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Complex

A complex number can be represented as a paired real number with imaginary unit; a+bi. Where a is real part, b is imaginary part and i is imaginary unit. Real a equals complex a+0i mathematically.

Complex object can be created as literal, and also by using Kernel#Complex, Complex::rect, Complex::polar or to_c method.

2+1i                 #=> (2+1i)
Complex(1)           #=> (1+0i)
Complex(2, 3)        #=> (2+3i)
Complex.polar(2, 3)  #=> (-1.9799849932008908+0.2822400161197344i)
3.to_c               #=> (3+0i)

You can also create complex object from floating-point numbers or strings.

Complex(0.3)         #=> (0.3+0i)
Complex('0.3-0.5i')  #=> (0.3-0.5i)
Complex('2/3+3/4i')  #=> ((2/3)+(3/4)*i)
Complex('1@2')       #=> (-0.4161468365471424+0.9092974268256817i)

0.3.to_c             #=> (0.3+0i)
'0.3-0.5i'.to_c      #=> (0.3-0.5i)
'2/3+3/4i'.to_c      #=> ((2/3)+(3/4)*i)
'1@2'.to_c           #=> (-0.4161468365471424+0.9092974268256817i)

A complex object is either an exact or an inexact number.

Complex(1, 1) / 2    #=> ((1/2)+(1/2)*i)
Complex(1, 1) / 2.0  #=> (0.5+0.5i)

Constants

I: The imaginary unit.

Public Class Methods

polar(abs[, arg]) → complex click to toggle source

Returns a complex object which denotes the given polar form.

Complex.polar(3, 0)            #=> (3.0+0.0i)
Complex.polar(3, Math::PI/2)   #=> (1.836909530733566e-16+3.0i)
Complex.polar(3, Math::PI)     #=> (-3.0+3.673819061467132e-16i)
Complex.polar(3, -Math::PI/2)  #=> (1.836909530733566e-16-3.0i)

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

    switch (rb_scan_args(argc, argv, "11", &abs, &arg)) {
      case 1:
        nucomp_real_check(abs);
        if (canonicalization) return abs;
        return nucomp_s_new_internal(klass, abs, ZERO);
      default:
        nucomp_real_check(abs);
        nucomp_real_check(arg);
        break;
    }
    return f_complex_polar(klass, abs, arg);
}

rect(real[, imag]) → complex click to toggle source

rectangular(real[, imag]) → complex

Returns a complex object which denotes the given rectangular form.

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

 
               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:
        nucomp_real_check(real);
        imag = ZERO;
        break;
      default:
        nucomp_real_check(real);
        nucomp_real_check(imag);
        break;
    }

    return nucomp_s_canonicalize_internal(klass, real, imag);
}

rectangular(real[, imag]) → complex click to toggle source

Returns a complex object which denotes the given rectangular form.

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

 
               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:
        nucomp_real_check(real);
        imag = ZERO;
        break;
      default:
        nucomp_real_check(real);
        nucomp_real_check(imag);
        break;
    }

    return nucomp_s_canonicalize_internal(klass, real, imag);
}

Public Instance Methods

cmp * numeric → complex click to toggle source

Performs multiplication.

Complex(2, 3)  * Complex(2, 3)   #=> (-5+12i)
Complex(900)   * Complex(1)      #=> (900+0i)
Complex(-2, 9) * Complex(-9, 2)  #=> (0-85i)
Complex(9, 8)  * 4               #=> (36+32i)
Complex(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;
        VALUE areal, aimag, breal, bimag;
        int arzero, aizero, brzero, bizero;

        get_dat2(self, other);

        arzero = f_zero_p(areal = adat->real);
        aizero = f_zero_p(aimag = adat->imag);
        brzero = f_zero_p(breal = bdat->real);
        bizero = f_zero_p(bimag = bdat->imag);
        real = f_sub(safe_mul(areal, breal, arzero, brzero),
                     safe_mul(aimag, bimag, aizero, bizero));
        imag = f_add(safe_mul(areal, bimag, arzero, bizero),
                     safe_mul(aimag, breal, aizero, brzero));

        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, '*');
}

cmp ** numeric → complex click to toggle source

Performs exponentiation.

