Understanding Numbers: Simplify life's mathematics. Decode the world around you.: 20 thought-provoking lessons (BUILD+BECOME) - Softcover

Freiberger, Marianne; Thomas, Rachel

Softcover

ISBN 10: 1781318158 ISBN 13: 9781781318157

Verlag: Quarto Publishing Plc, 2019

Inhaltsangabe

"Simple, clear explanations of twenty ways in which mathematics helps us to understand practical issues of everyday life. Suitable for teenagers and adults, and beautifully produced. Highly recommended!" - Professor Ian Stewart, bestselling author of The Cabinet of Mathematical Curiosities

Mathematics is an indispensable tool for life . From the systems that underpin our newsfeeds, through to the data analysis that informs our health and financial decisions, to the algorithms that power how we search online - mathematics is at the heart of how our modern world functions .

In 20 dip-in lessons, Understanding Numbers explains how and why mathematics fuels your world and arms you with the knowledge to make wiser choices in all areas of your life .

- Make sense of health statistics
- Understand the mathematics behind political voting systems
- Get to grips with how online search tools work
- Discover how mathematics can create the perfect city

Über diesen Titel

Über die Autorin bzw. den Autor

Rachel Thomas and Marianne Freiberger are the editors of Plus magazine, which publishes articles from the world's top mathematicians and science writers on topics as diverse as art, medicine, cosmology and sport.  Before joining Plus in 2001, Rachel was a math consultant in Australia working for government and industry. She recently edited the Gazette of the Australian Mathematical Society and has developed and taught science writing workshops for graduate students. She also created mathematical walking tours of London and Oxford and a virtual mathematical tour guide of the world for Maths in the City. Rachel obtained her MSc in Semigroup Theory in 1998 from the University of Western Australia. Marianne joined Plus in 2005 after completing a PhD and then a three year postdoc at Queen Mary, University of London. As a researcher she worked in complex dynamics and held various teaching engagements. In the world of math communication she has been Editor-in-Chief of the Mathscareers website and sometimes gives presentations to mathematicians about how to communicate their work to a wider audience, and to journalists about how to deal with math in the media.

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Understanding Numbers

Simplify Life's Mathematics. Decode the World Around You.

By Marianne Freiberger, Rachel Thomas

The Quarto Group

Copyright © 2019 Marianne Freiberger and Rachel Thomas
All rights reserved.

ISBN: 978-1-78131-815-7

Contents

INTRODUCTION, 
HOW TO USE THIS BOOK, 
01 HEALTH, 
01 Preventing disease, 
02 Detecting disease, 
03 Testing treatments, 
04 Statistically speaking, 
Toolkit 01–04, 
Further Learning, 
02 ENVIRONMENT, 
05 Architecture, 
06 Modelling traffic, 
07 Predicting the unpredictable, 
08 Changing climates, 
Toolkit 05–08, 
Further Learning, 
03 SOCIETY, 
09 Everyday averages, 
10 Voting, 
11 Evidence, 
12 Debts and savings, 
Toolkit 09–12, 
Further Learning, 
04 RELATIONSHIPS, 
13 The evolution of human kindness, 
14 How we search, 
15 The statistics of sex, 
16 Making relative sense, 
Toolkit 13–16, 
Further Learning, 
05 COMMUNICATION, 
17 Networks, 
18 Staying secure, 
19 Big data, 
20 Knowing where you are, 
Toolkit 17–20, 
Further Learning, 
Epilogue, 

CHAPTER 1

PREVENTING DISEASE

Infectious diseases are frightening. The 2014 Ebola outbreak claimed over 11,000 lives. Over 12,000 people died from swine flu in the US during the 2009 pandemic. The total death toll of AIDS to date is about 35 million.

It's not hard to see why infectious diseases can spread so quickly. Suppose that an infected person goes on to infect two other people during the course of their disease — not an unrealistic assumption if you consider the coughing and spluttering that goes on on public transport. A single infected person will infect another two people, giving a total of 1 + 2 = 3 infected people. The two newly infected people will infect another two each, giving a total of 1 + 2 + 4 = 7 infected people. The four newly infected people then infect another two each, giving a total of 1 + 2 + 4 + 8 = 15 people, and so on.

Continuing in this vein, you see that the number of infected people grows very fast. In fact, it grows exponentially. If each infected person infects their two victims within the first day of catching the disease, then it will only take 26 days to infect a population larger than that of the UK. And that's starting with a single infected individual. (You might want to work out for yourself that the number of infected people after n days would be 20 + 21 + 22 + ... + 2n-1 Luckily, you don't need to tap this long sum into a calculator to get the result: a sum of this form is always equal to 2n - 1).

The number 2 obviously plays an important role in this example. If an infected person infected more than two people a day, then the disease would spread a lot quicker. And if an infected person infected fewer than two people a day, the disease would spread more slowly. In fact, it turns out that the number 1 is the watershed in this context. If an infected person infects, on average, more than one other person, the number of sick people will grow beyond any bound, as long as nothing bars the path of the disease. If, on the other hand, an infected person infects fewer than one other person on average, the spread will eventually come to a halt of its own accord.

Epidemiologists, those tasked with analyzing the spread of disease, have a name for the number of individuals that are, on average, infected by a person who has a particular disease, assuming that all the population is susceptible to catch the disease: it's called the basic reproduction number of the disease. Looking up basic reproduction numbers of common diseases gives you a good idea of how dangerous they are. The basic reproduction number of Ebola is between 1.5 and 2.5. For AIDS it lies somewhere between 2 and 5. For influenza (the 1918 epidemic strain) it's between 2 and 3. And for measles it's between 12 and 18!

What is to be done in the face of such ferocious exponential growth? Epidemiologists use complex mathematical models to see how a disease might spread. Importantly these models can be used to test the effect an intervention, such as vaccination or perhaps a travel ban, might have. The results don't always chime with intuition and can lead to outraged headlines. But rest assured: the advice epidemiologists come up with is based on thorough mathematical investigation.

WHY DOES VACCINATION WORK?

To give an idea of how even some basic maths can help, let's turn to the sometimes contentious subject of vaccination. The idea behind vaccination is to make people immune to a disease by injecting them with a pathogen, but it isn't without problems. It can be difficult and costly to get hold of everyone in a population; some people may be put at risk because of underlying health problems; and others may flatly refuse to be vaccinated. Luckily, though, you don't need to vaccinate everybody in a population to ensure the disease eventually fizzles out. Here's a short calculation to show why.

Suppose you have vaccinated a proportion p of the community, so these people are now immune to the disease. This means a proportion 1 - p is still susceptible to catching the disease. The basic reproduction number, call it R, gives the number of people a sick person infects, on average, in a totally susceptible population. Since after vaccinating only a proportion, 1 - p of the population, are still susceptible, the reproduction number is now only a proportion, 1 - p, of what it was in a totally susceptible population: the basic reproduction number of R turns into an effective reproduction number of R × (1 - p). In order for the disease to eventually fizzle out, we'd like the effective reproduction number to be less than 1, so

R × (1 - p) < 1

A bit of rearranging will show that p, the proportion vaccinated, must therefore be at least 1-1/R:

1 - 1/R < p

In other words, to ensure the disease dies out, you need to vaccinate a proportion of at least 1 - 1/R of the population.

For a basic reproduction number of 2, you only need to vaccinate 1 - ½ = ½ of the population. If R is 3, the upper bound for influenza, you should vaccinate 1 - 1/3 = 2/3 of the population. Importantly, our calculation shows that not only vaccinated people benefit from vaccination. People who haven't been vaccinated do as well, because their overall risk of catching the disease has decreased. In this way people who cannot have a vaccination for whatever reason can still be protected. Vaccination isn't just for you, it's for everyone!

CHAPTER 2

DETECTING DISEASE

It seems obvious that knowledge is better than ignorance, that it is better to find out if you are likely to get a disease, rather than wait until you experience symptoms. But the example of screening for diseases illustrates that this isn't always the case. Screening programmes have both benefits (saving lives) and harms (which we will discuss below) – and these have to be balanced using careful statistical analysis. This is why screening programmes are so carefully researched before they are approved, and rigorously monitored to ensure the balance between benefit and harm is preserved.

The first thing to remember is that screening is not the same as diagnosis. Screening programmes check for well-understood markers that are a clear indication that a person is...

Details zur Ausgabe

• Verlag: Quarto Publishing Plc
• Erscheinungsdatum: 2019
• Sprache: Englisch
• ISBN 10: 1781318158
• ISBN 13: 9781781318157
• Einband: Taschenbuch
• Anzahl der Seiten: 160
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: WordUnited GmbH; distribution@wordunited.com; ; Theodorstrasse 105; DUSSELDORF; 40472; Deutschland