor s množenjem
Primjeri
1 1 |
Integer Ring Integer Ring |
Integer Ring Integer Ring |
1/2 1/2 |
12
|
Rational Field Rational Field |
1.00000000000000 1.00000000000000 |
Real Field with 53 bits of precision Real Field with 53 bits of precision |
Real Field with 80 bits of precision Real Field with 80 bits of precision |
1.0000000000000000000000 1.0000000000000000000000 |
Real Field with 80 bits of precision Real Field with 80 bits of precision |
Real Field with 53 bits of precision Real Field with 53 bits of precision |
Complex Field with 53 bits of precision Complex Field with 53 bits of precision |
Oznake skupova brojeva:
(0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000) (0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000) |
Vector space of dimension 5 over Real Field with 53 bits of precision Vector space of dimension 5 over Real Field with 53 bits of precision |
(0.00000000000000000000000, 0.00000000000000000000000, 0.00000000000000000000000, 0.00000000000000000000000, 0.00000000000000000000000) (0.00000000000000000000000, 0.00000000000000000000000, 0.00000000000000000000000, 0.00000000000000000000000, 0.00000000000000000000000) |
Vector space of dimension 5 over Real Field with 80 bits of precision Vector space of dimension 5 over Real Field with 80 bits of precision |
(0.000, 0.000, 0.000, 0.000, 0.000) (0.000, 0.000, 0.000, 0.000, 0.000) |
Vector space of dimension 5 over Real Field with 14 bits of precision Vector space of dimension 5 over Real Field with 14 bits of precision |
(1/2, 2/3, 3/4) (1/2, 2/3, 3/4) |
(12,23,34)
|
Vector space of dimension 3 over Rational Field Vector space of dimension 3 over Rational Field |
Vector space of dimension 2 over Real Field with 53 bits of precision Vector space of dimension 2 over Real Field with 53 bits of precision |
(0.500000000000000, 0.500000000000000) (0.500000000000000, 0.500000000000000) |
(13, 14, 15) Ambient free module of rank 3 over the principal ideal domain Integer Ring (13, 14, 15) Ambient free module of rank 3 over the principal ideal domain Integer Ring |
Ambient free module of rank 3 over the principal ideal domain Integer Ring Ambient free module of rank 3 over the principal ideal domain Integer Ring |
[ 1.00000000000000 0.000000000000000 0.000000000000000] [0.000000000000000 1.00000000000000 0.000000000000000] [0.000000000000000 0.000000000000000 1.00000000000000] [ 1.00000000000000 0.000000000000000 0.000000000000000] [0.000000000000000 1.00000000000000 0.000000000000000] [0.000000000000000 0.000000000000000 1.00000000000000] |
Full MatrixSpace of 3 by 3 dense matrices over Real Field with 53 bits of precision Full MatrixSpace of 3 by 3 dense matrices over Real Field with 53 bits of precision |
(0.94650602594839243598548−0.708789360397446770179030.604561034244539006781870.14120833919422517495153−0.43219527995240299331888−0.704836089489321002979570.342478034045933685707880.684451996074528184530110.803207047228001206742450.42730905416264094240416−0.998906404619708510491340.90340163985851571211700)
|
Full MatrixSpace of 4 by 3 dense matrices over Real Field with 80 bits of precision Full MatrixSpace of 4 by 3 dense matrices over Real Field with 80 bits of precision |
Cijeli brojevi
10 10 |
6 6 |
13/2 13/2 |
6 6 |
1 1 |
(6, 1) (6, 1) |
13 13 |
6.50000000000000 6.50000000000000 |
Real Field with 53 bits of precision Real Field with 53 bits of precision |
6.5 6.5 |
Real Field with 10 bits of precision Real Field with 10 bits of precision |
302875106592253 302875106592253 |
1/302875106592253 1/302875106592253 |
3.30169095523012e-15 3.30169095523012e-15 |
'1000111' '1000111' |
binarni prikaz:
(1000111)2=1⋅20+1⋅21+1⋅22+0⋅23+0⋅24+0⋅25+1⋅26=1+2+4+0⋅8+0⋅16+0⋅32+64=71
'-1000111' '-1000111' |
'11111111111111111111111110111001' '11111111111111111111111110111001' |
dvojni komplement:
negativni broj −x: 232−x
+(100000000000000000000000001000111)2
+(111111111111111111111111110111001)2
=(100000000000000000000000000000000)2
x−y=x+(−y): x+(232−y)=232−(y−x)
„Realni” brojevi
I am HAL Nine Thousand computer Production Number 3. I became operational at the Hal Plant in Urbana, Illinois, on January 12, 1997.
The quick brown fox jumps over the lazy dog.
The rain in Spain is mainly in the plain.
Dave—are you still here?
Did you know that the square root of 10 is 3.162277660168379?
