Arbitrary Precision Complex Numbers

AUTHORS:

  • William Stein (2006-01-26): complete rewrite
  • Joel B. Mohler (2006-12-16): naive rewrite into pyrex
  • William Stein(2007-01): rewrite of Mohler’s rewrite
  • Vincent Delecroix (2010-01): plot function
  • Travis Scrimshaw (2012-10-18): Added documentation for full coverage
  • Vincent Klein (2017-11-14) : add __mpc__() to class ComplexNumber. ComplexNumber constructor support gmpy2.mpc parameter.
class sage.rings.complex_number.CCtoCDF

Bases: sage.categories.map.Map

class sage.rings.complex_number.ComplexNumber

Bases: sage.structure.element.FieldElement

A floating point approximation to a complex number using any specified precision. Answers derived from calculations with such approximations may differ from what they would be if those calculations were performed with true complex numbers. This is due to the rounding errors inherent to finite precision calculations.

EXAMPLES:

sage: I = CC.0
sage: b = 1.5 + 2.5*I
sage: loads(b.dumps()) == b
True
additive_order()

Return the additive order of self.

EXAMPLES:

sage: CC(0).additive_order()
1
sage: CC.gen().additive_order()
+Infinity
agm(right, algorithm='optimal')

Return the Arithmetic-Geometric Mean (AGM) of self and right.

INPUT:

  • right (complex) – another complex number
  • algorithm (string, default “optimal”) – the algorithm to use (see below).

OUTPUT:

(complex) A value of the AGM of self and right. Note that this is a multi-valued function, and the algorithm used affects the value returned, as follows:

  • “pari”: Call the pari:agm function from the pari library.
  • “optimal”: Use the AGM sequence such that at each stage
    \((a,b)\) is replaced by \((a_1,b_1)=((a+b)/2,\pm\sqrt{ab})\) where the sign is chosen so that \(|a_1-b_1|\le|a_1+b_1|\), or equivalently \(\Re(b_1/a_1)\ge 0\). The resulting limit is maximal among all possible values.
  • “principal”: Use the AGM sequence such that at each stage
    \((a,b)\) is replaced by \((a_1,b_1)=((a+b)/2,\pm\sqrt{ab})\) where the sign is chosen so that \(\Re(b_1)\ge 0\) (the so-called principal branch of the square root).

The values \(AGM(a,0)\), \(AGM(0,a)\), and \(AGM(a,-a)\) are all taken to be 0.

EXAMPLES:

sage: a = CC(1,1)
sage: b = CC(2,-1)
sage: a.agm(b)
1.62780548487271 + 0.136827548397369*I
sage: a.agm(b, algorithm="optimal")
1.62780548487271 + 0.136827548397369*I
sage: a.agm(b, algorithm="principal")
1.62780548487271 + 0.136827548397369*I
sage: a.agm(b, algorithm="pari")
1.62780548487271 + 0.136827548397369*I

An example to show that the returned value depends on the algorithm parameter:

sage: a = CC(-0.95,-0.65)
sage: b = CC(0.683,0.747)
sage: a.agm(b, algorithm="optimal")
-0.371591652351761 + 0.319894660206830*I
sage: a.agm(b, algorithm="principal")
0.338175462986180 - 0.0135326969565405*I
sage: a.agm(b, algorithm="pari")
-0.371591652351761 + 0.319894660206830*I
sage: a.agm(b, algorithm="optimal").abs()
0.490319232466314
sage: a.agm(b, algorithm="principal").abs()
0.338446122230459
sage: a.agm(b, algorithm="pari").abs()
0.490319232466314
algdep(n, **kwds)

Return an irreducible polynomial of degree at most \(n\) which is approximately satisfied by this complex number.

ALGORITHM: Uses the PARI C-library algdep command.

INPUT: Type algdep? at the top level prompt. All additional parameters are passed onto the top-level algdep command.

EXAMPLES:

sage: C = ComplexField()
sage: z = (1/2)*(1 + sqrt(3.0) *C.0); z
0.500000000000000 + 0.866025403784439*I
sage: p = z.algdep(5); p
x^2 - x + 1
sage: p(z)
1.11022302462516e-16
algebraic_dependancy(*args, **kwds)

Deprecated: Use algebraic_dependency() instead. See trac ticket #22714 for details.

algebraic_dependency(n, **kwds)

Return an irreducible polynomial of degree at most \(n\) which is approximately satisfied by this complex number.

