Solving problems with complex numbers

Solving Problems with Complex Numbers

De Moivre’s Theorem

De Moivre’s Theorem states that for any complex number written in polar form, r(cos θ + isin θ), and any real number ‘n’, (cos θ + isin θ)^n = cos nθ + isin nθ.
You can use De Moivre’s theorem to raise complex numbers to a power. For example, to calculate (cos π/6 + isin π/6)^4, use the theorem to find cos (4π/6) + isin (4π/6).

Roots of Unity

The nth roots of unity are complex numbers that satisfy the equation z^n = 1. They form a regular polygon with n vertices on the unit circle in the complex plane.
You can calculate nth roots of unity by solving the equation cos 2kπ/n + i sin 2kπ/n = 1 for each ‘k’ from 0 to n-1.

Argand Diagram

An Argand Diagram is a plot on the complex number plane. It’s a graph where the x-axis is the real part of the number and the y-axis is the imaginary part of the number.
To plot a complex number on an Argand Diagram, mark a point with coordinates as the real and imaginary components of the number.
You can use an Argand Diagram to visually add or subtract complex numbers, as well as to see the product or quotient of two complex numbers.

Polar Form and Euler’s Formula

A complex number can be written either in Cartesian form (a + bi) or in Polar Form (r(cos θ + isin θ)), where ‘r’ is the magnitude of the number and ‘θ’ is the angle it forms with the positive real axis - also called the argument of the complex number.
Euler’s Formula states that for any real number ‘θ’, e^(iθ) = cos θ + isin θ. This is a useful formula for converting between Cartesian and Polar form.

Complex Roots

A complex number has ‘n’ distinct nth roots. They can be found by halving the angle formed on the Argand Diagram, and multiplying the magnitude by the nth root of the absolute value.
For example, the square roots of a complex number z = r(cos θ + isin θ) are ± √r (cos θ/2 + isin θ/2).

Solving Quadratic Equations

Solving a quadratic equation with complex coefficients involves the same process as with real coefficients, but you may end with solutions that are complex numbers.
If the discriminant of the quadratic equation is negative, this indicates that the roots are complex numbers. These can be found using the quadratic formula, substituting the square root of a negative number with ‘i’ times the square root of the corresponding positive number.

General Tips

For all calculations, remember that i^2 = -1.
Be comfortable with converting between different forms of a complex number, applying De Moivre’s theorem, and using Euler’s Formula.
Make sure to fully simplify your answers and, if required, present them in the correct form (Cartesian or Polar).