Log 10 to the base e is 0.43429481903252…
correction, that is log e to the base 10…
The reciprocal of 3 is 0.3333333333333333333…
2 times 2 is…2 times 2 is…
approximately 4.10101010101010…
I seem to be having difficulty…
—HAL, in 2001: A Space Odyssey
6.50000000000000 6.50000000000000 |
6.50000000000000 6.50000000000000 |
6.50000000000000 6.50000000000000 |
„Realne” se konstante (s točnošću zapisa od 53 bita) mogu pisati na pet načina:
(3.14000000000000, -1.41000000000000, 0.000000000000000, 0.510000000000000) (3.14000000000000, -1.41000000000000, 0.000000000000000, 0.510000000000000) |
(1.30000000000000e13, 1.30000000000000e-13) (1.30000000000000e13, 1.30000000000000e-13) |
(1.00000000000000, 0.500000000000000) (1.00000000000000, 0.500000000000000) |
(0.500000000000000, 0.333333333333333, 0.250000000000000, 0.0909090909090909, 0.0769230769230769) (0.500000000000000, 0.333333333333333, 0.250000000000000, 0.0909090909090909, 0.0769230769230769) |
(-1.0, -0.3333333333333333, -0.25) (-1.0, -0.3333333333333333, -0.25) |
Prva četiri načina daju brojeve koji pripadaju tipu RealField(53), a peti broj koji pripada tipu RealDoubleField:
Real Field with 53 bits of precision Real Field with 53 bits of precision Real Field with 53 bits of precision Real Field with 53 bits of precision Real Double Field Real Field with 53 bits of precision Real Field with 53 bits of precision Real Field with 53 bits of precision Real Field with 53 bits of precision Real Double Field |
Pomoću funkcija RealField()() i N() mogu se zadati „realni” brojevi drugih točnosti:
-1.20 -1.20 |
Real Field with 170 bits of precision Real Field with 170 bits of precision |
-1.2000000000000000000000000000000000000000000000000 -1.2000000000000000000000000000000000000000000000000 |
-1.2000000000000000000000000000000000000000000000000 -1.2000000000000000000000000000000000000000000000000 |
-1.2000000000000000000000000000000000000000000000000 -1.2000000000000000000000000000000000000000000000000 |
Realni brojevi (u matematici):
Racionalni brojevi:
13/7 13/7 |
1.85714285714286 1.85714285714286 |
18.5714285714286 18.5714285714286 |
18571.4285714286 18571.4285714286 |
0.0185714285714286 0.0185714285714286 |
0.0000185714285714286 0.0000185714285714286 |
1.85714285714286e-35 1.85714285714286e-35 |
Značajnim znamenkama broja x≠0 u zapisu sa zarezom nazivaju se prva znamenka slijeva različita od nule i sve znamenke desno od nje do kraja zapisa; za x=0 sve su znamenke u zapisu značajne.
0.000000000000000 0.000000000000000 |
Prikaz s pomičnim (plivajućim) zarezom (u engleskom: s plivajućom točkom — floating point)
dekadski sustav: x=±m⋅10e
m — normalizirana mantisa, 1≤m<10 ili m=0
e — eksponent
123456780000000000000,0=1,2345678⋅1020
123,45678=1,2345678⋅102
0,00012345678=1,2345678⋅10−4
0,000000000000000000012345678=1,2345678⋅10−20
normalizacija omogućava zapis najvećeg broja značajnih znamenaka ako je broj znamenaka u prikazu broja ograničen
binarni sustav (najjednostavnije): x=s⋅m⋅2e
s∈{−1,1}
1≤m<2 ili m=0
ako je m∈[1,2), prva je znamenka u binarnom prikazu uvijek 1 → skriveni bit
binarni sustav (u GNU MPFR-u i SageMath-u (RealField())): x=s⋅m⋅2e−p
p — točnost (broj bitova mantise)
2p−1≤m<2p ili m=0
−2B+1≤e≤2B−1
(B=30 na 32-bitnim računalima, B=62 na 64-bitnim računalima)
[GNU MPFR — GNU Multiple Precision Floating-Point Reliable Library; GNU (GNU's Not Unix!) — an extensive collection of free software]
-1.00000000000000 -1.00000000000000 |
(-1, 4503599627370496, -52, 1/4503599627370496) (-1, 4503599627370496, -52, 1/4503599627370496) |
-1 -1 |
0.333333333333333 0.333333333333333 |
(1, 6004799503160661, -54, 1/18014398509481984) (1, 6004799503160661, -54, 1/18014398509481984) |
600479950316066118014398509481984
|
600479950316066118014398509481984
|
0.100000000000000 0.100000000000000 |
(1, 7205759403792794, -56, 1/72057594037927936) (1, 7205759403792794, -56, 1/72057594037927936) |
360287970189639736028797018963968
|
360287970189639736028797018963968
|
-1.0000000000000000000000 -1.0000000000000000000000 |
Real Field with 80 bits of precision Real Field with 80 bits of precision |
(-1, 604462909807314587353088, -79, 1/604462909807314587353088) (-1, 604462909807314587353088, -79, 1/604462909807314587353088) |
-1 -1 |
-0.33333333333333333333333 -0.33333333333333333333333 |
(-1, 805950546409752783137451, -81, 1/2417851639229258349412352) (-1, 805950546409752783137451, -81, 1/2417851639229258349412352) |
−8059505464097527831374512417851639229258349412352
|
−8059505464097527831374512417851639229258349412352
|
(-0.33333333333333333333333, Real Field with 80 bits of precision) (-0.33333333333333333333333, Real Field with 80 bits of precision) |
(-0.333333333333333, Real Field with 53 bits of precision) (-0.333333333333333, Real Field with 53 bits of precision) |
