Complex Numbers

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

You will have to implement your own equality operator for the ComplexNumber object. This will pose the challenge of comparing two floating point numbers. It might be useful to use the method isApproximatelyEqual(to:absoluteTolerance:) which can be found in the Numerics library. Due to preformance issues will you only have access to the RealModule part. To import it simply write: import RealModule at the top of the file. A given tolerance of 0.00001 should be enough to pass the tests. The library is already imported in the project so it is just to import it in your file.

You are also free to implement your own method to compare the two complex numbers.

---

Source

Wikipedia