ALGORITHM: Uses the PARI C-library algdep command.

INPUT: Type algdep? at the top level prompt. All additional parameters are passed onto the top-level algdep command.

EXAMPLES:

sage: C = ComplexField()
sage: z = (1/2)*(1 + sqrt(3.0) *C.0); z
0.500000000000000 + 0.866025403784439*I
sage: p = z.algdep(5); p
x^2 - x + 1
sage: p(z)
1.11022302462516e-16
arccos()

Return the arccosine of self.

EXAMPLES:

sage: (1+CC(I)).arccos()
0.904556894302381 - 1.06127506190504*I
arccosh()

Return the hyperbolic arccosine of self.

EXAMPLES:

sage: (1+CC(I)).arccosh()
1.06127506190504 + 0.904556894302381*I
arccoth()

Return the hyperbolic arccotangent of self.

EXAMPLES:

sage: ComplexField(100)(1,1).arccoth()
0.40235947810852509365018983331 - 0.55357435889704525150853273009*I
arccsch()

Return the hyperbolic arccosecant of self.

EXAMPLES:

sage: ComplexField(100)(1,1).arccsch()
0.53063753095251782601650945811 - 0.45227844715119068206365839783*I
arcsech()

Return the hyperbolic arcsecant of self.

EXAMPLES:

sage: ComplexField(100)(1,1).arcsech()
0.53063753095251782601650945811 - 1.1185178796437059371676632938*I
arcsin()

Return the arcsine of self.

EXAMPLES:

sage: (1+CC(I)).arcsin()
0.666239432492515 + 1.06127506190504*I
arcsinh()

Return the hyperbolic arcsine of self.

EXAMPLES:

sage: (1+CC(I)).arcsinh()
1.06127506190504 + 0.666239432492515*I
arctan()

Return the arctangent of self.

EXAMPLES:

sage: (1+CC(I)).arctan()
1.01722196789785 + 0.402359478108525*I
arctanh()

Return the hyperbolic arctangent of self.

EXAMPLES:

sage: (1+CC(I)).arctanh()
0.402359478108525 + 1.01722196789785*I
arg()

See argument().

EXAMPLES:

sage: i = CC.0
sage: (i^2).arg()
3.14159265358979
argument()

The argument (angle) of the complex number, normalized so that \(-\pi < \theta \leq \pi\).

EXAMPLES:

sage: i = CC.0
sage: (i^2).argument()
3.14159265358979
sage: (1+i).argument()
0.785398163397448
sage: i.argument()
1.57079632679490
sage: (-i).argument()
-1.57079632679490
sage: (RR('-0.001') - i).argument()
-1.57179632646156
conjugate()

Return the complex conjugate of this complex number.

EXAMPLES:

sage: i = CC.0
sage: (1+i).conjugate()
1.00000000000000 - 1.00000000000000*I
cos()

Return the cosine of self.

EXAMPLES:

sage: (1+CC(I)).cos()
0.833730025131149 - 0.988897705762865*I
cosh()

Return the hyperbolic cosine of self.

EXAMPLES:

sage: (1+CC(I)).cosh()
0.833730025131149 + 0.988897705762865*I
cotan()

Return the cotangent of self.

EXAMPLES:

sage: (1+CC(I)).cotan()
0.217621561854403 - 0.868014142895925*I
sage: i = ComplexField(200).0
sage: (1+i).cotan()
0.21762156185440268136513424360523807352075436916785404091068 - 0.86801414289592494863584920891627388827343874994609327121115*I
sage: i = ComplexField(220).0
sage: (1+i).cotan()
0.21762156185440268136513424360523807352075436916785404091068124239 - 0.86801414289592494863584920891627388827343874994609327121115071646*I
coth()

Return the hyperbolic cotangent of self.

EXAMPLES:

sage: ComplexField(100)(1,1).coth()
0.86801414289592494863584920892 - 0.21762156185440268136513424361*I
csc()

Return the cosecant of self.

EXAMPLES:

sage: ComplexField(100)(1,1).csc()
0.62151801717042842123490780586 - 0.30393100162842645033448560451*I
csch()

Return the hyperbolic cosecant of self.

EXAMPLES:

sage: ComplexField(100)(1,1).csch()
0.30393100162842645033448560451 - 0.62151801717042842123490780586*I
dilog()

Returns the complex dilogarithm of self.

