"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
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Yes, you can access Understanding Numbers: Simplify life's mathematics. Decode the world around you. by Marianne Freiberger,Rachel Thomas in PDF and/or ePUB format, as well as other popular books in Mathematics & Game Theory. We have over one million books available in our catalogue for you to explore.
03TESTING TREATMENTS Has it worked? How can we be sure?
04STATISTICALLY SPEAKING Making sense of health statistics
Today all reputable medical research uses the scientific method. That’s good to know, but the resulting jungle of numbers and percentages can be hard to penetrate, let alone interpret.
Many people still instinctively think that bed rest is the best way to treat back pain. Indeed this was the treatment recommended by doctors until the mid-1990s. Then, in 1995, a team from Finland used a randomized controlled trial (RCT), the gold standard in evidence-based medicine, to turn this thinking on its head. The clear statistical arguments and rigorous mathematical design of RCTs provide the best known way to rule out bias and unknown factors to answer the question: what is the best treatment? Now patients with back pain are advised to stay active.
Today all reputable medical research uses the scientific method. That’s good to know, but the resulting jungle of numbers and percentages can be hard to penetrate, let alone interpret. What does a 5% overall risk of a disease mean for you, personally? How do you know a treatment really works? And should you spend money on costly alternative remedies that aren’t supported by your healthcare system or insurance?
Newspapers and media outlets have an important, and not always positive, role to play in this context. Health scares and miracle cures make great headlines, and they are easy to come by if you’re happy to massage the numbers to suit your purpose. Pharmaceutical companies also have an incentive to juggle the stats to suit their message. Even those in the medical profession don’t appear always to see through the number jungle.
So what is the poor lay person to do? In this chapter we look at four situations in which very personal healthcare decisions depend on numbers and percentages. The results aren’t always intuitive, so it’s good to know where those numbers come from and what exactly they mean.
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!
