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Library of Congress Cataloging-in-Publication Data

Cann, Alan.
Maths in biology/Alan J. Cann.
p. cm.
Includes bibliographical references (p.).
ISBN 0-471-49834-3 (cased) — ISBN 0-471-49835-1 (pbk.)
1. Biomathematics. I. Title.
QH323.5.C363 2002
570′.1′51 — dc21
2002028068

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0 471 49834 3 Hardback
0 471 49835 1 Paperback

Preface

This book arose from my own need for a text that I would be happy to recommend to my students. Although there is no particular shortage of volumes claiming to help biologists with mathematics, all those I am familiar with have one of two flaws. Either they are written by well-meaning mathematicians and pay scant attention to biology, or they are not appropriate for the level at which most of the problems lie – new college students who do not have much confidence in approaching mathematical problems, in spite of extensive prior exposure to mathematics in school.

I make no claim to be a mathematical genius. Indeed, I believe my struggle to explain the material in an easily accessible form is one of the strengths of this book, bringing me closer to the students I am trying to communicate with. I reject any charges of ‘dumbing down’ – anyone who has ever tried to help a panic-stricken student in the grip of maths phobia will know that a calming but not patronizing voice is an essential attribute in these circumstances. Throughout, my intention is to provide a highly accessible text for students who, with or without formal mathematics qualifications, are frightened by the perceived ‘difficulty’ of mathematics and unwilling, inept or inexperienced in applying mathematical skills. To accommodate these students, many of whom opt to undertake studies in biology in the belief (conscious or unconscious) that this is a way of pursuing a scientific career while avoiding mathematics, the ethos of the book is consciously informal and intended to be confidence-building.

The maths in this volume has been checked vigorously, but I cannot guarantee that the text is entirely free of numerical errors. In addition there may be some passages where the subject matter is not expressed as clearly as I would have hoped. I rely on readers to point these out to me – as I am sure they will.

Alan J. Cann
University of Leicester, UK.
alan.cann@le.ac.uk

1

Maths in Biology

Mathematics, from the Greek, manthano, ‘to learn’

Some people opt to undertake studies in biology in the belief (conscious or unconscious) that this is a way of pursuing a scientific career while avoiding maths. This book is designed to be accessible to students who, with or without formal mathematical qualifications, are frightened by the perceived ‘difficulty’ of maths and hence are unwilling to apply what mathematical skills they might have. Have you ever noticed when you have been taught how to solve a mathematical problem, that you still don’t know why you need to do a particular step? This is the root of many problems with maths, so this book will try explain the why of maths, in addition to the how. Sometimes, these explanations may seem unnecessary, but I urge you not to skip them – understanding why you need to do something is the key to remembering how to do it. The intention is to be informal and confidence-building to ensure that all readers will gain a general appreciation of basic mathematical, statistical and data handling techniques appropriate to biology. I will try to explain the jargon which confuses the non-numerically minded.

In subsequent chapters, we will look at manipulating numbers, units and conversions, molarities and dilutions, areas and volumes, exponents and logs and statistics. However, the basic advice in this chapter is really the most important part of the book, so please keep reading.

1.1. What can go wrong?

It is easy to make mistakes with maths. One answer looks much like another, so how can you tell if it is right or not? Look at some examples of the sort of mistakes it is all too easy to make. Everyone knows that numbers are meaningless without the units which define what they mean (more of this in Chapter 3). Even if we avoid the elementary mistake of forgetting this and giving an answer of ‘33.6’ (33.6 what? volts? metres? frogs?), things are not always simple. Consider the following questions:

An aquarium has internal dimensions of 100 * 45 * 45 cm. What is its volume in litres?

This is fairly simple. Calculate the volume in cubic centimetres then convert to litres. 1 litre = 1000 cm³ so divide by 1000:

However, life is not always that simple. If the same calculation is given in a different way, it is not as easy to answer:

An aquarium has internal dimensions of 39 * 18 * 18 inches. What is its volume in litres?

This is harder because the units in which the data is given and those in which the answer is required are from different systems of measurement. In real life, this happens all too frequently.

To avoid mistakes we need to convert the units so that they are consistent throughout. However, this means there are two ways to do the calculation:

In general, the best method is the one which requires fewer conversions and fewer steps (b). However, this depends on what conversion factors you have to hand – if you have to calculate a conversion factor from cubic inches to litres, it may be better to use (a). Note that the accuracy of conversions from one unit to another depends on the number of significant figures used. Significant figures are: ‘the minimum number of digits needed to write a given value (in scientific notation) without loss of accuracy’. The most significant figure is the left-most digit, the digit which is known most precisely. The least significant figure is the right-most digit, the digit which is known least precisely.

