This article introduces complex numbers, brings together into one bigger 'picture' some closely related elementary ideas like vectors and the exponential and trigonometric functions and. . . .

Challenge Level

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.

Challenge Level

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

In this article we are going to look at infinite continued fractions - continued fractions that do not terminate.

Challenge Level

Why is the modern piano tuned using an equal tempered scale and what has this got to do with logarithms?

Challenge Level

Investigate the effects of the half-lifes of the isotopes of cobalt on the mass of a mystery lump of the element.

Challenge Level

A function pyramid is a structure where each entry in the pyramid is determined by the two entries below it. Can you figure out how the pyramid is generated?

Challenge Level

Compares the size of functions f(n) for large values of n.

Challenge Level

How does the half-life of a drug affect the build up of medication in the body over time?

Challenge Level

If a sum invested gains 10% each year how long before it has doubled its value?

Challenge Level

Explore the properties of these two fascinating functions using trigonometry as a guide.

Challenge Level

Can you locate these values on this interactive logarithmic scale?

Challenge Level

In this question we push the pH formula to its theoretical limits.

Challenge Level

Investigate the mathematics behind blood buffers and derive the form of a titration curve.

Challenge Level

Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.

Challenge Level

What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.

Challenge Level

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

Challenge Level

Looking at small values of functions. Motivating the existence of the Taylor expansion.

Challenge Level

The equation a^x + b^x = 1 can be solved algebraically in special cases but in general it can only be solved by numerical methods.

Challenge Level

Is it true that a large integer m can be taken such that: 1 + 1/2 + 1/3 + ... +1/m > 100 ?

Challenge Level

This problem explores the biology behind Rudolph's glowing red nose.