class Complex

A Complex object houses a pair of values, given when the object is created as either rectangular coordinates or polar coordinates.

Rectangular Coordinates ↑

The rectangular coordinates of a complex number are called the real and imaginary parts; see Complex number definition.

You can create a Complex object from rectangular coordinates with:

Note that each of the stored parts may be a an instance one of the classes Complex, Float, Integer, or Rational; they may be retrieved:

The corresponding (computed) polar values may be retrieved:

Polar Coordinates ↑

The polar coordinates of a complex number are called the absolute and argument parts; see Complex polar plane.

In this class, the argument part in expressed radians (not degrees).

You can create a Complex object from polar coordinates with:

Note that each of the stored parts may be a an instance one of the classes Complex, Float, Integer, or Rational; they may be retrieved:

The corresponding (computed) rectangular values may be retrieved:

What’s Here ↑

First, what’s elsewhere:

Here, class Complex has methods for:

Creating Complex Objects ↑

  • ::polar: Returns a new Complex object based on given polar coordinates.

  • ::rect (and its alias ::rectangular): Returns a new Complex object based on given rectangular coordinates.

Querying ↑

  • abs (and its alias magnitude): Returns the absolute value for self.

  • arg (and its aliases angle and phase): Returns the argument (angle) for self in radians.

  • denominator: Returns the denominator of self.

  • finite?: Returns whether both self.real and self.image are finite.

  • hash: Returns the integer hash value for self.

  • imag (and its alias imaginary): Returns the imaginary value for self.

  • infinite?: Returns whether self.real or self.image is infinite.

  • numerator: Returns the numerator of self.

  • polar: Returns the array [self.abs, self.arg].

  • inspect: Returns a string representation of self.

  • real: Returns the real value for self.

  • real?: Returns false; for compatibility with Numeric#real?.

  • rect (and its alias rectangular): Returns the array [self.real, self.imag].

Comparing ↑

  • <=>: Returns whether self is less than, equal to, or greater than the given argument.

  • ==: Returns whether self is equal to the given argument.

Converting ↑

  • rationalize: Returns a Rational object whose value is exactly or approximately equivalent to that of self.real.

  • to_c: Returns self.

  • to_d: Returns the value as a BigDecimal object.

  • to_f: Returns the value of self.real as a Float, if possible.

  • to_i: Returns the value of self.real as an Integer, if possible.

  • to_r: Returns the value of self.real as a Rational, if possible.

  • to_s: Returns a string representation of self.

Performing Complex Arithmetic ↑

  • *: Returns the product of self and the given numeric.

  • **: Returns self raised to power of the given numeric.

  • +: Returns the sum of self and the given numeric.

  • -: Returns the difference of self and the given numeric.

  • -@: Returns the negation of self.

  • /: Returns the quotient of self and the given numeric.

  • abs2: Returns square of the absolute value (magnitude) for self.

  • conj (and its alias conjugate): Returns the conjugate of self.

  • fdiv: Returns Complex.rect(self.real/numeric, self.imag/numeric).

Working with JSON ↑

  • ::json_create: Returns a new Complex object, deserialized from the given serialized hash.

  • as_json: Returns a serialized hash constructed from self.

  • to_json: Returns a JSON string representing self.

These methods are provided by the JSON gem. To make these methods available:

require 'json/add/complex'

Constants

I

Equivalent to Complex.rect(0, 1):

Complex::I # => (0+1i)

Public Class Methods

polar(abs, arg = 0) → complex

Returns a new Complex object formed from the arguments, each of which must be an instance of Numeric, or an instance of one of its subclasses: Complex, Float, Integer, Rational. Argument arg is given in radians; see Polar Coordinates:

Complex.polar(3)        # => (3+0i)
Complex.polar(3, 2.0)   # => (-1.2484405096414273+2.727892280477045i)
Complex.polar(-3, -2.0) # => (1.2484405096414273+2.727892280477045i)

static VALUE
nucomp_s_polar(int argc, VALUE *argv, VALUE klass)
{
    VALUE abs, arg;

    argc = rb_scan_args(argc, argv, "11", &abs, &arg);
    abs = nucomp_real_check(abs);
    if (argc == 2) {
        arg = nucomp_real_check(arg);
    }
    else {
        arg = ZERO;
    }
    return f_complex_polar_real(klass, abs, arg);
}
rect(real, imag = 0) → complex