Significant figures are important when reporting scientific data because they give the reader an idea of how accurately data has been measured. Here are the rules:

Using the appropriate number of significant figures in calculations is important, since it prevents loss of accuracy. However, computers and calculators frequently give ridiculously large numbers of significant figures – way beyond the accuracy with which a measurement could be made. For this reason, and for ease of performing calculations (particularly when estimating, see below), it is often necessary to ‘round off’ the number of significant figures in a number. Note that this is ‘rounding off’, not ‘rounding up’, which leads to inaccuracy and errors. ‘Rounding up’ a digit which is followed by a 5 (e.g. 5.45 becomes 5.5) introduces errors in calculations because the digits one, two, three and four are ‘rounded down’ (four possibilities) but the digits five, six, seven, eight and nine are all ‘rounded up’ (five possibilities). ‘Rounding off’ avoids this error:

Examples

Round 123.456789 to three significant figures = 123 (rule a: round the number off)

Round 123.456789 to five significant figures = 123.46 (rule b: round the last digit up)

Round 123.456789 to four significant figures = 123.4 (rule c: make the last digit even)

Round 123.356789 to four significant figures = 123.4 (rule c: make the last digit even)

Round 123.456799 to eight significant figures = 123.45680 (note that 9 rounds up to 10, not down to 0).

1.2. Estimating

Calculators and computers spit out numbers at the press of a key, but are the answers right? Estimating is a vital skill if you wish to become confident and proficient with numbers. However, estimating and calculating are not the same thing and it is important to understand the difference. Where calculation attempts to produce the most accurate answer possible (within limits of experimental error), estimation deliberately avoids accuracy in order to simplify working out the answer.

If you have used a computer or calculator to calculate an answer, it is best to work out the estimate in your head or on a scrap of paper in order to check for any errors you may have introduced by using the machine. This is why estimation involves simplifying the calculation – an estimate is not meant to be accurate, but it should be easy to calculate and a reliable check. Aside from performing the calculation, estimating is the most important part of ensuring that answers to problems are correct. Some calculations in biology are complex and involve many steps (Chapter 5). Estimating is particularly important here to ensure the answer looks sensible. Manipulation of numbers and equations may not give a numerical answer but a mathematical term (e.g. 3y – 2). Here, the trick is to check your answer by substituting back into the original equation to see if it works (Chapter 2).

1.3. How to use this book

If you have been told to use this book as part of a particular course, you had better follow the instructions given by whoever is running the course. Other than that, you can use this book however you want. Some people may want to read though all (or most) of the chapters in order. Others may skip sections and dip into chapters that they feel they need. Either way is fine, as long as you can solve problems consistently and accurately and, most importantly, that you gain the knowledge and confidence to start to try to work out possible answers.

1.4. Mathematical conventions used in this book

To make them easier to read, numbers with more than four digits are split into groups of three digits separated by spaces (not commas), e.g. 9 999 999 is nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine. I have also chosen to use the asterisk (*) as a multiplication sign rather than ‘×’ or a dot, since these are sometimes confusing when written.

2

Manipulating Numbers

Algebra (from the Arabic, al-jabr, ‘the reduction’) – a form of maths where symbols are used to represent numbers

Arithmetic is concerned with the effect of operations (e.g. addition, multiplication, etc.) on specified numbers. In algebra, operations are applied to variables rather than specific numbers. Why? Here is a classic example:

John is 10 years old. His father is 35 years old. After how many years will the father be twice as old as the son?

You could try to find the answer by experimenting with different numbers, but this is laborious. The better way is to treat this an algebra problem and write the problem as an equation which we can then solve.

Let the father be twice as old as the son in x years time. The son will then be (10 + x) years old and the father will be (35 + x) year old:

Therefore,

Simplify this by subtracting x from each side to keep the equation balanced:

Simplify by subtracting 20 from each side to keep the equation balanced:

2.1. Manipulating numbers

To manipulate numbers, you need to know the rules. In mathematics, this is known as the ‘Order of Operations’ – an internationally agreed set of arbitrary rules which allows mathematicians the world over to arrive at the same answers to problems: