Week 1: Chance and chaos

Introduction

Mathematics has long been associated with certainty. For example, the equation 2 + 2 = 4 has been a synonym for evident truth since at least the middle of the 16th century, whereas the equation 2 + 2 = 5 has become a synonym for the opposite (most notably in George Orwell’s dystopian novel Nineteen Eighty-Four). But, as you will see, there are parts of mathematics where certainty is a commodity in short supply.

The two mathematical ideas you’ll explore this week are chance and chaos – both of which will be demonstrated through the rolling of dice.

Here’s Marcus to introduce this week’s topic.

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Transcript: Video 1 Introduction

MARCUS DU SAUTOY

If I roll a dice, can I know how it will land before it comes to rest? This week, we’re going to tackle the challenge of looking into the future. Can we predict the future before it becomes the present? Such an ability gives anyone with such knowledge unrivalled power! We begin with the mathematics of probability – an amazing tool that allows us to navigate the uncertainties of everything from games of chance to the behaviour of particles of gas. But perhaps we don’t need to rely on probabilities. Post-Newton, the equations of mathematics and physics gave us the hope that we could know the destiny of the universe in all its complexity. But as you will find out in this week, the mathematics of chaos theory – discovered at the beginning of the twentieth century – places impenetrable limitations on what we can know about the future. The devil, it appears, is in the detail.

End transcript: Video 1 Introduction

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Video 1 Introduction

By the end of this week, you should be able to:

• understand the basic concept of probability
• explain the use of probability in the context of rolling dice
• understand the notion of mathematical chaos, and what is meant by sensitivity to initial conditions
• explain the problem of the stability of the solar system, and how it can be modelled mathematically
• appreciate the difficulties inherent in predicting chaotic systems, like weather forecasting.

The Open University would really appreciate a few minutes of your time to tell us about yourself and your expectations for the course before you begin, in our optional start-of-course survey [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] . Participation will be completely confidential and we will not pass on your details to others.