Returns a new Complex object formed from the arguments, each of which must be an instance of Numeric, or an instance of one of its subclasses: Complex, Float, Integer, Rational; see Rectangular Coordinates:

Complex.rect(3)             # => (3+0i)
Complex.rect(3, Math::PI)   # => (3+3.141592653589793i)
Complex.rect(-3, -Math::PI) # => (-3-3.141592653589793i)

Complex.rectangular is an alias for Complex.rect.

static VALUE
nucomp_s_new(int argc, VALUE *argv, VALUE klass)
{
    VALUE real, imag;

    switch (rb_scan_args(argc, argv, "11", &real, &imag)) {
      case 1:
        real = nucomp_real_check(real);
        imag = ZERO;
        break;
      default:
        real = nucomp_real_check(real);
        imag = nucomp_real_check(imag);
        break;
    }

    return nucomp_s_new_internal(klass, real, imag);
}
rect(real, imag = 0) → complex

Returns a new Complex object formed from the arguments, each of which must be an instance of Numeric, or an instance of one of its subclasses: Complex, Float, Integer, Rational; see Rectangular Coordinates:

Complex.rect(3)             # => (3+0i)
Complex.rect(3, Math::PI)   # => (3+3.141592653589793i)
Complex.rect(-3, -Math::PI) # => (-3-3.141592653589793i)

Complex.rectangular is an alias for Complex.rect.

static VALUE
nucomp_s_new(int argc, VALUE *argv, VALUE klass)
{
    VALUE real, imag;

    switch (rb_scan_args(argc, argv, "11", &real, &imag)) {
      case 1:
        real = nucomp_real_check(real);
        imag = ZERO;
        break;
      default:
        real = nucomp_real_check(real);
        imag = nucomp_real_check(imag);
        break;
    }

    return nucomp_s_new_internal(klass, real, imag);
}

Public Instance Methods

complex * numeric → new_complex

Returns the product of self and numeric:

Complex.rect(2, 3)  * Complex.rect(2, 3)  # => (-5+12i)
Complex.rect(900)   * Complex.rect(1)     # => (900+0i)
Complex.rect(-2, 9) * Complex.rect(-9, 2) # => (0-85i)
Complex.rect(9, 8)  * 4                   # => (36+32i)
Complex.rect(20, 9) * 9.8                 # => (196.0+88.2i)

VALUE
rb_complex_mul(VALUE self, VALUE other)
{
    if (RB_TYPE_P(other, T_COMPLEX)) {
        VALUE real, imag;
        get_dat2(self, other);

        comp_mul(adat->real, adat->imag, bdat->real, bdat->imag, &real, &imag);

        return f_complex_new2(CLASS_OF(self), real, imag);
    }
    if (k_numeric_p(other) && f_real_p(other)) {
        get_dat1(self);

        return f_complex_new2(CLASS_OF(self),
                              f_mul(dat->real, other),
                              f_mul(dat->imag, other));
    }
    return rb_num_coerce_bin(self, other, '*');
}
complex ** numeric → new_complex

Returns self raised to power numeric:

Complex.rect(0, 1) ** 2            # => (-1+0i)
Complex.rect(-8) ** Rational(1, 3) # => (1.0000000000000002+1.7320508075688772i)

VALUE
rb_complex_pow(VALUE self, VALUE other)
{
    if (k_numeric_p(other) && k_exact_zero_p(other))
        return f_complex_new_bang1(CLASS_OF(self), ONE);

    if (RB_TYPE_P(other, T_RATIONAL) && RRATIONAL(other)->den == LONG2FIX(1))
        other = RRATIONAL(other)->num; /* c14n */

    if (RB_TYPE_P(other, T_COMPLEX)) {
        get_dat1(other);

        if (k_exact_zero_p(dat->imag))
            other = dat->real; /* c14n */
    }

    if (other == ONE) {
        get_dat1(self);
        return nucomp_s_new_internal(CLASS_OF(self), dat->real, dat->imag);
    }

