# struct Complex

- Complex
- Struct
- Value
- Object

##
Overview

A complex number is a number represented in the form a + bi. In this form, a and b are real numbers, and i is an imaginary number such as i² = -1. The a is the real part of the number, and the b is the imaginary part of the number.

NOTE To use `Complex`, you must explicitly import it with `require "complex"`

    require "complex"

    Complex.new(1, 0)   # => 1.0 + 0.0.i
    Complex.new(5, -12) # => 5.0 - 12.0.i

    1.to_c # => 1.0 + 0.0.i
    1.i    # => 0.0 + 1.0.i

##
Defined in:

complex.cr

##
Constructors

- .additive_identity : self
- .multiplicative_identity : self
- .new(real : Number, imag : Number = 0)
- .new(c : Complex)
- .zero : Complex

  Returns the number `0` in complex form.

##
Instance Method Summary

- #*(other : Complex) : Complex

  Multiplies `self` by other.

- #*(other : Number) : Complex

  Multiplies `self` by other.

- #+(other : Complex) : Complex

  Adds the value of `self` to other.

- #+(other : Number) : Complex

  Adds the value of `self` to other.

- #+ : Complex

  Returns `self`.

- #-(other : Complex) : Complex

  Removes the value of other from `self`.

- #-(other : Number) : Complex

  Removes the value of other from `self`.

- #- : Complex

  Returns the opposite of `self`.

- #/(other : Complex) : Complex

  Divides `self` by other.

- #/(other : Number) : Complex

  Divides `self` by other.

- #==(other : Complex)

  Determines whether `self` equals other or not.

- #==(other : Number)

  Determines whether `self` equals other or not.

- #==(other)

  Determines whether `self` equals other or not.

- #abs : Float64

  Returns the absolute value of this complex number in a number form, using the Pythagorean theorem.

- #abs2 : Float64

  Returns the square of absolute value in a number form.

- #clone
- #conj : Complex

  Returns the conjugate of `self`.

- #hash(hasher)

  See `Object#hash(hasher)`

- #imag : Float64

  Returns the imaginary part.

- #inspect(io : IO) : Nil

  Writes this complex object to an io, surrounded by parentheses.

- #inv : Complex

  Returns the inverse of `self`.

- #phase : Float64

  Returns the phase of `self`.

- #polar : Tuple(Float64, Float64)

  Returns a `Tuple` with the `#abs` value and the `#phase`.

- #real : Float64

  Returns the real part.

- #round(digits = 0) : Complex

  Rounds to the nearest digits.

- #sign : Complex

  Returns the complex sign of `self`.

- #to_c

  Returns `self`.

- #to_f

  See `#to_f64`.

- #to_f32(*args, **options)
- #to_f32(*args, **options, &)
- #to_f64 : Float64

  Returns the value as a `Float64` if possible (the imaginary part should be exactly zero), raises otherwise.

- #to_i

  See `#to_i32`.

- #to_i128(*args, **options)
- #to_i128(*args, **options, &)
- #to_i16(*args, **options)
- #to_i16(*args, **options, &)
- #to_i32(*args, **options)
- #to_i32(*args, **options, &)
- #to_i64(*args, **options)
- #to_i64(*args, **options, &)
- #to_i8(*args, **options)
- #to_i8(*args, **options, &)
- #to_s(io : IO) : Nil

  Writes this complex object to an io.

- #to_u128(*args, **options)
- #to_u128(*args, **options, &)
- #to_u16(*args, **options)
- #to_u16(*args, **options, &)
- #to_u32(*args, **options)
- #to_u32(*args, **options, &)
- #to_u64(*args, **options)
- #to_u64(*args, **options, &)
- #to_u8(*args, **options)
- #to_u8(*args, **options, &)
- #zero? : Bool

  Returns `true` if the complex number is zero.

### Instance methods inherited from struct `Struct`

==(other) : Bool ==, hash(hasher) hash, inspect(io : IO) : Nil inspect, pretty_print(pp) : Nil pretty_print, to_s(io : IO) : Nil to_s

### Instance methods inherited from struct `Value`

==(other : JSON::Any)
==(other : YAML::Any)
==(other) ==, dup dup

### Instance methods inherited from class `Object`

! : Bool !, !=(other) !=, !~(other) !~, ==(other) ==, ===(other : JSON::Any)
===(other : YAML::Any)
===(other) ===, =~(other) =~, as(type : Class) as, as?(type : Class) as?, class class, dup dup, hash(hasher)
hash hash, in?(collection : Object) : Bool
in?(*values : Object) : Bool in?, inspect(io : IO) : Nil
inspect : String inspect, is_a?(type : Class) : Bool is_a?, itself itself, nil? : Bool nil?, not_nil!(message)
not_nil! not_nil!, pretty_inspect(width = 79, newline = "\n", indent = 0) : String pretty_inspect, pretty_print(pp : PrettyPrint) : Nil pretty_print, responds_to?(name : Symbol) : Bool responds_to?, tap(&) tap, to_json(io : IO) : Nil
to_json : String to_json, to_pretty_json(indent : String = " ") : String
to_pretty_json(io : IO, indent : String = " ") : Nil to_pretty_json, to_s(io : IO) : Nil
to_s : String to_s, to_yaml(io : IO) : Nil
to_yaml : String to_yaml, try(&) try, unsafe_as(type : T.class) forall T unsafe_as