točnost (precision) prikaza s pomičnim zarezom ovisi o broju bitova mantise (binarnih značajnih znamenaka)
veza između broja značajnih znamenaka u binarnom i u dekadskom zapisu:
9007199254740991 9007199254740992 9007199254740991 9007199254740992 |
9999999999999999 10000000000000000 9999999999999999 10000000000000000 |
9007199254740992 10000000000000000 9007199254740992 10000000000000000 |
15.9545897701910 15.9545897701910 |
53.1508495181978 53.1508495181978 |
|
16 16 |
53 53 |
24 24 |
80 80 |
Real Field with 53 bits of precision Real Field with 53 bits of precision |
1.3333333333333333 1.33333333333333 1.3333333333333333 1.33333333333333 |
15 15 |
7 7 |
2.71828182845905 2.718282 2.71828182845905 2.718282 |
2.71828 2.71828 |
Traceback (click to the left of this block for traceback) ... TypeError: cannot approximate to a precision of 27 bits, use at most 24 bits Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_6.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("TiAoZWUsIGRpZ2l0cyA9IDcp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpqkbP9v/___code___.py", line 3, in <module> exec compile(u'N (ee, digits = _sage_const_7 ) File "", line 1, in <module> File "/opt/SageMath/local/lib/python2.7/site-packages/sage/misc/functional.py", line 1419, in numerical_approx return n(prec, algorithm=algorithm) File "sage/structure/element.pyx", line 850, in sage.structure.element.Element.numerical_approx (build/cythonized/sage/structure/element.c:7829) File "sage/arith/numerical_approx.pyx", line 65, in sage.arith.numerical_approx.numerical_approx_generic (build/cythonized/sage/arith/numerical_approx.c:2666) TypeError: cannot approximate to a precision of 27 bits, use at most 24 bits |
2.718282 2.718282 |
Real Field with 47 bits of precision Real Field with 47 bits of precision |
strojna točnost (machine epsilon) — najmanji pozitivni broj ϵm koji pribrojen broju 1 daje broj različit od 1: 1+ϵm≠1
najmanji broj veći od 1: ˉx=(1,000…001)2=1+2−(p−1)=1+ϵm⇒ϵm=2−(p−1)
Real Field with 53 bits of precision Real Field with 53 bits of precision |
1/4503599627370496 1/4503599627370496 |
False False |
True True |
2.22044604925031e-16 2.22044604925031e-16 |
1.11022302462516e-16 1.11022302462516e-16 |
|
0 : 1.00000000000000 1 : 0.500000000000000 2 : 0.250000000000000 3 : 0.125000000000000 4 : 0.0625000000000000 5 : 0.0312500000000000 6 : 0.0156250000000000 7 : 0.00781250000000000 8 : 0.00390625000000000 9 : 0.00195312500000000 10 : 0.000976562500000000 11 : 0.000488281250000000 12 : 0.000244140625000000 13 : 0.000122070312500000 14 : 0.0000610351562500000 15 : 0.0000305175781250000 16 : 0.0000152587890625000 17 : 7.62939453125000e-6 18 : 3.81469726562500e-6 19 : 1.90734863281250e-6 20 : 9.53674316406250e-7 21 : 4.76837158203125e-7 22 : 2.38418579101562e-7 23 : 1.19209289550781e-7 24 : 5.96046447753906e-8 25 : 2.98023223876953e-8 26 : 1.49011611938477e-8 27 : 7.45058059692383e-9 28 : 3.72529029846191e-9 29 : 1.86264514923096e-9 30 : 9.31322574615479e-10 31 : 4.65661287307739e-10 32 : 2.32830643653870e-10 33 : 1.16415321826935e-10 34 : 5.82076609134674e-11 35 : 2.91038304567337e-11 36 : 1.45519152283669e-11 37 : 7.27595761418343e-12 38 : 3.63797880709171e-12 39 : 1.81898940354586e-12 40 : 9.09494701772928e-13 41 : 4.54747350886464e-13 42 : 2.27373675443232e-13 43 : 1.13686837721616e-13 44 : 5.68434188608080e-14 45 : 2.84217094304040e-14 46 : 1.42108547152020e-14 47 : 7.10542735760100e-15 48 : 3.55271367880050e-15 49 : 1.77635683940025e-15 50 : 8.88178419700125e-16 51 : 4.44089209850063e-16 52 : 2.22044604925031e-16 2.22044604925031e-16 0 : 1.00000000000000 1 : 0.500000000000000 2 : 0.250000000000000 3 : 0.125000000000000 4 : 0.0625000000000000 5 : 0.0312500000000000 6 : 0.0156250000000000 7 : 0.00781250000000000 8 : 0.00390625000000000 9 : 0.00195312500000000 10 : 0.000976562500000000 11 : 0.000488281250000000 12 : 0.000244140625000000 13 : 0.000122070312500000 14 : 0.0000610351562500000 15 : 0.0000305175781250000 16 : 0.0000152587890625000 17 : 7.62939453125000e-6 18 : 3.81469726562500e-6 19 : 1.90734863281250e-6 20 : 9.53674316406250e-7 21 : 4.76837158203125e-7 22 : 2.38418579101562e-7 23 : 1.19209289550781e-7 24 : 5.96046447753906e-8 25 : 2.98023223876953e-8 26 : 1.49011611938477e-8 27 : 7.45058059692383e-9 28 : 3.72529029846191e-9 29 : 1.86264514923096e-9 30 : 9.31322574615479e-10 31 : 4.65661287307739e-10 32 : 2.32830643653870e-10 33 : 1.16415321826935e-10 34 : 5.82076609134674e-11 35 : 2.91038304567337e-11 36 : 1.45519152283669e-11 37 : 7.27595761418343e-12 38 : 3.63797880709171e-12 39 : 1.81898940354586e-12 40 : 9.09494701772928e-13 41 : 4.54747350886464e-13 42 : 2.27373675443232e-13 43 : 1.13686837721616e-13 44 : 5.68434188608080e-14 45 : 2.84217094304040e-14 46 : 1.42108547152020e-14 47 : 7.10542735760100e-15 48 : 3.55271367880050e-15 49 : 1.77635683940025e-15 50 : 8.88178419700125e-16 51 : 4.44089209850063e-16 52 : 2.22044604925031e-16 2.22044604925031e-16 |