The complex dilogarithm, or Spence’s function, is defined by

\[Li_2(z) = - \int_0^z \frac{\log|1-\zeta|}{\zeta} d(\zeta) = \sum_{k=1}^\infty \frac{z^k}{k}\]

Note that the series definition can only be used for \(|z| < 1\).

EXAMPLES:

sage: a = ComplexNumber(1,0)
sage: a.dilog()
1.64493406684823
sage: float(pi^2/6)
1.6449340668482262
sage: b = ComplexNumber(0,1)
sage: b.dilog()
-0.205616758356028 + 0.915965594177219*I
sage: c = ComplexNumber(0,0)
sage: c.dilog()
0.000000000000000
eta(omit_frac=False)

Return the value of the Dedekind \(\eta\) function on self, intelligently computed using \(\mathbb{SL}(2,\ZZ)\) transformations.

The \(\eta\) function is

\[\eta(z) = e^{\pi i z / 12} \prod_{n=1}^{\infty}(1-e^{2\pi inz})\]

INPUT:

  • self – element of the upper half plane (if not, raises a ValueError).
  • omit_frac – (bool, default: False), if True, omit the \(e^{\pi i z / 12}\) factor.

OUTPUT: a complex number

ALGORITHM: Uses the PARI C library.

EXAMPLES:

First we compute \(\eta(1+i)\):

sage: i = CC.0
sage: z = 1+i; z.eta()
0.742048775836565 + 0.198831370229911*I

We compute eta to low precision directly from the definition:

sage: z = 1 + i; z.eta()
0.742048775836565 + 0.198831370229911*I
sage: pi = CC(pi)        # otherwise we will get a symbolic result.
sage: exp(pi * i * z / 12) * prod([1-exp(2*pi*i*n*z) for n in range(1,10)])
0.742048775836565 + 0.198831370229911*I

The optional argument allows us to omit the fractional part:

sage: z = 1 + i
sage: z.eta(omit_frac=True)
0.998129069925959
sage: prod([1-exp(2*pi*i*n*z) for n in range(1,10)])
0.998129069925958 + 4.59099857829247e-19*I

We illustrate what happens when \(z\) is not in the upper half plane:

sage: z = CC(1)
sage: z.eta()
Traceback (most recent call last):
...
ValueError: value must be in the upper half plane

You can also use functional notation:

sage: eta(1+CC(I))
0.742048775836565 + 0.198831370229911*I
exp()

Compute \(e^z\) or \(\exp(z)\).

EXAMPLES:

sage: i = ComplexField(300).0
sage: z = 1 + i
sage: z.exp()
1.46869393991588515713896759732660426132695673662900872279767567631093696585951213872272450 + 2.28735528717884239120817190670050180895558625666835568093865811410364716018934540926734485*I
gamma()

Return the Gamma function evaluated at this complex number.

EXAMPLES:

sage: i = ComplexField(30).0
sage: (1+i).gamma()
0.49801567 - 0.15494983*I
sage: CC(-1).gamma()
Infinity
gamma_inc(t)

Return the incomplete Gamma function evaluated at this complex number.

EXAMPLES:

sage: C, i = ComplexField(30).objgen()
sage: (1+i).gamma_inc(2 + 3*i)  # abs tol 2e-10
0.0020969149 - 0.059981914*I
sage: (1+i).gamma_inc(5)
-0.0013781309 + 0.0065198200*I
sage: C(2).gamma_inc(1 + i)
0.70709210 - 0.42035364*I
sage: CC(2).gamma_inc(5)
0.0404276819945128
imag()

Return imaginary part of self.

EXAMPLES:

sage: i = ComplexField(100).0
sage: z = 2 + 3*i
sage: x = z.imag(); x
3.0000000000000000000000000000
sage: x.parent()
Real Field with 100 bits of precision
sage: z.imag_part()
3.0000000000000000000000000000
imag_part()

Return imaginary part of self.

EXAMPLES:

sage: i = ComplexField(100).0
sage: z = 2 + 3*i
sage: x = z.imag(); x
3.0000000000000000000000000000
sage: x.parent()
Real Field with 100 bits of precision
sage: z.imag_part()
3.0000000000000000000000000000
is_NaN()

Check if self is not-a-number.

EXAMPLES:

sage: CC(1, 2).is_NaN()
False
sage: CC(NaN).is_NaN()
True
sage: CC(NaN,2).log().is_NaN()
True
is_imaginary()

Return True if self is imaginary, i.e. has real part zero.

