Instructions

A complex number is expressed in the form z = a + b * i, where:

• a is the real part (a real number),

• b is the imaginary part (also a real number), and

• i is the imaginary unit satisfying i^2 = -1.

Operations on Complex Numbers

Conjugate

The conjugate of the complex number z = a + b * i is given by:

zc = a - b * i

Absolute Value

The absolute value (or modulus) of z is defined as:

|z| = sqrt(a^2 + b^2)

The square of the absolute value is computed as the product of z and its conjugate zc:

|z|^2 = z * zc = a^2 + b^2

Addition

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:

z1 + z2 = (a + b * i) + (c + d * i)
        = (a + c) + (b + d) * i

Subtraction

The difference of two complex numbers is obtained by subtracting their respective parts:

z1 - z2 = (a + b * i) - (c + d * i)
        = (a - c) + (b - d) * i

Multiplication

The product of two complex numbers is defined as:

z1 * z2 = (a + b * i) * (c + d * i)
        = (a * c - b * d) + (b * c + a * d) * i

Reciprocal

The reciprocal of a non-zero complex number is given by:

1 / z = 1 / (a + b * i)
      = a / (a^2 + b^2) - b / (a^2 + b^2) * i

Division

The division of one complex number by another is given by:

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

Exponentiation

Raising e (the base of the natural logarithm) to a complex exponent can be expressed using Euler's formula:

e^(a + b * i) = e^a * e^(b * i)
              = e^a * (cos(b) + i * sin(b))

Implementation Requirements

Given that you should not use built-in support for complex numbers, implement the following operations:

• 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