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

 
               static VALUE
nucomp_expt(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 (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)) {
        if (f_gt_p(other, ZERO)) {
            VALUE x, z;
            long n;

            x = self;
            z = x;
            n = FIX2LONG(other) - 1;

            while (n) {
                long q, r;

                while (1) {
                    get_dat1(x);

                    q = n / 2;
                    r = n % 2;

                    if (r)
                        break;

                    x = nucomp_s_new_internal(CLASS_OF(self),
                                       f_sub(f_mul(dat->real, dat->real),
                                             f_mul(dat->imag, dat->imag)),
                                       f_mul(f_mul(TWO, dat->real), dat->imag));
                    n = q;
                }
                z = f_mul(z, x);
                n--;
            }
            return z;
        }
        return f_expt(f_reciprocal(self), rb_int_uminus(other));
    }
    if (k_numeric_p(other) && f_real_p(other)) {
        VALUE r, theta;

        if (RB_TYPE_P(other, T_BIGNUM))
            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);
}

cmp + numeric → complex click to toggle source

Performs addition.

Complex(2, 3)  + Complex(2, 3)   #=> (4+6i)
Complex(900)   + Complex(1)      #=> (901+0i)
Complex(-2, 9) + Complex(-9, 2)  #=> (-11+11i)
Complex(9, 8)  + 4               #=> (13+8i)
Complex(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, '+');
}

cmp - numeric → complex click to toggle source

Performs subtraction.

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

 
               static VALUE
nucomp_sub(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, '-');
}

-cmp → complex click to toggle source

Returns negation of the value.

-Complex(1, 2)  #=> (-1-2i)

 
               static VALUE
nucomp_negate(VALUE self)
{
  get_dat1(self);
  return f_complex_new2(CLASS_OF(self),
                        f_negate(dat->real), f_negate(dat->imag));
}

cmp / numeric → complex click to toggle source

quo(numeric) → complex

Performs division.

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

 
               static VALUE
nucomp_div(VALUE self, VALUE other)
{
    return f_divide(self, other, f_quo, id_quo);
}

cmp == object → true or false click to toggle source

Returns true if cmp equals object numerically.

Complex(2, 3)  == Complex(2, 3)   #=> true
Complex(5)     == 5               #=> true
Complex(0)     == 0.0             #=> true
Complex('1/3') == 0.33            #=> false
Complex('1/2') == '1/2'           #=> false

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

        return f_boolcast(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 f_boolcast(f_eqeq_p(dat->real, other) && f_zero_p(dat->imag));
    }
    return f_boolcast(f_eqeq_p(other, self));
}

abs → real click to toggle source

Returns the absolute part of its polar form.

Complex(-1).abs         #=> 1
Complex(3.0, -4.0).abs  #=> 5.0

 
               static VALUE
nucomp_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);
}

abs2 → real click to toggle source

Returns square of the absolute value.

Complex(-1).abs2         #=> 1
Complex(3.0, -4.0).abs2  #=> 25.0

 
               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 → float click to toggle source

Returns the angle part of its polar form.

Complex.polar(3, Math::PI/2).arg  #=> 1.5707963267948966

 
               static VALUE
nucomp_arg(VALUE self)
{
    get_dat1(self);
    return rb_math_atan2(dat->imag, dat->real);
}

arg → float click to toggle source

Returns the angle part of its polar form.

Complex.polar(3, Math::PI/2).arg  #=> 1.5707963267948966

 
               static VALUE
nucomp_arg(VALUE self)
{
    get_dat1(self);
    return rb_math_atan2(dat->imag, dat->real);
}

conj → complex click to toggle source

conjugate → complex

Returns the complex conjugate.

