A short introduction to complex numbers written primarily for students aged 14 to 19.

Choose some complex numbers and mark them by points on a graph. Multiply your numbers by i once, twice, three times, four times, ..., n times? What happens?

Track the roots of quadratic equations as you move the corresponding graphs and discover the transitions from real to complex roots.

Explore changes in solutions to cubic equations as you change the graph of the cubic polynomial. Track the real and complex roots.

Evaluate without a calculator: (5 sqrt2 + 7)^{1/3} - (5 sqrt2 - 7)^1/3}.

Make a conjecture about the curved track taken by the complex roots of a quadratic equation and use complex conjugates to prove your conjecture.