# Complex Numbers in Erlang

#### Implement complex numbers.

1 | ```
exercism fetch erlang 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.

You also need to implement a `equal/2`

function. For this you can consider
two numbers as equal, when the difference of each component is less than 0.005.

## Running tests

In order to run the tests, issue the following command from the exercise directory:

For running the tests provided, `rebar3`

is used as it is the official build and
dependency management tool for erlang now. Please refer to the tracks installation
instructions on how to do that.

In order to run the tests, you can issue the following command from the exercise directory.

1 |
```
$ rebar3 eunit
``` |

### Test versioning

Each problem defines a macro `TEST_VERSION`

in the test file and
verifies that the solution defines and exports a function `test_version`

returning that same value.

To make tests pass, add the following to your solution:

The benefit of this is that reviewers can see against which test version an iteration was written if, for example, a previously posted solution does not solve the current problem or passes current tests.

## Questions?

For detailed information about the Erlang track, please refer to the help page on the Exercism site. This covers the basic information on setting up the development environment expected by the exercises.

## 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.