# Complex Numbers in C#

#### Implement complex numbers.

1 | ```
exercism fetch csharp 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.

### Submitting Exercises

Note that, when trying to submit an exercise, make sure the exercise file that you're submitting is in the `exercism/csharp/<exerciseName>`

directory.

For example, if you're submitting `bob.cs`

for the Bob exercise, the submit command would be something like `exercism submit <path_to_exercism_dir>/csharp/bob/bob.cs`

.

## 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.