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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.

In ruby, you can create complex object with Complex, ::rect, ::polar or to_c method.

```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)
```

I

### 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);
arg = ZERO;
break;
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.

```
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.

```
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.

```
static VALUE
nucomp_mul(VALUE self, VALUE other)
{
if (k_complex_p(other)) {
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, '*');
}
```
cmp ** numeric → complex click to toggle source

Performs exponentiation.

For example:

```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 (k_rational_p(other) && f_one_p(f_denominator(other)))
other = f_numerator(other); /* c14n */

if (k_complex_p(other)) {
get_dat1(other);

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

if (k_complex_p(other)) {
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 (k_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 = f_complex_new2(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), f_negate(other));
}
if (k_numeric_p(other) && f_real_p(other)) {
VALUE r, theta;

if (k_bignum_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);
}
```
cmp + numeric → complex click to toggle source

```
static VALUE
{
}
```
cmp - numeric → complex click to toggle source

Performs subtraction.

```
static VALUE
nucomp_sub(VALUE self, VALUE other)
{
}
```
-cmp → complex click to toggle source

Returns negation of the value.

```
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.

For example:

```Complex(10.0) / 3  #=> (3.3333333333333335+(0/1)*i)
Complex(10)   / 3  #=> ((10/3)+(0/1)*i)  # not (3+0i)
```
```
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.

```
static VALUE
nucomp_eqeq_p(VALUE self, VALUE other)
{
if (k_complex_p(other)) {
get_dat2(self, other);

}
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_eqeq_p(other, self);
}
```
abs → real click to toggle source

Returns the absolute part of its polar form.

```
static VALUE
nucomp_abs(VALUE self)
{
get_dat1(self);

if (f_zero_p(dat->real)) {
VALUE a = f_abs(dat->imag);
if (k_float_p(dat->real) && !k_float_p(dat->imag))
a = f_to_f(a);
return a;
}
if (f_zero_p(dat->imag)) {
VALUE a = f_abs(dat->real);
if (!k_float_p(dat->real) && k_float_p(dat->imag))
a = f_to_f(a);
return a;
}
return m_hypot(dat->real, dat->imag);
}
```
abs2 → real click to toggle source

Returns square of the absolute value.

```
static VALUE
nucomp_abs2(VALUE self)
{
get_dat1(self);
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 m_atan2_bang(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 m_atan2_bang(dat->imag, dat->real);
}
```
conj → complex click to toggle source
conjugate → complex

Returns the complex conjugate.

```
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.

```
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.

For example:

```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);
}
```
imag → real click to toggle source
imaginary → real

Returns the imaginary part.

```
static VALUE
nucomp_imag(VALUE self)
{
get_dat1(self);
return dat->imag;
}
```
imaginary → real click to toggle source

Returns the imaginary part.

```
static VALUE
nucomp_imag(VALUE self)
{
get_dat1(self);
return dat->imag;
}
```
inspect → string click to toggle source

Returns the value as a string for inspection.

```
static VALUE
nucomp_inspect(VALUE self)
{
VALUE s;

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

return s;
}
```
magnitude → real click to toggle source

Returns the absolute part of its polar form.

```
static VALUE
nucomp_abs(VALUE self)
{
get_dat1(self);

if (f_zero_p(dat->real)) {
VALUE a = f_abs(dat->imag);
if (k_float_p(dat->real) && !k_float_p(dat->imag))
a = f_to_f(a);
return a;
}
if (f_zero_p(dat->imag)) {
VALUE a = f_abs(dat->real);
if (!k_float_p(dat->real) && k_float_p(dat->imag))
a = f_to_f(a);
return a;
}
return m_hypot(dat->real, dat->imag);
}
```
numerator → numeric click to toggle source

Returns the numerator.

For example:

```    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 m_atan2_bang(dat->imag, dat->real);
}
```
polar → array click to toggle source

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

```
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.

For example:

```Complex(10.0) / 3  #=> (3.3333333333333335+(0/1)*i)
Complex(10)   / 3  #=> ((10/3)+(0/1)*i)  # not (3+0i)
```
```
static VALUE
nucomp_div(VALUE self, VALUE other)
{
return f_divide(self, other, f_quo, id_quo);
}
```
rationalize([eps]) → rational click to toggle source

If the imaginary part is exactly 0, returns the real part as a Rational, otherwise a RangeError is raised.

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

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

if (k_inexact_p(dat->imag) || f_nonzero_p(dat->imag)) {
VALUE s = f_to_s(self);
rb_raise(rb_eRangeError, "can't convert %s into Rational",
StringValuePtr(s));
}
return rb_funcall2(dat->real, rb_intern("rationalize"), argc, argv);
}
```
real → real click to toggle source

Returns the real part.

```
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].

```
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].

```
static VALUE
nucomp_rect(VALUE self)
{
get_dat1(self);
return rb_assoc_new(dat->real, dat->imag);
}
```
to_f → float click to toggle source

Returns the value as a float if possible.

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

if (k_inexact_p(dat->imag) || f_nonzero_p(dat->imag)) {
VALUE s = f_to_s(self);
rb_raise(rb_eRangeError, "can't convert %s into Float",
StringValuePtr(s));
}
return f_to_f(dat->real);
}
```
to_i → integer click to toggle source

Returns the value as an integer if possible.

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

if (k_inexact_p(dat->imag) || f_nonzero_p(dat->imag)) {
VALUE s = f_to_s(self);
rb_raise(rb_eRangeError, "can't convert %s into Integer",
StringValuePtr(s));
}
return f_to_i(dat->real);
}
```
to_r → rational click to toggle source

If the imaginary part is exactly 0, returns the real part as a Rational, otherwise a RangeError is raised.

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

if (k_inexact_p(dat->imag) || f_nonzero_p(dat->imag)) {
VALUE s = f_to_s(self);
rb_raise(rb_eRangeError, "can't convert %s into Rational",
StringValuePtr(s));
}
return f_to_r(dat->real);
}
```
to_s → string click to toggle source

Returns the value as a string.

```
static VALUE
nucomp_to_s(VALUE self)
{
return f_format(self, f_to_s);
}
```
conj → complex click to toggle source
conjugate → complex

Returns the complex conjugate.

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