    VALUE result = complex_pow_for_special_angle(self, other);
    if (!UNDEF_P(result)) return result;

    if (RB_TYPE_P(other, T_COMPLEX)) {
        VALUE r, theta, nr, ntheta;

        get_dat1(other);

        r = f_abs(self);
        theta = f_arg(self);

        nr = m_exp_bang(f_sub(f_mul(dat->real, m_log_bang(r)),
                              f_mul(dat->imag, theta)));
        ntheta = f_add(f_mul(theta, dat->real),
                       f_mul(dat->imag, m_log_bang(r)));
        return f_complex_polar(CLASS_OF(self), nr, ntheta);
    }
    if (FIXNUM_P(other)) {
        long n = FIX2LONG(other);
        if (n == 0) {
            return nucomp_s_new_internal(CLASS_OF(self), ONE, ZERO);
        }
        if (n < 0) {
            self = f_reciprocal(self);
            other = rb_int_uminus(other);
            n = -n;
        }
        {
            get_dat1(self);
            VALUE xr = dat->real, xi = dat->imag, zr = xr, zi = xi;

            if (f_zero_p(xi)) {
                zr = rb_num_pow(zr, other);
            }
            else if (f_zero_p(xr)) {
                zi = rb_num_pow(zi, other);
                if (n & 2) zi = f_negate(zi);
                if (!(n & 1)) {
                    VALUE tmp = zr;
                    zr = zi;
                    zi = tmp;
                }
            }
            else {
                while (--n) {
                    long q, r;

                    for (; q = n / 2, r = n % 2, r == 0; n = q) {
                        VALUE tmp = f_sub(f_mul(xr, xr), f_mul(xi, xi));
                        xi = f_mul(f_mul(TWO, xr), xi);
                        xr = tmp;
                    }
                    comp_mul(zr, zi, xr, xi, &zr, &zi);
                }
            }
            return nucomp_s_new_internal(CLASS_OF(self), zr, zi);
        }
    }
    if (k_numeric_p(other) && f_real_p(other)) {
        VALUE r, theta;

        if (RB_BIGNUM_TYPE_P(other))
            rb_warn("in a**b, b may be too big");

        r = f_abs(self);
        theta = f_arg(self);

        return f_complex_polar(CLASS_OF(self), f_expt(r, other),
                               f_mul(theta, other));
    }
    return rb_num_coerce_bin(self, other, id_expt);
}
complex + numeric → new_complex

Returns the sum of self and numeric:

Complex.rect(2, 3)  + Complex.rect(2, 3)  # => (4+6i)
Complex.rect(900)   + Complex.rect(1)     # => (901+0i)
Complex.rect(-2, 9) + Complex.rect(-9, 2) # => (-11+11i)
Complex.rect(9, 8)  + 4                   # => (13+8i)
Complex.rect(20, 9) + 9.8                 # => (29.8+9i)

VALUE
rb_complex_plus(VALUE self, VALUE other)
{
    if (RB_TYPE_P(other, T_COMPLEX)) {
        VALUE real, imag;

        get_dat2(self, other);

        real = f_add(adat->real, bdat->real);
        imag = f_add(adat->imag, bdat->imag);

        return f_complex_new2(CLASS_OF(self), real, imag);
    }
    if (k_numeric_p(other) && f_real_p(other)) {
        get_dat1(self);

        return f_complex_new2(CLASS_OF(self),
                              f_add(dat->real, other), dat->imag);
    }
    return rb_num_coerce_bin(self, other, '+');
}
complex - numeric → new_complex

Returns the difference of self and numeric:

Complex.rect(2, 3)  - Complex.rect(2, 3)  # => (0+0i)
Complex.rect(900)   - Complex.rect(1)     # => (899+0i)
Complex.rect(-2, 9) - Complex.rect(-9, 2) # => (7+7i)
Complex.rect(9, 8)  - 4                   # => (5+8i)
Complex.rect(20, 9) - 9.8                 # => (10.2+9i)

VALUE
rb_complex_minus(VALUE self, VALUE other)
{
    if (RB_TYPE_P(other, T_COMPLEX)) {
        VALUE real, imag;

        get_dat2(self, other);

        real = f_sub(adat->real, bdat->real);
        imag = f_sub(adat->imag, bdat->imag);

        return f_complex_new2(CLASS_OF(self), real, imag);
    }
    if (k_numeric_p(other) && f_real_p(other)) {
        get_dat1(self);

        return f_complex_new2(CLASS_OF(self),
                              f_sub(dat->real, other), dat->imag);
    }
    return rb_num_coerce_bin(self, other, '-');
}
-complex → new_complex

