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exercises/practice/complex-numbers/.docs/instructions.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`. | ||
A **complex number** is expressed in the form `z = a + b * i`, where: | ||
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`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. | ||
- `a` is the **real part** (a real number), | ||
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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`. | ||
- `b` is the **imaginary part** (also a real number), and | ||
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Multiplication result is by definition | ||
`(a + i * b) * (c + i * d) = (a * c - b * d) + (b * c + a * d) * i`. | ||
- `i` is the **imaginary unit** satisfying `i^2 = -1`. | ||
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The reciprocal of a non-zero complex number is | ||
`1 / (a + i * b) = a/(a^2 + b^2) - b/(a^2 + b^2) * i`. | ||
## Operations on Complex Numbers | ||
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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`. | ||
### Conjugate | ||
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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)`. | ||
The conjugate of the complex number `z = a + b * i` is given by: | ||
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Implement the following operations: | ||
```text | ||
zc = a - b * i | ||
``` | ||
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- addition, subtraction, multiplication and division of two complex numbers, | ||
- conjugate, absolute value, exponent of a given complex number. | ||
### Absolute Value | ||
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Assume the programming language you are using does not have an implementation of complex numbers. | ||
The absolute value (or modulus) of `z` is defined as: | ||
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```text | ||
|z| = sqrt(a^2 + b^2) | ||
``` | ||
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The square of the absolute value is computed as the product of `z` and its conjugate `zc`: | ||
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```text | ||
|z|^2 = z * zc = a^2 + b^2 | ||
``` | ||
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### Addition | ||
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The sum of two complex numbers `z1 = a + b * i` and `z2 = c + d * i` is computed by adding their real and imaginary parts separately: | ||
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```text | ||
z1 + z2 = (a + b * i) + (c + d * i) | ||
= (a + c) + (b + d) * i | ||
``` | ||
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### Subtraction | ||
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The difference of two complex numbers is obtained by subtracting their respective parts: | ||
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```text | ||
z1 - z2 = (a + b * i) - (c + d * i) | ||
= (a - c) + (b - d) * i | ||
``` | ||
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### Multiplication | ||
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The product of two complex numbers is defined as: | ||
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```text | ||
z1 * z2 = (a + b * i) * (c + d * i) | ||
= (a * c - b * d) + (b * c + a * d) * i | ||
``` | ||
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### Reciprocal | ||
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The reciprocal of a non-zero complex number is given by: | ||
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```text | ||
1 / z = 1 / (a + b * i) | ||
= a / (a^2 + b^2) - b / (a^2 + b^2) * i | ||
``` | ||
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### Division | ||
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The division of one complex number by another is given by: | ||
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```text | ||
z1 / z2 = z1 * (1 / z2) | ||
= (a + b * i) / (c + d * i) | ||
= (a * c + b * d) / (c^2 + d^2) + (b * c - a * d) / (c^2 + d^2) * i | ||
``` | ||
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### Exponentiation | ||
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Raising _e_ (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula: | ||
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```text | ||
e^(a + b * i) = e^a * e^(b * i) | ||
= e^a * (cos(b) + i * sin(b)) | ||
``` | ||
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## Implementation Requirements | ||
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Given that you should not use built-in support for complex numbers, implement the following operations: | ||
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- **addition** of two complex numbers | ||
- **subtraction** of two complex numbers | ||
- **multiplication** of two complex numbers | ||
- **division** of two complex numbers | ||
- **conjugate** of a complex number | ||
- **absolute value** of a complex number | ||
- **exponentiation** of _e_ (the base of the natural logarithm) to a complex number |
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