EXAMPLES:

sage: CC(1.23*i).is_imaginary()
True
sage: CC(1+i).is_imaginary()
False
is_infinity()

Check if self is \(\infty\).

EXAMPLES:

sage: CC(1, 2).is_infinity()
False
sage: CC(0, oo).is_infinity()
True
is_integer()

Return True if self is a integer

EXAMPLES:

sage: CC(3).is_integer()
True
sage: CC(1,2).is_integer()
False
is_negative_infinity()

Check if self is \(-\infty\).

EXAMPLES:

sage: CC(1, 2).is_negative_infinity()
False
sage: CC(-oo, 0).is_negative_infinity()
True
sage: CC(0, -oo).is_negative_infinity()
False
is_positive_infinity()

Check if self is \(+\infty\).

EXAMPLES:

sage: CC(1, 2).is_positive_infinity()
False
sage: CC(oo, 0).is_positive_infinity()
True
sage: CC(0, oo).is_positive_infinity()
False
is_real()

Return True if self is real, i.e. has imaginary part zero.

EXAMPLES:

sage: CC(1.23).is_real()
True
sage: CC(1+i).is_real()
False
is_square()

This function always returns true as \(\CC\) is algebraically closed.

EXAMPLES:

sage: a = ComplexNumber(2,1)
sage: a.is_square()
True

\(\CC\) is algebraically closed, hence every element is a square:

sage: b = ComplexNumber(5)
sage: b.is_square()
True
log(base=None)

Complex logarithm of \(z\) with branch chosen as follows: Write \(z = \rho e^{i \theta}\) with \(-\pi < \theta <= pi\). Then \(\mathrm{log}(z) = \mathrm{log}(\rho) + i \theta\).

Warning

Currently the real log is computed using floats, so there is potential precision loss.

EXAMPLES:

sage: a = ComplexNumber(2,1)
sage: a.log()
0.804718956217050 + 0.463647609000806*I
sage: log(a.abs())
0.804718956217050
sage: a.argument()
0.463647609000806
sage: b = ComplexNumber(float(exp(42)),0)
sage: b.log()
41.99999999999971
sage: c = ComplexNumber(-1,0)
sage: c.log()
3.14159265358979*I

The option of a base is included for compatibility with other logs:

sage: c = ComplexNumber(-1,0)
sage: c.log(2)
4.53236014182719*I

If either component (real or imaginary) of the complex number is NaN (not a number), log will return the complex NaN:

sage: c = ComplexNumber(NaN,2)
sage: c.log()
NaN + NaN*I
multiplicative_order()

Return the multiplicative order of this complex number, if known, or raise a NotImplementedError.

EXAMPLES:

sage: C.<i> = ComplexField()
sage: i.multiplicative_order()
4
sage: C(1).multiplicative_order()
1
sage: C(-1).multiplicative_order()
2
sage: C(i^2).multiplicative_order()
2
sage: C(-i).multiplicative_order()
4
sage: C(2).multiplicative_order()
+Infinity
sage: w = (1+sqrt(-3.0))/2; w
0.500000000000000 + 0.866025403784439*I
sage: abs(w)
1.00000000000000
sage: w.multiplicative_order()
Traceback (most recent call last):
...
NotImplementedError: order of element not known
norm()

Returns the norm of this complex number.

If \(c = a + bi\) is a complex number, then the norm of \(c\) is defined as the product of \(c\) and its complex conjugate:

\[\text{norm}(c) = \text{norm}(a + bi) = c \cdot \overline{c} = a^2 + b^2.\]

The norm of a complex number is different from its absolute value. The absolute value of a complex number is defined to be the square root of its norm. A typical use of the complex norm is in the integral domain \(\ZZ[i]\) of Gaussian integers, where the norm of each Gaussian integer \(c = a + bi\) is defined as its complex norm.

EXAMPLES:

This indeed acts as the square function when the imaginary component of self is equal to zero:

sage: a = ComplexNumber(2,1)
sage: a.norm()
5.00000000000000
sage: b = ComplexNumber(4.2,0)
sage: b.norm()
17.6400000000000
sage: b^2
17.6400000000000
nth_root(n, all=False)

The \(n\)-th root function.

INPUT:

  • all - bool (default: False); if True, return a list of all \(n\)-th roots.