0 : 1.00 1 : 0.500 2 : 0.250 3 : 0.125 4 : 0.0625 5 : 0.0312 6 : 0.0156 7 : 0.00781 8 : 0.00391 9 : 0.00195 10 : 0.000977 11 : 0.000488 12 : 0.000244 13 : 0.000122 0.000122 0 : 1.00 1 : 0.500 2 : 0.250 3 : 0.125 4 : 0.0625 5 : 0.0312 6 : 0.0156 7 : 0.00781 8 : 0.00391 9 : 0.00195 10 : 0.000977 11 : 0.000488 12 : 0.000244 13 : 0.000122 0.000122 |
1.6543612251060553497428e-24 1.6543612251060553497428e-24 |
1/604462909807314587353088 1/604462909807314587353088 |
1.6543612251060553497428e-24 1.6543612251060553497428e-24 |
jedinica na posljednjem mjestu (unit in the last place, ULP) — za (prikazivi) broj x>0, razmak između x i prvoga prikazivog većeg broja
2.22044604925031e-16 2.22044604925031e-16 |
1.77635683940025e-15 1.77635683940025e-15 |
1.42108547152020e-14 1.42108547152020e-14 |
False False |
True True |
False False |
True True |
True True |
True True |
False False |
True True |
False False |
|
-10 : 0.000976562500000000 : 2.16840434497101e-19 -9 : 0.00195312500000000 : 4.33680868994202e-19 -8 : 0.00390625000000000 : 8.67361737988404e-19 -7 : 0.00781250000000000 : 1.73472347597681e-18 -6 : 0.0156250000000000 : 3.46944695195361e-18 -5 : 0.0312500000000000 : 6.93889390390723e-18 -4 : 0.0625000000000000 : 1.38777878078145e-17 -3 : 0.125000000000000 : 2.77555756156289e-17 -2 : 0.250000000000000 : 5.55111512312578e-17 -1 : 0.500000000000000 : 1.11022302462516e-16 0 : 1.00000000000000 : 2.22044604925031e-16 1 : 2.00000000000000 : 4.44089209850063e-16 2 : 4.00000000000000 : 8.88178419700125e-16 3 : 8.00000000000000 : 1.77635683940025e-15 4 : 16.0000000000000 : 3.55271367880050e-15 5 : 32.0000000000000 : 7.10542735760100e-15 6 : 64.0000000000000 : 1.42108547152020e-14 7 : 128.000000000000 : 2.84217094304040e-14 8 : 256.000000000000 : 5.68434188608080e-14 9 : 512.000000000000 : 1.13686837721616e-13 10 : 1024.00000000000 : 2.27373675443232e-13 -10 : 0.000976562500000000 : 2.16840434497101e-19 -9 : 0.00195312500000000 : 4.33680868994202e-19 -8 : 0.00390625000000000 : 8.67361737988404e-19 -7 : 0.00781250000000000 : 1.73472347597681e-18 -6 : 0.0156250000000000 : 3.46944695195361e-18 -5 : 0.0312500000000000 : 6.93889390390723e-18 -4 : 0.0625000000000000 : 1.38777878078145e-17 -3 : 0.125000000000000 : 2.77555756156289e-17 -2 : 0.250000000000000 : 5.55111512312578e-17 -1 : 0.500000000000000 : 1.11022302462516e-16 0 : 1.00000000000000 : 2.22044604925031e-16 1 : 2.00000000000000 : 4.44089209850063e-16 2 : 4.00000000000000 : 8.88178419700125e-16 3 : 8.00000000000000 : 1.77635683940025e-15 4 : 16.0000000000000 : 3.55271367880050e-15 5 : 32.0000000000000 : 7.10542735760100e-15 6 : 64.0000000000000 : 1.42108547152020e-14 7 : 128.000000000000 : 2.84217094304040e-14 8 : 256.000000000000 : 5.68434188608080e-14 9 : 512.000000000000 : 1.13686837721616e-13 10 : 1024.00000000000 : 2.27373675443232e-13 |
-10 : 0.00097656250000000000000000 : 1.6155871338926321774832e-27 -9 : 0.0019531250000000000000000 : 3.2311742677852643549664e-27 -8 : 0.0039062500000000000000000 : 6.4623485355705287099329e-27 -7 : 0.0078125000000000000000000 : 1.2924697071141057419866e-26 -6 : 0.015625000000000000000000 : 2.5849394142282114839732e-26 -5 : 0.031250000000000000000000 : 5.1698788284564229679463e-26 -4 : 0.062500000000000000000000 : 1.0339757656912845935893e-25 -3 : 0.12500000000000000000000 : 2.0679515313825691871785e-25 -2 : 0.25000000000000000000000 : 4.1359030627651383743570e-25 -1 : 0.50000000000000000000000 : 8.2718061255302767487141e-25 0 : 1.0000000000000000000000 : 1.6543612251060553497428e-24 1 : 2.0000000000000000000000 : 3.3087224502121106994856e-24 2 : 4.0000000000000000000000 : 6.6174449004242213989713e-24 3 : 8.0000000000000000000000 : 1.3234889800848442797943e-23 4 : 16.000000000000000000000 : 2.6469779601696885595885e-23 5 : 32.000000000000000000000 : 5.2939559203393771191770e-23 6 : 64.000000000000000000000 : 1.0587911840678754238354e-22 7 : 128.00000000000000000000 : 2.1175823681357508476708e-22 8 : 256.00000000000000000000 : 4.2351647362715016953416e-22 9 : 512.00000000000000000000 : 8.4703294725430033906832e-22 10 : 1024.0000000000000000000 : 1.6940658945086006781366e-21 -10 : 0.00097656250000000000000000 : 1.6155871338926321774832e-27 -9 : 0.0019531250000000000000000 : 3.2311742677852643549664e-27 -8 : 0.0039062500000000000000000 : 6.4623485355705287099329e-27 -7 : 0.0078125000000000000000000 : 1.2924697071141057419866e-26 -6 : 0.015625000000000000000000 : 2.5849394142282114839732e-26 -5 : 0.031250000000000000000000 : 5.1698788284564229679463e-26 -4 : 0.062500000000000000000000 : 1.0339757656912845935893e-25 -3 : 0.12500000000000000000000 : 2.0679515313825691871785e-25 -2 : 0.25000000000000000000000 : 4.1359030627651383743570e-25 -1 : 0.50000000000000000000000 : 8.2718061255302767487141e-25 0 : 1.0000000000000000000000 : 1.6543612251060553497428e-24 1 : 2.0000000000000000000000 : 3.3087224502121106994856e-24 2 : 4.0000000000000000000000 : 6.6174449004242213989713e-24 3 : 8.0000000000000000000000 : 1.3234889800848442797943e-23 4 : 16.000000000000000000000 : 2.6469779601696885595885e-23 5 : 32.000000000000000000000 : 5.2939559203393771191770e-23 6 : 64.000000000000000000000 : 1.0587911840678754238354e-22 7 : 128.00000000000000000000 : 2.1175823681357508476708e-22 8 : 256.00000000000000000000 : 4.2351647362715016953416e-22 9 : 512.00000000000000000000 : 8.4703294725430033906832e-22 10 : 1024.0000000000000000000 : 1.6940658945086006781366e-21 |
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50 : 1.12589990684262e15 : 0.250000000000000 51 : 2.25179981368525e15 : 0.500000000000000 52 : 4.50359962737050e15 : 1.00000000000000 53 : 9.00719925474099e15 : 2.00000000000000 54 : 1.80143985094820e16 : 4.00000000000000 55 : 3.60287970189640e16 : 8.00000000000000 56 : 7.20575940379279e16 : 16.0000000000000 57 : 1.44115188075856e17 : 32.0000000000000 58 : 2.88230376151712e17 : 64.0000000000000 59 : 5.76460752303423e17 : 128.000000000000 60 : 1.15292150460685e18 : 256.000000000000 61 : 2.30584300921369e18 : 512.000000000000 62 : 4.61168601842739e18 : 1024.00000000000 63 : 9.22337203685478e18 : 2048.00000000000 64 : 1.84467440737096e19 : 4096.00000000000 65 : 3.68934881474191e19 : 8192.00000000000 66 : 7.37869762948382e19 : 16384.0000000000 67 : 1.47573952589676e20 : 32768.0000000000 68 : 2.95147905179353e20 : 65536.0000000000 69 : 5.90295810358706e20 : 131072.000000000 70 : 1.18059162071741e21 : 262144.000000000 50 : 1.12589990684262e15 : 0.250000000000000 51 : 2.25179981368525e15 : 0.500000000000000 52 : 4.50359962737050e15 : 1.00000000000000 53 : 9.00719925474099e15 : 2.00000000000000 54 : 1.80143985094820e16 : 4.00000000000000 55 : 3.60287970189640e16 : 8.00000000000000 56 : 7.20575940379279e16 : 16.0000000000000 57 : 1.44115188075856e17 : 32.0000000000000 58 : 2.88230376151712e17 : 64.0000000000000 59 : 5.76460752303423e17 : 128.000000000000 60 : 1.15292150460685e18 : 256.000000000000 61 : 2.30584300921369e18 : 512.000000000000 62 : 4.61168601842739e18 : 1024.00000000000 63 : 9.22337203685478e18 : 2048.00000000000 64 : 1.84467440737096e19 : 4096.00000000000 65 : 3.68934881474191e19 : 8192.00000000000 66 : 7.37869762948382e19 : 16384.0000000000 67 : 1.47573952589676e20 : 32768.0000000000 68 : 2.95147905179353e20 : 65536.0000000000 69 : 5.90295810358706e20 : 131072.000000000 70 : 1.18059162071741e21 : 262144.000000000 |
0 : 1.00 : 0.000122 1 : 2.00 : 0.000244 2 : 4.00 : 0.000488 3 : 8.00 : 0.000977 4 : 16.0 : 0.00195 5 : 32.0 : 0.00391 6 : 64.0 : 0.00781 7 : 128. : 0.0156 8 : 256. : 0.0312 9 : 512. : 0.0625 10 : 1020. : 0.125 0 : 1.00 : 0.000122 1 : 2.00 : 0.000244 2 : 4.00 : 0.000488 3 : 8.00 : 0.000977 4 : 16.0 : 0.00195 5 : 32.0 : 0.00391 6 : 64.0 : 0.00781 7 : 128. : 0.0156 8 : 256. : 0.0312 9 : 512. : 0.0625 10 : 1020. : 0.125 |
|
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zbrajanje „realnih” brojeva nije asocijativno:
|
False False |
False False |
1.00000000000000 1.0000000000000002 1.00000000000000 1.0000000000000002 |
True True |
1.00000000000000 1.0000000000000000 1.00000000000000 1.0000000000000000 |
True 1.00000000000000 True 1.00000000000000 |
False 1.00000000000000 False 1.00000000000000 |
zbrajanje brojeva tipa RealField(14):
3+2=(11000000000000)2×2−12+(10000000000000)2×2−12:
+(111000000000000)2×2−12
+(110000000000000)2×2−12
=(101000000000000)2×2−12
=(110100000000000)2×2−11 … normalizacija
3+34=(11000000000000)2×2−12+(11000000000000)2×2−14:
+(11000000000000)2×2−12
+(00110000000000)2×2−12 … pomicanje mantise −12−(−14)=2 mjesta udesno
=(11110000000000)2×2−12
1+12ulp(1)=(10000000000000)2×2−13+(10000000000000)2×2−27:
+(10000000000000|1)2×2−13
+(00000000000000|1)2×2−13 … pomicanje mantise −13−(−27)=14 mjesta udesno
=(10000000000000|1)2×2−13
≈(10000000000000|1)2×2−13
12ulp(1)+12ulp(1)=(10000000000000)2×2−27+(10000000000000)2×2−27:
+(110000000000000)2×2−27
+(110000000000000)2×2−27
=(100000000000000)2×2−27
=(110000000000000)2×2−26
1+ulp(1)=(10000000000000)2×2−13+(10000000000000)2×2−26:
+(11000000000000)2×2−13
+(00000000000001)2×2−13 … pomicanje mantise −13−(−26)=13 mjesta udesno
=(10000000000001)2×2−13
(0.333, 0.667, -0.333, -0.667) (0.333, 0.667, -0.333, -0.667) |
(0.333, 0.666, -0.334, -0.667) (0.333, 0.666, -0.334, -0.667) |
