# 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:

• Class Complex inherits (directly or indirectly) from classes Numeric and Object.

• Includes (indirectly) module Comparable.

Here, class Complex has methods for:

### Comparing¶ ↑

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

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

### 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 click to toggle source

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 click to toggle source

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 click to toggle source

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 click to toggle source

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);

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 click to toggle source

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)));
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 click to toggle source

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);

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),
}
return rb_num_coerce_bin(self, other, '+');
}```
complex - numeric → new_complex click to toggle source

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);

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 click to toggle source

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 click to toggle source

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 click to toggle source

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);
}
}
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 click to toggle source

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);

}
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 click to toggle source

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 click to toggle source

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);
f_mul(dat->imag, dat->imag));
}```
angle()

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
```
Alias for: arg
arg → float click to toggle source

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 -> complex click to toggle source

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

```Complex.rect(1, 2).conj # => (1-2i)
```
```VALUE
rb_complex_conjugate(VALUE self)
{
get_dat1(self);
return f_complex_new2(CLASS_OF(self), dat->real, f_negate(dat->imag));
}```
Also aliased as: conj
denominator → integer click to toggle source

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 click to toggle source

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 click to toggle source

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
```
```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 click to toggle source

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 -> numeric click to toggle source

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.
```
```VALUE
rb_complex_imag(VALUE self)
{
get_dat1(self);
return dat->imag;
}```
Also aliased as: imag
infinite? → 1 or nil click to toggle source

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
```
```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 click to toggle source

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

return s;
}```
magnitude()

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.
```
Alias for: abs
numerator → new_complex click to toggle source

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()

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
```
Alias for: arg
polar → array click to toggle source

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

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

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 click to toggle source

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 click to toggle source

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 click to toggle source

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 click to toggle source

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 -> array click to toggle source

Returns the array `[self.real, self.imag]`:

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

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

```Complex.polar(1.0, 1.0).rect # => [0.5403023058681398, 0.8414709848078965]
```

`Complex#rectangular` is an alias for `Complex#rect`.

```static VALUE
nucomp_rect(VALUE self)
{
get_dat1(self);
return rb_assoc_new(dat->real, dat->imag);
}```
Also aliased as: rect, rect
to_c → self click to toggle source

Returns `self`.

```static VALUE
nucomp_to_c(VALUE self)
{
return self;
}```
to_f → float click to toggle source

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 click to toggle source

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 click to toggle source

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 click to toggle source

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_String);
}```