# Complex Numbers in Haskell

#### Implement complex numbers.

1 | ```
exercism fetch haskell complex-numbers
``` |

# Complex Numbers

A complex number is a number in the form `a + b * i`

where `a`

and `b`

are real and `i`

satisfies `i^2 = -1`

.

`a`

is called the real part and `b`

is called the imaginary part of `z`

.
The conjugate of the number `a + b * i`

is the number `a - b * i`

.
The absolute value of a complex number `z = a + b * i`

is a real number `|z| = sqrt(a^2 + b^2)`

. The square of the absolute value `|z|^2`

is the result of multiplication of `z`

by its complex conjugate.

The sum/difference of two complex numbers involves adding/subtracting their real and imaginary parts separately:
`(a + i * b) + (c + i * d) = (a + c) + (b + d) * i`

,
`(a + i * b) - (c + i * d) = (a - c) + (b - d) * i`

.

Multiplication result is by definition
`(a + i * b) * (c + i * d) = (a * c - b * d) + (b * c + a * d) * i`

.

The reciprocal of a non-zero complex number is
`1 / (a + i * b) = a/(a^2 + b^2) - b/(a^2 + b^2) * i`

.

Dividing a complex number `a + i * b`

by another `c + i * d`

gives:
`(a + i * b) / (c + i * d) = (a * c + b * d)/(c^2 + d^2) + (b * c - a * d)/(c^2 + d^2) * i`

.

Exponent of a complex number can be expressed as
`exp(a + i * b) = exp(a) * exp(i * b)`

,
and the last term is given by Euler's formula `exp(i * b) = cos(b) + i * sin(b)`

.

Implement the following operations:

- addition, subtraction, multiplication and division of two complex numbers,
- conjugate, absolute value, exponent of a given complex number.

Assume the programming language you are using does not have an implementation of complex numbers.

## Getting Started

For installation and learning resources, refer to the exercism help page.

## Running the tests

To run the test suite, execute the following command:

1 |
```
stack test
``` |

#### If you get an error message like this...

1 |
```
No .cabal file found in directory
``` |

You are probably running an old stack version and need to upgrade it.

#### Otherwise, if you get an error message like this...

1 2 |
No compiler found, expected minor version match with... Try running "stack setup" to install the correct GHC... |

Just do as it says and it will download and install the correct compiler version:

1 |
```
stack setup
``` |

## Running *GHCi*

If you want to play with your solution in GHCi, just run the command:

1 |
```
stack ghci
``` |

## Feedback, Issues, Pull Requests

The exercism/haskell repository on GitHub is the home for all of the Haskell exercises.

If you have feedback about an exercise, or want to help implementing a new one, head over there and create an issue. We'll do our best to help you!

## Source

Wikipedia https://en.wikipedia.org/wiki/Complex_number

## Submitting Incomplete Solutions

It's possible to submit an incomplete solution so you can see how others have completed the exercise.