# struct Complex

## 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.

``````require "complex"

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

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

complex.cr

## Constructor Detail

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

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 #

def to_c #

Returns `self`.

def to_f #

See `#to_f64`.

def to_f32 : Float32 #

Returns the value as a `Float32` if possible (the imaginary part should be exactly zero), raises otherwise.

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_i16(*args, **options) #

def to_i16(*args, **options, &) #

def to_i32(*args, **options) #

def to_i32(*args, **options, &) #

def to_i64 : Int64 #

Returns the value as an `Int64` if possible (the imaginary part should be exactly zero), raises otherwise.

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_u16(*args, **options) #

def to_u16(*args, **options, &) #

def to_u32(*args, **options) #

def to_u32(*args, **options, &) #

def to_u64 : UInt64 #

Returns the value as an `UInt64` if possible (the imaginary part should be exactly zero), raises otherwise.

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``````