Notes on using ComplexRoots.xls
This spreadsheet allows students to explore the complex roots of a number geometrically leading to a discovery of, or better understanding of, a method of calculating the roots.
A number of methods are possible so some intervention by the teacher to clarify and encourage a consistent method is probably useful at the end of Part1 and again after the first few questions of Part2.
Background:
Theory of complex numbers including their representation in polar form and/or exponential form (rCisq or reiq) and geometrical interpretation using an Argand diagram.
The ability to convert from polar or exponential form to Cartesian form.
Questions for students.
Part 1: The roots of unity (unity means 1).
Set the real part to 1 and the imaginary part to zero. Vary the number of roots from 2 to 8.
Write down the four 4th roots of unity in Cartesian form.
Now look at the sixth roots. Write down the polar coordinates of each root.
Write down the 6 roots as complex numbers in polar or exponential form.
Write down the six roots in Cartesian (x + iy) form. Use the spread sheet to check your answers (If you click on the point that represents a root its coordinates will be shown).
Repeat this process to find the cube roots and 5th of unity in Cartesian form.
Part 2: The roots of other complex numbers.
Set the real part to 4 and the imaginary part to zero. Find the sixth roots of 4 in Cartesian form (hint: think about how you could use work you have already done).
Set the real part to -4 and the imaginary part to zero. Find the fifth roots of -4 in Cartesian form.
Set the real part to zero and the imaginary part to 4. Find the cube roots of 4i in Cartesian form.
Set the real part to zero and the imaginary part to -4. Find the fifth roots of –4i in Cartesian form.
Set the real part to 1 and the imaginary part to 1. Find the sixth roots of 1+ i in Cartesian form.
Set the real part to 6 and the imaginary part to -8. Find the seventh roots of 6–8i in Cartesian form.
Set your own question.