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.