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

 
               static VALUE
nucomp_conj(VALUE self)
{
    get_dat1(self);
    return f_complex_new2(CLASS_OF(self), dat->real, f_negate(dat->imag));
}

conjugate → complex click to toggle source

Returns the complex conjugate.

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

 
               static VALUE
nucomp_conj(VALUE self)
{
    get_dat1(self);
    return f_complex_new2(CLASS_OF(self), dat->real, f_negate(dat->imag));
}

denominator → integer click to toggle source

Returns the denominator (lcm of both denominator - real and imag).

See numerator.

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

fdiv(numeric) → complex click to toggle source

Performs division as each part is a float, never returns a float.

Complex(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 click to toggle source

Returns true if cmp's magnitude is finite number, oterwise returns false.

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

    if (f_finite_p(dat->real) && f_finite_p(dat->imag)) {
        return Qtrue;
    }
    return Qfalse;
}

imag → real click to toggle source

imaginary → real

Returns the imaginary part.

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

 
               static VALUE
nucomp_imag(VALUE self)
{
    get_dat1(self);
    return dat->imag;
}

imaginary → real click to toggle source

Returns the imaginary part.

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

 
               static VALUE
nucomp_imag(VALUE self)
{
    get_dat1(self);
    return dat->imag;
}

infinite? → nil or 1 or -1 click to toggle source

Returns values corresponding to the value of cmp's magnitude:

finite: nil
+Infinity: +1

For example:

   (1+1i).infinite?                   #=> nil
   (Float::INFINITY + 1i).infinite?   #=> 1

 
               static VALUE
rb_complex_infinite_p(VALUE self)
{
    VALUE magnitude = nucomp_abs(self);

    if (FINITE_TYPE_P(magnitude)) {
        return Qnil;
    }
    if (RB_FLOAT_TYPE_P(magnitude)) {
        const double f = RFLOAT_VALUE(magnitude);
        if (isinf(f)) {
            return INT2FIX(f < 0 ? -1 : 1);
        }
        return Qnil;
    }
    else {
        return rb_funcall(magnitude, id_infinite_p, 0);
    }
}

inspect → string click to toggle source

Returns the value as a string for inspection.

Complex(2).inspect                       #=> "(2+0i)"
Complex('-8/6').inspect                  #=> "((-4/3)+0i)"
Complex('1/2i').inspect                  #=> "(0+(1/2)*i)"
Complex(0, Float::INFINITY).inspect      #=> "(0+Infinity*i)"
Complex(Float::NAN, Float::NAN).inspect  #=> "(NaN+NaN*i)"

 
               static VALUE
nucomp_inspect(VALUE self)
{
    VALUE s;

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

    return s;
}

magnitude → real click to toggle source

Returns the absolute part of its polar form.

Complex(-1).abs         #=> 1
Complex(3.0, -4.0).abs  #=> 5.0

 
               static VALUE
nucomp_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);
}

numerator → numeric click to toggle source

Returns the numerator.

    1   2       3+4i  <-  numerator
    - + -i  ->  ----
    2   3        6    <-  denominator

c = Complex('1/2+2/3i')  #=> ((1/2)+(2/3)*i)
n = c.numerator          #=> (3+4i)
d = c.denominator        #=> 6
n / d                    #=> ((1/2)+(2/3)*i)
Complex(Rational(n.real, d), Rational(n.imag, d))
                         #=> ((1/2)+(2/3)*i)

See denominator.

 
               static VALUE
nucomp_numerator(VALUE self)
{
    VALUE cd;

    get_dat1(self);

    cd = f_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 → float click to toggle source

Returns the angle part of its polar form.

