# class 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.

You can create a Complex object explicitly with:

You can convert certain objects to Complex objects with:

`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;

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]) → 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:
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]) → 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:
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

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;
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.

```Complex('i') ** 2              #=> (-1+0i)
Complex(-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 (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);
}```
cmp + numeric → complex click to toggle source

```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);

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, '+');
}```
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)
```
```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, '-');
}```
-cmp → complex click to toggle source

Returns negation of the value.

```-Complex(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));
}```
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)
```
```VALUE
rb_complex_div(VALUE self, VALUE other)
{
return f_divide(self, other, f_quo, id_quo);
}```
cmp <=> object → 0, 1, -1, or nil click to toggle source

If `cmp`‘s imaginary part is zero, and `object` is also a real number (or a `Complex` number where the imaginary part is zero), compare the real part of `cmp` to object. Otherwise, return nil.

```Complex(2, 3)  <=> Complex(2, 3)   #=> nil
Complex(2, 3)  <=> 1               #=> nil
Complex(2)     <=> 1               #=> 1
Complex(2)     <=> 2               #=> 0
Complex(2)     <=> 3               #=> -1
```
```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;
}```
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);

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

Returns the angle part of its polar form.

```Complex.polar(3, Math::PI/2).arg  #=> 1.5707963267948966
```
Alias for: arg
arg → float click to toggle source

Returns the angle part of its polar form.

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

```Complex(1, 2).conjugate  #=> (1-2i)
```
Alias for: conjugate
conjugate -> complex click to toggle source

Returns the complex conjugate.

```Complex(1, 2).conjugate  #=> (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 (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 real and imaginary parts are both finite numbers, otherwise returns `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() click to toggle source
```static VALUE
nucomp_hash(VALUE self)
{
return ST2FIX(rb_complex_hash(self));
}```
imag → real

Returns the imaginary part.

```Complex(7).imaginary      #=> 0
Complex(9, -4).imaginary  #=> -4
```
Alias for: imaginary
imaginary -> real click to toggle source

Returns the imaginary part.

```Complex(7).imaginary      #=> 0
Complex(9, -4).imaginary  #=> -4
```
```VALUE
rb_complex_imag(VALUE self)
{
get_dat1(self);
return dat->imag;
}```
Also aliased as: imag
infinite? → nil or 1 click to toggle source

Returns `1` if `cmp`‘s real or imaginary part is an infinite number, otherwise returns `nil`.

```For example:

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

Returns the absolute part of its polar form.

```Complex(-1).abs         #=> 1
Complex(3.0, -4.0).abs  #=> 5.0
```
Alias for: abs
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 = 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 → float

Returns the angle part of its polar form.

```Complex.polar(3, Math::PI/2).arg  #=> 1.5707963267948966
```
Alias for: arg
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)
```
```VALUE
rb_complex_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_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 → real click to toggle source

Returns the real part.

```Complex(7).real      #=> 7
Complex(9, -4).real  #=> 9
```
```VALUE
rb_complex_real(VALUE self)
{
get_dat1(self);
return dat->real;
}```
Complex(1).real? → false click to toggle source
Complex(1, 2).real? → false

Returns false, even if the complex number has no imaginary part.

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

Returns a complex object which denotes the given rectangular form.

```Complex.rectangular(1, 2)  #=> (1+2i)
```
Alias for: rectangular
rectangular -> array click to toggle source

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);
}```
Also aliased as: rect, rect
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);
}```