Complex numbers

problem
Favourite

Three by One

Age

16 to 18

Challenge level

1 out of 3

There are many different methods to solve this geometrical problem - how many can you find?
$Complex partial fractions$

problem
Favourite

Complex Partial Fractions

Age

16 to 18

Challenge level

2 out of 3

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.
problem

Roots and Coefficients

Age

16 to 18

Challenge level

1 out of 3

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?
problem

Sextet

Age

16 to 18

Challenge level

1 out of 3

Investigate x to the power n plus 1 over x to the power n when x plus 1 over x equals 1.
problem

Complex Rotations

Age

16 to 18

Challenge level

2 out of 3

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?
problem

Footprints

Age

16 to 18

Challenge level

2 out of 3

Make a footprint pattern using only reflections.
problem

Pumping the Power

Age

16 to 18

Challenge level

2 out of 3

What is an AC voltage? How much power does an AC power source supply?
problem

Complex Sine

Age

16 to 18

Challenge level

3 out of 3

Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.
problem

Impedance Can Be Complex!

Age

16 to 18

Challenge level

3 out of 3

Put your complex numbers and calculus to the test with this impedance calculation.
problem

Opening the Door

Age

14 to 18

Challenge level

1 out of 3

What happens when we add together two complex numbers?