# 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``````

complex.cr

## Instance Method Summary

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### Instance methods inherited from class `Object`

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## Constructor Detail

def self.new(real : Number, imag : Number) #

def self.zero : Complex #

Returns the number `0` in complex form.

## Instance Method Detail

def *(other : Number) #

Multiplies `self` by other.

def *(other : Complex) #

Multiplies `self` by other.

def +(other : Number) #

Adds the value of `self` to other.

def +(other : Complex) #

Adds the value of `self` to other.

def - #

Returns the opposite of self.

def -(other : Number) #

Removes the value from other to self.

def -(other : Complex) #

Removes the value from other to self.

def /(other : Number) #

Divides `self` by other.

def /(other : 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 #

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

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

def abs2 #

Returns the square of absolute value in a number form.

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

def clone #

def conj #

Returns the conjugate of self.

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

def exp #

Calculates the exp of self.

``Complex.new(4, 2).exp # => -22.720847417619233 + 49.645957334580565i``

def hash(hasher) #

See `Object#hash(hasher)`

def imag : Float64 #

Returns the image part of self.

def inspect(io : IO) #

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

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

def inv #

Returns the inverse of self.

def log #

Calculates the log of self.

def log10 #

Calculates the log10 of self.

def log2 #

Calculates the log2 of self.

def phase #

Returns the phase of self.

def polar #

Returns a tuple with the abs value and the phase.

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

def real : Float64 #

Returns the real part of self.

def sign #

def sqrt #

`Complex#sqrt` was inspired by the following blog post of Pavel Panchekha on floating point precision.

def to_s(io : IO) #

Write this complex object to an io.

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

def zero? : Bool #