(0.334, 0.667, -0.333, -0.666) (0.334, 0.667, -0.333, -0.666) |
(0.333, 0.666, -0.333, -0.666) (0.333, 0.666, -0.333, -0.666) |
(0.334, 0.667, -0.334, -0.667) (0.334, 0.667, -0.334, -0.667) |
zbrajanja i zaokruživanja:
(True, True) (True, True) |
(True, False) (True, False) |
(False, True) (False, True) |
(True, True) (True, True) |
(False, False) (False, False) |
(apsolutna) greška zaokruživanja:
„intervalna” aritmetika:
(3.14159265358979, 2.71828182845904, 0.650623412682596) (3.14159265358979, 2.71828182845904, 0.650623412682596) |
(3.14159265358980, 2.71828182845905, 0.650623412682597) (3.14159265358980, 2.71828182845905, 0.650623412682597) |
... realni broj √π−r leži u intervalu između „realnih” brojeva sd i su
ispitivanje jednakosti „realnih” brojeva
1.00000000000000 1.00000000000000 |
False False |
^C Traceback (click to the left of this block for traceback) ... __SAGE__ ^C Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_3.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cyA9IDAuCndoaWxlIChzICE9IDEuKSA6CiAgICBzICs9IDAuMQpz"),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpQnoHQZ/___code___.py", line 5, in <module> s += _sage_const_0p1 File "src/cysignals/signals.pyx", line 320, in cysignals.signals.python_check_interrupt KeyboardInterrupt __SAGE__ |
2.64211950131177e6 2.64211950131177e6 |
'0.10000000000000001' '0.20000000000000001' '0.30000000000000004' '0.40000000000000002' '0.50000000000000000' '0.59999999999999998' '0.69999999999999996' '0.79999999999999993' '0.89999999999999991' '0.99999999999999989' '1.0999999999999999' '1.2000000000000000' '0.10000000000000001' '0.20000000000000001' '0.30000000000000004' '0.40000000000000002' '0.50000000000000000' '0.59999999999999998' '0.69999999999999996' '0.79999999999999993' '0.89999999999999991' '0.99999999999999989' '1.0999999999999999' '1.2000000000000000' |
0.100000000000000 0.100000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000 0.100000000000000 0.10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 |
0.00011001100110011001100110011001100110011001100110011010 0.000110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011001100110011001100110011001100110\ 01100110011001100110011001100110011001100110011010 0.00011001100110011001100110011001100110011001100110011010 0.00011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011001100110011010 |
0.10000000000000001 0.100000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000\ 01 0.10000000000000001 0.1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 |
0.100000000000000005551115123125782702118158340454101562500000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000\ 00000000000000000000000000000000000000000000000000000000000000000000000000000000 0.10000000000000000555111512312578270211815834045410156250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 |
Traceback (click to the left of this block for traceback) ... TypeError: cannot approximate to a precision of 1325 bits, use at most 53 bits Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_87.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("TiAoYTUzLCAxMzI1KQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpqZQ98Y/___code___.py", line 3, in <module> exec compile(u'N (a53, _sage_const_1325 ) File "", line 1, in <module> File "/opt/SageMath/local/lib/python2.7/site-packages/sage/misc/functional.py", line 1419, in numerical_approx return n(prec, algorithm=algorithm) File "sage/structure/element.pyx", line 850, in sage.structure.element.Element.numerical_approx (build/cythonized/sage/structure/element.c:7829) File "sage/arith/numerical_approx.pyx", line 65, in sage.arith.numerical_approx.numerical_approx_generic (build/cythonized/sage/arith/numerical_approx.c:2666) TypeError: cannot approximate to a precision of 1325 bits, use at most 53 bits |
(4.00000000000000−1.00000000000000−1.000000000000000.0000000000000000.000000000000000−1.000000000000004.00000000000000−1.00000000000000−1.000000000000000.000000000000000−1.00000000000000−1.000000000000004.00000000000000−1.00000000000000−1.000000000000000.000000000000000−1.00000000000000−1.000000000000004.00000000000000−1.000000000000000.0000000000000000.000000000000000−1.00000000000000−1.000000000000004.00000000000000)
|
(0.3238866396761130.1417004048583000.1538461538461540.08906882591093120.06072874493927120.1417004048583000.3744939271255060.1923076923076920.1639676113360320.08906882591093120.1538461538461540.1923076923076920.4230769230769230.1923076923076920.1538461538461540.08906882591093120.1639676113360320.1923076923076920.3744939271255060.1417004048583000.06072874493927120.08906882591093120.1538461538461540.1417004048583000.323886639676113)
|
za inverznu bi matricu trebalo vrijediti AA−1=A−1A=I, ali...
