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.

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).

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.

In this exercise, complex numbers are represented as a tuple-pair containing the real and imaginary parts.

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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Complex Numbers

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