### Class methods inherited from class `Object`

from_json(string_or_io, root : String)
from_json(string_or_io) from_json, from_yaml(string_or_io : String | IO) from_yaml

### Macros inherited from class `Object`

class_getter(*names, &block) class_getter, class_getter!(*names) class_getter!, class_getter?(*names, &block) class_getter?, class_property(*names, &block) class_property, class_property!(*names) class_property!, class_property?(*names, &block) class_property?, class_setter(*names) class_setter, def_clone def_clone, def_equals(*fields) def_equals, def_equals_and_hash(*fields) def_equals_and_hash, def_hash(*fields) def_hash, delegate(*methods, to object) delegate, forward_missing_to(delegate) forward_missing_to, getter(*names, &block) getter, getter!(*names) getter!, getter?(*names, &block) getter?, property(*names, &block) property, property!(*names) property!, property?(*names, &block) property?, setter(*names) setter

##
Constructor Detail

def self.additive_identity : self #
def self.multiplicative_identity : self #
def self.new(real : Number, imag : Number = 0) #
def self.new(c : Complex) #
def self.zero : Complex #

Returns the number `0` in complex form.

##
Instance Method Detail

def *(other : Complex) : Complex #

Multiplies `self` by other.

def *(other : Number) : Complex #

Multiplies `self` by other.

def +(other : Complex) : Complex #

Adds the value of `self` to other.

def +(other : Number) : Complex #

Adds the value of `self` to other.

def + : Complex #

Returns `self`.

def -(other : Complex) : Complex #

Removes the value of other from `self`.

def -(other : Number) : Complex #

Removes the value of other from `self`.

def - : Complex #

Returns the opposite of `self`.

def /(other : Complex) : Complex #

Divides `self` by other.

def /(other : Number) : Complex #

Divides `self` by other.

def ==(other : Complex) #

Determines whether `self` equals other or not.

def ==(other : Number) #

Determines whether `self` equals other or not.

def ==(other) #

Determines whether `self` equals other or not.

def abs : Float64 #

Returns the absolute value of this complex number in a number form, using the Pythagorean theorem.

    require "complex"

    Complex.new(42, 2).abs  # => 42.04759208325728
    Complex.new(-42, 2).abs # => 42.04759208325728

def abs2 : Float64 #

Returns the square of absolute value in a number form.

    require "complex"

    Complex.new(42, 2).abs2 # => 1768

def clone #
def conj : Complex #

Returns the conjugate of `self`.

    require "complex"

    Complex.new(42, 2).conj  # => 42.0 - 2.0.i
    Complex.new(42, -2).conj # => 42.0 + 2.0.i

def hash(hasher) #

See `Object#hash(hasher)`

def imag : Float64 #

Returns the imaginary part.

def inspect(io : IO) : Nil #

Writes this complex object to an io, surrounded by parentheses.

    require "complex"

    Complex.new(42, 2).inspect # => "(42.0 + 2.0i)"

def inv : Complex #

Returns the inverse of `self`.

def phase : Float64 #

Returns the phase of `self`.

def polar : Tuple(Float64, Float64) #

Returns a `Tuple` with the `#abs` value and the `#phase`.

    require "complex"

    Complex.new(42, 2).polar # => {42.047592083257278, 0.047583103276983396}

def real : Float64 #

Returns the real part.

def round(digits = 0) : Complex #

Rounds to the nearest digits.

def sign : Complex #

Returns the complex sign of `self`.

If `self` is non-zero, the returned value has the same phase as `self` and absolute value `1.0`. If `self` is zero, returns `self`.

The returned value's real and imaginary components always have the same signs as the respective components of `self`.

    require "complex"

    Complex.new(7, -24).sign        # => (0.28 - 0.96.i)
    Complex.new(1.0 / 0.0, 24).sign # => (1.0 + 0.0.i)
    Complex.new(-0.0, +0.0).sign    # => (-0.0 + 0.0.i)

def to_c #

Returns `self`.

def to_f #

See `#to_f64`.

def to_f32(*args, **options) #
def to_f32(*args, **options, &) #
def to_f64 : Float64 #

Returns the value as a `Float64` if possible (the imaginary part should be exactly zero), raises otherwise.

def to_i #

See `#to_i32`.

def to_i128(*args, **options) #
def to_i128(*args, **options, &) #
def to_i16(*args, **options) #
def to_i16(*args, **options, &) #
def to_i32(*args, **options) #
def to_i32(*args, **options, &) #
def to_i64(*args, **options) #
def to_i64(*args, **options, &) #
def to_i8(*args, **options) #
def to_i8(*args, **options, &) #
def to_s(io : IO) : Nil #

Writes this complex object to an io.

    require "complex"

    Complex.new(42, 2).to_s # => "42.0 + 2.0i"

def to_u128(*args, **options) #
def to_u128(*args, **options, &) #
def to_u16(*args, **options) #
def to_u16(*args, **options, &) #
def to_u32(*args, **options) #
def to_u32(*args, **options, &) #
def to_u64(*args, **options) #
def to_u64(*args, **options, &) #
def to_u8(*args, **options) #
def to_u8(*args, **options, &) #
def zero? : Bool #

Returns `true` if the complex number is zero. This means the real and imaginary are both zero.

    require "complex"

    Complex.new(0, 0).zero? # => true
    Complex.new(1, 0).zero? # => false
    Complex.new(0, 1).zero? # => false