False False |
False False |
False False |
... naime, kako „realna” aritmetika na računalu nije matematička realna aritmetika beskonačne točnosti, komponente matrice AA−1 koje bi trebale biti nule, nisu nule:
(1.00000000000000−2.77555756156289×10−170.000000000000000−2.77555756156289×10−17−2.77555756156289×10−17−5.55111512312578×10−171.00000000000000−1.11022302462516×10−16−5.55111512312578×10−170.0000000000000006.93889390390723×10−18−8.32667268468867×10−171.00000000000000−2.77555756156289×10−170.000000000000000−2.08166817117217×10−178.32667268468867×10−172.77555756156289×10−171.000000000000000.000000000000000−2.77555756156289×10−170.0000000000000000.000000000000000−1.11022302462516×10−161.00000000000000)
|
... štoviše, nisu ni sve „jedinice” jedinice, iako su ispisane kao jedinice:
1 4503599627370497/4503599627370496 1 9007199254740991/9007199254740992 9007199254740991/9007199254740992 4503599627370495/4503599627370496 1 4503599627370497/4503599627370496 1 9007199254740991/9007199254740992 9007199254740991/9007199254740992 4503599627370495/4503599627370496 |
1.0000000000000002 1.0000000000000000 0.99999999999999989 0.99999999999999989 0.99999999999999978 1.0000000000000002 1.0000000000000000 0.99999999999999989 0.99999999999999989 0.99999999999999978 |
neki (ali nikako svi) ispravni načini ispitivanja „jednakosti”:
1.49011611938477e-8 1.49011611938477e-8 |
[1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] |
[1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] |
[1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] |
... ili, varijacija na temu:
[1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] [1 1 1 1 1] |
{(0, 0): 2.22044604925031e-16, (0, 1): -2.77555756156289e-17, (0, 2): 0.000000000000000, (0, 3): -2.77555756156289e-17, (0, 4): -2.77555756156289e-17, (1, 0): -5.55111512312578e-17, (1, 1): 0.000000000000000, (1, 2): -1.11022302462516e-16, (1, 3): -5.55111512312578e-17, (1, 4): 0.000000000000000, (2, 0): 6.93889390390723e-18, (2, 1): -8.32667268468867e-17, (2, 2): -1.11022302462516e-16, (2, 3): -2.77555756156289e-17, (2, 4): 0.000000000000000, (3, 0): -2.08166817117217e-17, (3, 1): 8.32667268468867e-17, (3, 2): 2.77555756156289e-17, (3, 3): -1.11022302462516e-16, (3, 4): 0.000000000000000, (4, 0): -2.77555756156289e-17, (4, 1): 0.000000000000000, (4, 2): 0.000000000000000, (4, 3): -1.11022302462516e-16, (4, 4): -2.22044604925031e-16} {(0, 0): 2.22044604925031e-16, (0, 1): -2.77555756156289e-17, (0, 2): 0.000000000000000, (0, 3): -2.77555756156289e-17, (0, 4): -2.77555756156289e-17, (1, 0): -5.55111512312578e-17, (1, 1): 0.000000000000000, (1, 2): -1.11022302462516e-16, (1, 3): -5.55111512312578e-17, (1, 4): 0.000000000000000, (2, 0): 6.93889390390723e-18, (2, 1): -8.32667268468867e-17, (2, 2): -1.11022302462516e-16, (2, 3): -2.77555756156289e-17, (2, 4): 0.000000000000000, (3, 0): -2.08166817117217e-17, (3, 1): 8.32667268468867e-17, (3, 2): 2.77555756156289e-17, (3, 3): -1.11022302462516e-16, (3, 4): 0.000000000000000, (4, 0): -2.77555756156289e-17, (4, 1): 0.000000000000000, (4, 2): 0.000000000000000, (4, 3): -1.11022302462516e-16, (4, 4): -2.22044604925031e-16} |
25 25 |
True True |
{} {} |
0 0 |
True True |
... ili:
True True |
True True |
... ili, varijacije:
True True |
True True |
True True |
infinity & NaN (Not a Number)
+infinity +infinity |
Traceback (click to the left of this block for traceback) ... ZeroDivisionError: rational division by zero Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_112.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("MS8w"),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmp3fMu04/___code___.py", line 3, in <module> exec compile(u'_sage_const_1 /_sage_const_0 File "", line 1, in <module> File "sage/rings/integer.pyx", line 2047, in sage.rings.integer.Integer.__div__ (build/cythonized/sage/rings/integer.c:14042) ZeroDivisionError: rational division by zero |
-infinity -infinity |
-infinity -infinity |
-Infinity -Infinity |
+Infinity +Infinity |
−∞
|
The Infinity Ring Real Field with 53 bits of precision The Infinity Ring Real Field with 53 bits of precision |
Real Field with 53 bits of precision Symbolic Ring Real Field with 53 bits of precision Symbolic Ring |
True True |
True True |
+Infinity +Infinity |
+infinity +infinity |
+Infinity +Infinity |
+Infinity +Infinity |
+Infinity +Infinity |
+Infinity +Infinity |
+infinity +infinity |
Traceback (click to the left of this block for traceback) ... TypeError: unsupported operand parent(s) for ^: 'The Infinity Ring' and 'The Infinity Ring' Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_129.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Ml5vbw=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpPRg_2E/___code___.py", line 3, in <module> exec compile(u'_sage_const_2 **oo File "", line 1, in <module> File "sage/rings/integer.pyx", line 2213, in sage.rings.integer.Integer.__pow__ (build/cythonized/sage/rings/integer.c:14884) File "sage/structure/coerce.pyx", line 1163, in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:9859) File "sage/structure/element.pyx", line 2038, in sage.structure.element.Element.__pow__ (build/cythonized/sage/structure/element.c:14037) File "sage/structure/element.pyx", line 2079, in sage.structure.element.Element._pow_ (build/cythonized/sage/structure/element.c:14381) TypeError: unsupported operand parent(s) for ^: 'The Infinity Ring' and 'The Infinity Ring' |
+infinity +infinity |
NaN NaN |
Traceback (click to the left of this block for traceback) ... sage.rings.infinity.SignError: cannot add infinity to minus infinity Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_132.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("b28gLSBvbw=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpHCRCdg/___code___.py", line 2, in <module> exec compile(u'oo - oo File "", line 1, in <module> File "sage/structure/element.pyx", line 1359, in sage.structure.element.Element.__sub__ (build/cythonized/sage/structure/element.c:11339) File "sage/structure/element.pyx", line 2371, in sage.structure.element.ModuleElement._sub_ (build/cythonized/sage/structure/element.c:15388) File "/opt/SageMath/local/lib/python2.7/site-packages/sage/rings/infinity.py", line 429, in _sub_ raise SignError("cannot add infinity to minus infinity") sage.rings.infinity.SignError: cannot add infinity to minus infinity |