Returns the negation of self, which is the negation of each of its parts:

-Complex.rect(1, 2)   # => (-1-2i)
-Complex.rect(-1, -2) # => (1+2i)

VALUE
rb_complex_uminus(VALUE self)
{
    get_dat1(self);
    return f_complex_new2(CLASS_OF(self),
                          f_negate(dat->real), f_negate(dat->imag));
}
complex / numeric → new_complex

Returns the quotient of self and numeric:

Complex.rect(2, 3)  / Complex.rect(2, 3)  # => (1+0i)
Complex.rect(900)   / Complex.rect(1)     # => (900+0i)
Complex.rect(-2, 9) / Complex.rect(-9, 2) # => ((36/85)-(77/85)*i)
Complex.rect(9, 8)  / 4                   # => ((9/4)+2i)
Complex.rect(20, 9) / 9.8                 # => (2.0408163265306123+0.9183673469387754i)

VALUE
rb_complex_div(VALUE self, VALUE other)
{
    return f_divide(self, other, f_quo, id_quo);
}
complex <=> object → -1, 0, 1, or nil

Returns:

  • self.real <=> object.real if both of the following are true:

    • self.imag == 0.

    • object.imag == 0. # Always true if object is numeric but not complex.

  • nil otherwise.

Examples:

Complex.rect(2) <=> 3                  # => -1
Complex.rect(2) <=> 2                  # => 0
Complex.rect(2) <=> 1                  # => 1
Complex.rect(2, 1) <=> 1               # => nil # self.imag not zero.
Complex.rect(1) <=> Complex.rect(1, 1) # => nil # object.imag not zero.
Complex.rect(1) <=> 'Foo'              # => nil # object.imag not defined.

static VALUE
nucomp_cmp(VALUE self, VALUE other)
{
    if (!k_numeric_p(other)) {
        return rb_num_coerce_cmp(self, other, idCmp);
    }
    if (!nucomp_real_p(self)) {
        return Qnil;
    }
    if (RB_TYPE_P(other, T_COMPLEX)) {
        if (nucomp_real_p(other)) {
            get_dat2(self, other);
            return rb_funcall(adat->real, idCmp, 1, bdat->real);
        }
    }
    else {
        get_dat1(self);
        if (f_real_p(other)) {
            return rb_funcall(dat->real, idCmp, 1, other);
        }
        else {
            return rb_num_coerce_cmp(dat->real, other, idCmp);
        }
    }
    return Qnil;
}
complex == object → true or false

Returns true if self.real == object.real and self.imag == object.imag:

Complex.rect(2, 3)  == Complex.rect(2.0, 3.0) # => true

static VALUE
nucomp_eqeq_p(VALUE self, VALUE other)
{
    if (RB_TYPE_P(other, T_COMPLEX)) {
        get_dat2(self, other);

        return RBOOL(f_eqeq_p(adat->real, bdat->real) &&
                          f_eqeq_p(adat->imag, bdat->imag));
    }
    if (k_numeric_p(other) && f_real_p(other)) {
        get_dat1(self);

        return RBOOL(f_eqeq_p(dat->real, other) && f_zero_p(dat->imag));
    }
    return RBOOL(f_eqeq_p(other, self));
}
abs → float

Returns the absolute value (magnitude) for self; see polar coordinates:

Complex.polar(-1, 0).abs # => 1.0

If self was created with rectangular coordinates, the returned value is computed, and may be inexact:

Complex.rectangular(1, 1).abs # => 1.4142135623730951 # The square root of 2.

