68 lines
2.7 KiB
Markdown
68 lines
2.7 KiB
Markdown
# Complex Numbers
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A complex number is a number in the form `a + b * i` where `a` and `b` are real and `i` satisfies `i^2 = -1`.
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`a` is called the real part and `b` is called the imaginary part of `z`.
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The conjugate of the number `a + b * i` is the number `a - b * i`.
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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.
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The sum/difference of two complex numbers involves adding/subtracting their real and imaginary parts separately:
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`(a + i * b) + (c + i * d) = (a + c) + (b + d) * i`,
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`(a + i * b) - (c + i * d) = (a - c) + (b - d) * i`.
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Multiplication result is by definition
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`(a + i * b) * (c + i * d) = (a * c - b * d) + (b * c + a * d) * i`.
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The reciprocal of a non-zero complex number is
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`1 / (a + i * b) = a/(a^2 + b^2) - b/(a^2 + b^2) * i`.
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Dividing a complex number `a + i * b` by another `c + i * d` gives:
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`(a + i * b) / (c + i * d) = (a * c + b * d)/(c^2 + d^2) + (b * c - a * d)/(c^2 + d^2) * i`.
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Raising e to a complex exponent can be expressed as `e^(a + i * b) = e^a * e^(i * b)`, the last term of which is given by Euler's formula `e^(i * b) = cos(b) + i * sin(b)`.
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Implement the following operations:
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- addition, subtraction, multiplication and division of two complex numbers,
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- conjugate, absolute value, exponent of a given complex number.
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Assume the programming language you are using does not have an implementation of complex numbers.
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## Getting Started
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Make sure you have read the "Guides" section of the
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[C track][c-track] on the Exercism site. This covers
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the basic information on setting up the development environment expected
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by the exercises.
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## Passing the Tests
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Get the first test compiling, linking and passing by following the [three
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rules of test-driven development][3-tdd-rules].
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The included makefile can be used to create and run the tests using the `test`
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task.
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make test
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Create just the functions you need to satisfy any compiler errors and get the
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test to fail. Then write just enough code to get the test to pass. Once you've
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done that, move onto the next test.
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As you progress through the tests, take the time to refactor your
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implementation for readability and expressiveness and then go on to the next
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test.
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Try to use standard C99 facilities in preference to writing your own
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low-level algorithms or facilities by hand.
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## Source
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Wikipedia [https://en.wikipedia.org/wiki/Complex_number](https://en.wikipedia.org/wiki/Complex_number)
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## Submitting Incomplete Solutions
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It's possible to submit an incomplete solution so you can see how others have completed the exercise.
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[c-track]: https://exercism.io/my/tracks/c
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[3-tdd-rules]: http://butunclebob.com/ArticleS.UncleBob.TheThreeRulesOfTdd
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