Complex.polar(3, Math::PI/2).arg  #=> 1.5707963267948966

 
               static VALUE
nucomp_arg(VALUE self)
{
    get_dat1(self);
    return rb_math_atan2(dat->imag, dat->real);
}

polar → array click to toggle source

Returns an array; [cmp.abs, cmp.arg].

Complex(1, 2).polar  #=> [2.23606797749979, 1.1071487177940904]

 
               static VALUE
nucomp_polar(VALUE self)
{
    return rb_assoc_new(f_abs(self), f_arg(self));
}

cmp / numeric → complex click to toggle source

quo(numeric) → complex

Performs division.

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

 
               static VALUE
nucomp_div(VALUE self, VALUE other)
{
    return f_divide(self, other, f_quo, id_quo);
}

rationalize([eps]) → rational click to toggle source

Returns the value as a rational if possible (the imaginary part should be exactly zero).

Complex(1.0/3, 0).rationalize  #=> (1/3)
Complex(1, 0.0).rationalize    # RangeError
Complex(1, 2).rationalize      # RangeError

See to_r.

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

    rb_scan_args(argc, argv, "01", NULL);

    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 → real click to toggle source

Returns the real part.

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

 
               static VALUE
nucomp_real(VALUE self)
{
    get_dat1(self);
    return dat->real;
}

real? → false click to toggle source

Returns false.

 
               static VALUE
nucomp_false(VALUE self)
{
    return Qfalse;
}

rect → array click to toggle source

rectangular → array

Returns an array; [cmp.real, cmp.imag].

Complex(1, 2).rectangular  #=> [1, 2]

 
               static VALUE
nucomp_rect(VALUE self)
{
    get_dat1(self);
    return rb_assoc_new(dat->real, dat->imag);
}

rect → array click to toggle source

rectangular → array

Returns an array; [cmp.real, cmp.imag].

Complex(1, 2).rectangular  #=> [1, 2]

 
               static VALUE
nucomp_rect(VALUE self)
{
    get_dat1(self);
    return rb_assoc_new(dat->real, dat->imag);
}

to_c → self click to toggle source

Returns self.

Complex(2).to_c      #=> (2+0i)
Complex(-8, 6).to_c  #=> (-8+6i)

 
               static VALUE
nucomp_to_c(VALUE self)
{
    return self;
}

to_f → float click to toggle source

Returns the value as a float if possible (the imaginary part should be exactly zero).

Complex(1, 0).to_f    #=> 1.0
Complex(1, 0.0).to_f  # RangeError
Complex(1, 2).to_f    # RangeError

 
               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 click to toggle source

Returns the value as an integer if possible (the imaginary part should be exactly zero).

Complex(1, 0).to_i    #=> 1
Complex(1, 0.0).to_i  # RangeError
Complex(1, 2).to_i    # RangeError

 
               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 click to toggle source

Returns the value as a rational if possible (the imaginary part should be exactly zero).

Complex(1, 0).to_r    #=> (1/1)
Complex(1, 0.0).to_r  # RangeError
Complex(1, 2).to_r    # RangeError

See rationalize.

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

    if (!k_exact_zero_p(dat->imag)) {
        rb_raise(rb_eRangeError, "can't convert %"PRIsVALUE" into Rational",
                 self);
    }
    return f_to_r(dat->real);
}

to_s → string click to toggle source

Returns the value as a string.

Complex(2).to_s                       #=> "2+0i"
Complex('-8/6').to_s                  #=> "-4/3+0i"
Complex('1/2i').to_s                  #=> "0+1/2i"
Complex(0, Float::INFINITY).to_s      #=> "0+Infinity*i"
Complex(Float::NAN, Float::NAN).to_s  #=> "NaN+NaN*i"

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

conj → complex click to toggle source

conjugate → complex

Returns the complex conjugate.

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

 
               static VALUE
nucomp_conj(VALUE self)
{
    get_dat1(self);
    return f_complex_new2(CLASS_OF(self), dat->real, f_negate(dat->imag));
}

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