VALUE
rb_complex_abs(VALUE self)
{
    get_dat1(self);

    if (f_zero_p(dat->real)) {
        VALUE a = f_abs(dat->imag);
        if (RB_FLOAT_TYPE_P(dat->real) && !RB_FLOAT_TYPE_P(dat->imag))
            a = f_to_f(a);
        return a;
    }
    if (f_zero_p(dat->imag)) {
        VALUE a = f_abs(dat->real);
        if (!RB_FLOAT_TYPE_P(dat->real) && RB_FLOAT_TYPE_P(dat->imag))
            a = f_to_f(a);
        return a;
    }
    return rb_math_hypot(dat->real, dat->imag);
}
Also aliased as: magnitude
abs2 → float

Returns square of the absolute value (magnitude) for self; see polar coordinates:

Complex.polar(2, 2).abs2 # => 4.0

If self was created with rectangular coordinates, the returned value is computed, and may be inexact:

Complex.rectangular(1.0/3, 1.0/3).abs2 # => 0.2222222222222222

static VALUE
nucomp_abs2(VALUE self)
{
    get_dat1(self);
    return f_add(f_mul(dat->real, dat->real),
                 f_mul(dat->imag, dat->imag));
}
angle
Alias for: arg
arg → float

Returns the argument (angle) for self in radians; see polar coordinates:

Complex.polar(3, Math::PI/2).arg  # => 1.57079632679489660

If self was created with rectangular coordinates, the returned value is computed, and may be inexact:

Complex.polar(1, 1.0/3).arg # => 0.33333333333333326

VALUE
rb_complex_arg(VALUE self)
{
    get_dat1(self);
    return rb_math_atan2(dat->imag, dat->real);
}
Also aliased as: angle, phase
conj → complex

Returns the conjugate of self, Complex.rect(self.imag, self.real):

Complex.rect(1, 2).conj # => (1-2i)
Alias for: conjugate
conjugate
Also aliased as: conj
denominator → integer

Returns the denominator of self, which is the least common multiple of self.real.denominator and self.imag.denominator:

Complex.rect(Rational(1, 2), Rational(2, 3)).denominator # => 6

Note that n.denominator of a non-rational numeric is 1.

Related: Complex#numerator.

static VALUE
nucomp_denominator(VALUE self)
{
    get_dat1(self);
    return rb_lcm(f_denominator(dat->real), f_denominator(dat->imag));
}
fdiv(numeric) → new_complex

Returns Complex.rect(self.real/numeric, self.imag/numeric):

Complex.rect(11, 22).fdiv(3) # => (3.6666666666666665+7.333333333333333i)

static VALUE
nucomp_fdiv(VALUE self, VALUE other)
{
    return f_divide(self, other, f_fdiv, id_fdiv);
}
finite? → true or false

Returns true if both self.real.finite? and self.imag.finite? are true, false otherwise:

Complex.rect(1, 1).finite?               # => true
Complex.rect(Float::INFINITY, 0).finite? # => false

Related: Numeric#finite?, Float#finite?.

static VALUE
rb_complex_finite_p(VALUE self)
{
    get_dat1(self);

    return RBOOL(f_finite_p(dat->real) && f_finite_p(dat->imag));
}
hash → integer

Returns the integer hash value for self.

Two Complex objects created from the same values will have the same hash value (and will compare using eql?):

Complex.rect(1, 2).hash == Complex.rect(1, 2).hash # => true

static VALUE
nucomp_hash(VALUE self)
{
    return ST2FIX(rb_complex_hash(self));
}
imag → numeric

Returns the imaginary value for self:

Complex.rect(7).imag     # => 0
Complex.rect(9, -4).imag # => -4

If self was created with polar coordinates, the returned value is computed, and may be inexact:

Complex.polar(1, Math::PI/4).imag # => 0.7071067811865476 # Square root of 2.
Alias for: imaginary
imaginary
Also aliased as: imag
infinite? → 1 or nil

Returns 1 if either self.real.infinite? or self.imag.infinite? is true, nil otherwise:

Complex.rect(Float::INFINITY, 0).infinite? # => 1
Complex.rect(1, 1).infinite?               # => nil

Related: Numeric#infinite?, Float#infinite?.

static VALUE
rb_complex_infinite_p(VALUE self)
{
    get_dat1(self);

    if (!f_infinite_p(dat->real) && !f_infinite_p(dat->imag)) {
        return Qnil;
    }
    return ONE;
}
inspect → string

Returns a string representation of self:

Complex.rect(2).inspect                      # => "(2+0i)"
Complex.rect(-8, 6).inspect                  # => "(-8+6i)"
Complex.rect(0, Rational(1, 2)).inspect      # => "(0+(1/2)*i)"
Complex.rect(0, Float::INFINITY).inspect     # => "(0+Infinity*i)"
Complex.rect(Float::NAN, Float::NAN).inspect # => "(NaN+NaN*i)"

static VALUE
nucomp_inspect(VALUE self)
{
    VALUE s;

    s = rb_usascii_str_new2("(");
    f_format(self, s, rb_inspect);
    rb_str_cat2(s, ")");

    return s;
}
magnitude
Alias for: abs
numerator → new_complex

Returns the Complex object created from the numerators of the real and imaginary parts of self, after converting each part to the lowest common denominator of the two:

c = Complex.rect(Rational(2, 3), Rational(3, 4)) # => ((2/3)+(3/4)*i)
c.numerator                                      # => (8+9i)

In this example, the lowest common denominator of the two parts is 12; the two converted parts may be thought of as Rational(8, 12) and Rational(9, 12), whose numerators, respectively, are 8 and 9; so the returned value of c.numerator is Complex.rect(8, 9).

Related: Complex#denominator.

static VALUE
nucomp_numerator(VALUE self)
{
    VALUE cd;

    get_dat1(self);

    cd = nucomp_denominator(self);
    return f_complex_new2(CLASS_OF(self),
                          f_mul(f_numerator(dat->real),
                                f_div(cd, f_denominator(dat->real))),
                          f_mul(f_numerator(dat->imag),
                                f_div(cd, f_denominator(dat->imag))));
}
phase
Alias for: arg
polar → array

Returns the array [self.abs, self.arg]:

Complex.polar(1, 2).polar # => [1.0, 2.0]

See Polar Coordinates.

If self was created with rectangular coordinates, the returned value is computed, and may be inexact:

Complex.rect(1, 1).polar # => [1.4142135623730951, 0.7853981633974483]

static VALUE
nucomp_polar(VALUE self)
{
    return rb_assoc_new(f_abs(self), f_arg(self));
}
complex / numeric → new_complex

Returns the quotient of self and numeric:

Complex.rect(2, 3)  / Complex.rect(2, 3)  # => (1+0i)
Complex.rect(900)   / Complex.rect(1)     # => (900+0i)
Complex.rect(-2, 9) / Complex.rect(-9, 2) # => ((36/85)-(77/85)*i)
Complex.rect(9, 8)  / 4                   # => ((9/4)+2i)
Complex.rect(20, 9) / 9.8                 # => (2.0408163265306123+0.9183673469387754i)

VALUE
rb_complex_div(VALUE self, VALUE other)
{
    return f_divide(self, other, f_quo, id_quo);
}
rationalize(epsilon = nil) → rational

Returns a Rational object whose value is exactly or approximately equivalent to that of self.real.

With no argument epsilon given, returns a Rational object whose value is exactly equal to that of self.real.rationalize:

Complex.rect(1, 0).rationalize              # => (1/1)
Complex.rect(1, Rational(0, 1)).rationalize # => (1/1)
Complex.rect(3.14159, 0).rationalize        # => (314159/100000)

With argument epsilon given, returns a Rational object whose value is exactly or approximately equal to that of self.real to the given precision:

Complex.rect(3.14159, 0).rationalize(0.1)          # => (16/5)
Complex.rect(3.14159, 0).rationalize(0.01)         # => (22/7)
Complex.rect(3.14159, 0).rationalize(0.001)        # => (201/64)
Complex.rect(3.14159, 0).rationalize(0.0001)       # => (333/106)
Complex.rect(3.14159, 0).rationalize(0.00001)      # => (355/113)
Complex.rect(3.14159, 0).rationalize(0.000001)     # => (7433/2366)
Complex.rect(3.14159, 0).rationalize(0.0000001)    # => (9208/2931)
Complex.rect(3.14159, 0).rationalize(0.00000001)   # => (47460/15107)
Complex.rect(3.14159, 0).rationalize(0.000000001)  # => (76149/24239)
Complex.rect(3.14159, 0).rationalize(0.0000000001) # => (314159/100000)
Complex.rect(3.14159, 0).rationalize(0.0)          # => (3537115888337719/1125899906842624)

Related: Complex#to_r.

static VALUE
nucomp_rationalize(int argc, VALUE *argv, VALUE self)
{
    get_dat1(self);

    rb_check_arity(argc, 0, 1);

    if (!k_exact_zero_p(dat->imag)) {
       rb_raise(rb_eRangeError, "can't convert %"PRIsVALUE" into Rational",
                self);
    }
    return rb_funcallv(dat->real, id_rationalize, argc, argv);
}
real → numeric

Returns the real value for self:

Complex.rect(7).real     # => 7
Complex.rect(9, -4).real # => 9

If self was created with polar coordinates, the returned value is computed, and may be inexact:

Complex.polar(1, Math::PI/4).real # => 0.7071067811865476 # Square root of 2.