EXAMPLES:

sage: a = CC(27)
sage: a.nth_root(3)
3.00000000000000
sage: a.nth_root(3, all=True)
[3.00000000000000, -1.50000000000000 + 2.59807621135332*I, -1.50000000000000 - 2.59807621135332*I]
sage: a = ComplexField(20)(2,1)
sage: [r^7 for r in a.nth_root(7, all=True)]
[2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0001*I, 2.0000 + 1.0001*I]
plot(**kargs)

Plots this complex number as a point in the plane

The accepted options are the ones of point2d(). Type point2d.options to see all options.

Note

Just wraps the sage.plot.point.point2d method

EXAMPLES:

You can either use the indirect:

sage: z = CC(0,1)
sage: plot(z)
Graphics object consisting of 1 graphics primitive

or the more direct:

sage: z = CC(0,1)
sage: z.plot()
Graphics object consisting of 1 graphics primitive
prec()

Return precision of this complex number.

EXAMPLES:

sage: i = ComplexField(2000).0
sage: i.prec()
2000
real()

Return real part of self.

EXAMPLES:

sage: i = ComplexField(100).0
sage: z = 2 + 3*i
sage: x = z.real(); x
2.0000000000000000000000000000
sage: x.parent()
Real Field with 100 bits of precision
sage: z.real_part()
2.0000000000000000000000000000
real_part()

Return real part of self.

EXAMPLES:

sage: i = ComplexField(100).0
sage: z = 2 + 3*i
sage: x = z.real(); x
2.0000000000000000000000000000
sage: x.parent()
Real Field with 100 bits of precision
sage: z.real_part()
2.0000000000000000000000000000
sec()

Return the secant of self.

EXAMPLES:

sage: ComplexField(100)(1,1).sec()
0.49833703055518678521380589177 + 0.59108384172104504805039169297*I
sech()

Return the hyperbolic secant of self.

EXAMPLES:

sage: ComplexField(100)(1,1).sech()
0.49833703055518678521380589177 - 0.59108384172104504805039169297*I
sin()

Return the sine of self.

EXAMPLES:

sage: (1+CC(I)).sin()
1.29845758141598 + 0.634963914784736*I
sinh()

Return the hyperbolic sine of self.

EXAMPLES:

sage: (1+CC(I)).sinh()
0.634963914784736 + 1.29845758141598*I
sqrt(all=False)

The square root function, taking the branch cut to be the negative real axis.

INPUT:

  • all - bool (default: False); if True, return a list of all square roots.

EXAMPLES:

sage: C.<i> = ComplexField(30)
sage: i.sqrt()
0.70710678 + 0.70710678*I
sage: (1+i).sqrt()
1.0986841 + 0.45508986*I
sage: (C(-1)).sqrt()
1.0000000*I
sage: (1 + 1e-100*i).sqrt()^2
1.0000000 + 1.0000000e-100*I
sage: i = ComplexField(200).0
sage: i.sqrt()
0.70710678118654752440084436210484903928483593768847403658834 + 0.70710678118654752440084436210484903928483593768847403658834*I
str(base=10, istr='I', **kwds)

Return a string representation of self.

INPUT:

  • base – (default: 10) base for output
  • istr – (default: I) String representation of the complex unit
  • **kwds – other arguments to pass to the str() method of the real numbers in the real and imaginary parts.

EXAMPLES:

sage: a = CC(pi + I*e)
sage: a
3.14159265358979 + 2.71828182845905*I
sage: a.str(truncate=True)
'3.14159265358979 + 2.71828182845905*I'
sage: a.str()
'3.1415926535897931 + 2.7182818284590451*I'
sage: a.str(base=2)
'11.001001000011111101101010100010001000010110100011000 + 10.101101111110000101010001011000101000101011101101001*I'
sage: CC(0.5 + 0.625*I).str(base=2)
'0.10000000000000000000000000000000000000000000000000000 + 0.10100000000000000000000000000000000000000000000000000*I'
sage: a.str(base=16)
'3.243f6a8885a30 + 2.b7e151628aed2*I'
sage: a.str(base=36)
'3.53i5ab8p5fc + 2.puw5nggjf8f*I'
sage: CC(0)
0.000000000000000
sage: CC.0.str(istr='%i')
'1.0000000000000000*%i'
tan()

Return the tangent of self.

EXAMPLES:

sage: (1+CC(I)).tan()
0.271752585319512 + 1.08392332733869*I
tanh()

Return the hyperbolic tangent of self.

EXAMPLES:

sage: (1+CC(I)).tanh()
1.08392332733869 + 0.271752585319512*I
zeta()

Return the Riemann zeta function evaluated at this complex number.

