48 lines
2.0 KiB
Markdown
48 lines
2.0 KiB
Markdown
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# Complex Numbers
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Welcome to Complex Numbers on Exercism's Elixir Track.
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If you need help running the tests or submitting your code, check out `HELP.md`.
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## Instructions
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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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In this exercise, complex numbers are represented as a tuple-pair containing the real and imaginary parts.
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For example, the real number `1` is `{1, 0}`, the imaginary number `i` is `{0, 1}` and the complex number `4+3i` is `{4, 3}`.
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## Source
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### Created by
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- @jiegillet
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### Based on
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Wikipedia - https://en.wikipedia.org/wiki/Complex_number
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