VALUE
rb_complex_real(VALUE self)
{
    get_dat1(self);
    return dat->real;
}
real? → false

Returns false; for compatibility with Numeric#real?.

static VALUE
nucomp_real_p_m(VALUE self)
{
    return Qfalse;
}
rect → array

Returns a new Complex object formed from the arguments, each of which must be an instance of Numeric, or an instance of one of its subclasses: Complex, Float, Integer, Rational; see Rectangular Coordinates:

Complex.rect(3)             # => (3+0i)
Complex.rect(3, Math::PI)   # => (3+3.141592653589793i)
Complex.rect(-3, -Math::PI) # => (-3-3.141592653589793i)

Complex.rectangular is an alias for Complex.rect.

Alias for: rectangular
rectangular
Also aliased as: rect, rect
to_c → self

Returns self.

static VALUE
nucomp_to_c(VALUE self)
{
    return self;
}
to_f → float

Returns the value of self.real as a Float, if possible:

Complex.rect(1, 0).to_f              # => 1.0
Complex.rect(1, Rational(0, 1)).to_f # => 1.0

Raises RangeError if self.imag is not exactly zero (either Integer(0) or Rational(0, n)).

static VALUE
nucomp_to_f(VALUE self)
{
    get_dat1(self);

    if (!k_exact_zero_p(dat->imag)) {
        rb_raise(rb_eRangeError, "can't convert %"PRIsVALUE" into Float",
                 self);
    }
    return f_to_f(dat->real);
}
to_i → integer

Returns the value of self.real as an Integer, if possible:

Complex.rect(1, 0).to_i              # => 1
Complex.rect(1, Rational(0, 1)).to_i # => 1

Raises RangeError if self.imag is not exactly zero (either Integer(0) or Rational(0, n)).

static VALUE
nucomp_to_i(VALUE self)
{
    get_dat1(self);

    if (!k_exact_zero_p(dat->imag)) {
        rb_raise(rb_eRangeError, "can't convert %"PRIsVALUE" into Integer",
                 self);
    }
    return f_to_i(dat->real);
}
to_r → rational

Returns the value of self.real as a Rational, if possible:

Complex.rect(1, 0).to_r              # => (1/1)
Complex.rect(1, Rational(0, 1)).to_r # => (1/1)
Complex.rect(1, 0.0).to_r            # => (1/1)

Raises RangeError if self.imag is not exactly zero (either Integer(0) or Rational(0, n)) and self.imag.to_r is not exactly zero.

Related: Complex#rationalize.

static VALUE
nucomp_to_r(VALUE self)
{
    get_dat1(self);

    if (RB_FLOAT_TYPE_P(dat->imag) && FLOAT_ZERO_P(dat->imag)) {
        /* Do nothing here */
    }
    else if (!k_exact_zero_p(dat->imag)) {
        VALUE imag = rb_check_convert_type_with_id(dat->imag, T_RATIONAL, "Rational", idTo_r);
        if (NIL_P(imag) || !k_exact_zero_p(imag)) {
            rb_raise(rb_eRangeError, "can't convert %"PRIsVALUE" into Rational",
                     self);
        }
    }
    return f_to_r(dat->real);
}
to_s → string

Returns a string representation of self:

Complex.rect(2).to_s                      # => "2+0i"
Complex.rect(-8, 6).to_s                  # => "-8+6i"
Complex.rect(0, Rational(1, 2)).to_s      # => "0+1/2i"
Complex.rect(0, Float::INFINITY).to_s     # => "0+Infinity*i"
Complex.rect(Float::NAN, Float::NAN).to_s # => "NaN+NaN*i"

static VALUE
nucomp_to_s(VALUE self)
{
    return f_format(self, rb_usascii_str_new2(""), rb_String);
}