NaN NaN |
NaN NaN |
Traceback (click to the left of this block for traceback) ... sage.rings.infinity.SignError: cannot multiply infinity by zero Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_135.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("MCAqIG9v"),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpJKBbmc/___code___.py", line 3, in <module> exec compile(u'_sage_const_0 * oo File "", line 1, in <module> File "sage/rings/integer.pyx", line 1989, in sage.rings.integer.Integer.__mul__ (build/cythonized/sage/rings/integer.c:13725) File "sage/structure/coerce.pyx", line 1163, in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:9859) File "sage/structure/element.pyx", line 1517, in sage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:11977) File "sage/structure/element.pyx", line 2548, in sage.structure.element.RingElement._mul_ (build/cythonized/sage/structure/element.c:17481) File "/opt/SageMath/local/lib/python2.7/site-packages/sage/rings/infinity.py", line 1369, in _mul_ raise SignError("cannot multiply infinity by zero") sage.rings.infinity.SignError: cannot multiply infinity by zero |
NaN NaN |
Traceback (click to the left of this block for traceback) ... sage.rings.infinity.SignError: cannot multiply infinity by zero Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_137.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("b28gLyBvbw=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmp8XKd5C/___code___.py", line 2, in <module> exec compile(u'oo / oo File "", line 1, in <module> File "sage/structure/element.pyx", line 1647, in sage.structure.element.Element.__div__ (build/cythonized/sage/structure/element.c:12537) File "sage/structure/element.pyx", line 2649, in sage.structure.element.RingElement._div_ (build/cythonized/sage/structure/element.c:18126) File "/opt/SageMath/local/lib/python2.7/site-packages/sage/rings/infinity.py", line 481, in _div_ return self * ~other File "sage/structure/element.pyx", line 1519, in sage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:12012) File "sage/structure/coerce.pyx", line 1163, in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:9859) File "sage/structure/element.pyx", line 1517, in sage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:11977) File "sage/structure/element.pyx", line 2548, in sage.structure.element.RingElement._mul_ (build/cythonized/sage/structure/element.c:17481) File "/opt/SageMath/local/lib/python2.7/site-packages/sage/rings/infinity.py", line 455, in _mul_ raise SignError("cannot multiply infinity by zero") sage.rings.infinity.SignError: cannot multiply infinity by zero |
NaN NaN |
True True |
False False |
False False |
False False |
True True |
True True |
False False |
True True |
False False |
False False |
False False |
False False |
|
|
0 -8.99999999999999 -1.89188589083065 1 74.2610504415619 1.99848639858029 2 -49414.0501674566 -1.99999999651880 3 1.41943459326446e13 2.00000000000000 4 -infinity NaN 5 NaN NaN + NaN*I 6 NaN + NaN*I NaN + NaN*I 7 NaN + NaN*I NaN + NaN*I 8 NaN + NaN*I NaN + NaN*I 9 NaN + NaN*I NaN + NaN*I NaN + NaN*I 0 -8.99999999999999 -1.89188589083065 1 74.2610504415619 1.99848639858029 2 -49414.0501674566 -1.99999999651880 3 1.41943459326446e13 2.00000000000000 4 -infinity NaN 5 NaN NaN + NaN*I 6 NaN + NaN*I NaN + NaN*I 7 NaN + NaN*I NaN + NaN*I 8 NaN + NaN*I NaN + NaN*I 9 NaN + NaN*I NaN + NaN*I NaN + NaN*I |
|
0 -8.99999999999999 -1.89188589083065 1 74.2610504415619 1.99848639858029 2 -49414.0501674566 -1.99999999651880 3 1.41943459326446e13 2.00000000000000 4 -infinity NaN -infinity 0 -8.99999999999999 -1.89188589083065 1 74.2610504415619 1.99848639858029 2 -49414.0501674566 -1.99999999651880 3 1.41943459326446e13 2.00000000000000 4 -infinity NaN -infinity |
... i još dvije sitnice (ili pitanje dosljednosti):
1.00000000000000*I 1.00000000000000*I |
Traceback (click to the left of this block for traceback) ... ValueError: negative number -1.00000000000000 does not have a square root in the real field Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_150.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("c3FydCAoLTEuLCBleHRlbmQgPSBGYWxzZSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpXavP9e/___code___.py", line 3, in <module> exec compile(u'sqrt (-_sage_const_1p , extend = False) File "", line 1, in <module> File "/opt/SageMath/local/lib/python2.7/site-packages/sage/functions/other.py", line 870, in sqrt return x.sqrt(*args, **kwds) File "sage/rings/real_mpfr.pyx", line 4129, in sage.rings.real_mpfr.RealNumber.sqrt (build/cythonized/sage/rings/real_mpfr.c:26558) ValueError: negative number -1.00000000000000 does not have a square root in the real field |
3.14159265358979*I 3.14159265358979*I |
Traceback (click to the left of this block for traceback) ... TypeError: __call__() got an unexpected keyword argument 'extend' Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_152.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("bG9nICgxLiwgZXh0ZW5kID0gRmFsc2Up"),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpPLp2_c/___code___.py", line 3, in <module> exec compile(u'log (_sage_const_1p , extend = False) File "", line 1, in <module> File "/opt/SageMath/local/lib/python2.7/site-packages/sage/functions/log.py", line 436, in log return ln(args[0], **kwds) File "sage/symbolic/function.pyx", line 897, in sage.symbolic.function.BuiltinFunction.__call__ (build/cythonized/sage/symbolic/function.cpp:10405) TypeError: __call__() got an unexpected keyword argument 'extend' |
|