EXAMPLES:

sage: i = ComplexField(30).gen()
sage: z = 1 + i
sage: z.zeta()
0.58215806 - 0.92684856*I
sage: zeta(z)
0.58215806 - 0.92684856*I

sage: CC(1).zeta()
Infinity
class sage.rings.complex_number.RRtoCC

Bases: sage.categories.map.Map

EXAMPLES:

sage: from sage.rings.complex_number import RRtoCC
sage: RRtoCC(RR, CC)
Natural map:
  From: Real Field with 53 bits of precision
  To:   Complex Field with 53 bits of precision
sage.rings.complex_number.cmp_abs(a, b)

Returns -1, 0, or 1 according to whether \(|a|\) is less than, equal to, or greater than \(|b|\).

Optimized for non-close numbers, where the ordering can be determined by examining exponents.

EXAMPLES:

sage: from sage.rings.complex_number import cmp_abs
sage: cmp_abs(CC(5), CC(1))
1
sage: cmp_abs(CC(5), CC(4))
1
sage: cmp_abs(CC(5), CC(5))
0
sage: cmp_abs(CC(5), CC(6))
-1
sage: cmp_abs(CC(5), CC(100))
-1
sage: cmp_abs(CC(-100), CC(1))
1
sage: cmp_abs(CC(-100), CC(100))
0
sage: cmp_abs(CC(-100), CC(1000))
-1
sage: cmp_abs(CC(1,1), CC(1))
1
sage: cmp_abs(CC(1,1), CC(2))
-1
sage: cmp_abs(CC(1,1), CC(1,0.99999))
1
sage: cmp_abs(CC(1,1), CC(1,-1))
0
sage: cmp_abs(CC(0), CC(1))
-1
sage: cmp_abs(CC(1), CC(0))
1
sage: cmp_abs(CC(0), CC(0))
0
sage: cmp_abs(CC(2,1), CC(1,2))
0
sage.rings.complex_number.create_ComplexNumber(s_real, s_imag=None, pad=0, min_prec=53)

Return the complex number defined by the strings s_real and s_imag as an element of ComplexField(prec=n), where \(n\) potentially has slightly more (controlled by pad) bits than given by \(s\).

INPUT:

  • s_real – a string that defines a real number (or something whose string representation defines a number)
  • s_imag – a string that defines a real number (or something whose string representation defines a number)
  • pad – an integer at least 0.
  • min_prec – number will have at least this many bits of precision, no matter what.

EXAMPLES:

sage: ComplexNumber('2.3')
2.30000000000000
sage: ComplexNumber('2.3','1.1')
2.30000000000000 + 1.10000000000000*I
sage: ComplexNumber(10)
10.0000000000000
sage: ComplexNumber(10,10)
10.0000000000000 + 10.0000000000000*I
sage: ComplexNumber(1.000000000000000000000000000,2)
1.00000000000000000000000000 + 2.00000000000000000000000000*I
sage: ComplexNumber(1,2.000000000000000000000)
1.00000000000000000000 + 2.00000000000000000000*I
sage: sage.rings.complex_number.create_ComplexNumber(s_real=2,s_imag=1)
2.00000000000000 + 1.00000000000000*I
sage.rings.complex_number.is_ComplexNumber(x)

Returns True if x is a complex number. In particular, if x is of the ComplexNumber type.

EXAMPLES:

sage: from sage.rings.complex_number import is_ComplexNumber
sage: a = ComplexNumber(1,2); a
1.00000000000000 + 2.00000000000000*I
sage: is_ComplexNumber(a)
True
sage: b = ComplexNumber(1); b
1.00000000000000
sage: is_ComplexNumber(b)
True

Note that the global element I is of type SymbolicConstant. However, elements of the class ComplexField_class are of type ComplexNumber:

sage: c = 1 + 2*I
sage: is_ComplexNumber(c)
False
sage: d = CC(1 + 2*I)
sage: is_ComplexNumber(d)
True
sage.rings.complex_number.make_ComplexNumber0(fld, mult_order, re, im)

Create a complex number for pickling.

EXAMPLES:

sage: a = CC(1 + I)
sage: loads(dumps(a)) == a # indirect doctest
True
sage.rings.complex_number.set_global_complex_round_mode(n)

Set the global complex rounding mode.

Warning

Do not call this function explicitly. The default rounding mode is n = 0.

EXAMPLES:

sage: sage.rings.complex_number.set_global_complex_round_mode(0)