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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Numerical ability is an essential skill for everyone studying the biological sciences but many students are frightened by the 'perceived' difficulty of mathematics, and are nervous about applying mathematical skills in their chosen field of study. Having taught introductory maths and statistics for many years, Alan Cann understands these challenges and just how invaluable an accessible, confidence building textbook could be to the fearful student. Unable to find a book pitched at the right level, that concentrated on why numerical skills are useful to biologists, he wrote his own. The result is Maths from Scratch for Biologists , a highly instructive, informal text that explains step by step how and why you need to tackle maths within the biological sciences. Features: * An accessible, jargon-busting approach to help readers master basic mathematical, statistical and data handling techniques in biology * Numerous end of chapter problems to reinforce key concepts and encourage students to test their newly acquired skills through practise * A handy, time-saving glossary * A supplementary website with numerous problems and self-test exercises
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Seitenzahl: 219
Veröffentlichungsjahr: 2013
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Contents
Preface
1 Maths in Biology
1.1 What can go wrong?
1.2 Estimating
1.3 How to use this book
1.4 Mathematical conventions used in this book
2 Manipulating Numbers
2.1 Manipulating numbers
2.2 Solving equations
2.3 Why do you need to know all this?
2.4 Fractions
2.5 The number 1
2.6 Lowest common multiple and greatest common factor
2.7 Adding and subtracting fractions
2.8 Multiplying fractions
2.9 Dividing fractions
2.10 Fractions, decimals and percentages
2.11 Ratios and proportions
3 Units and Conversions
3.1 The SI system of units
3.2 SI prefixes
3.3 SI usage
3.4 Measuring energy
3.5 Temperature
4 Molarities and Dilutions
4.1 Avogadro’s number
4.2 Molecular weight
4.3 Solutions
4.4 Spectroscopy
4.5 Dilutions
5 Areas and Volumes
5.1 Geometry
5.2 Calculating areas and volumes
6 Exponents and Logs
6.1 Exponents
6.2 Exponential functions
6.3 Logarithms
7 Introduction to Statistics
7.1 What is statistics?
7.2 Statistical variables
7.3 Statistical methods
7.4 Frequency distributions
7.5 Frequency distribution graphs
8 Descriptive Statistics
8.1 Populations and samples
8.2 The central tendency
8.3 Variability
8.4 Standard error
8.5 Confidence intervals
8.6 Parametric and non-parametric statistics
8.7 Choosing an appropriate statistical test
8.8 Exploratory data analysis
9 Probability
9.1 Probability theory
9.2 Replacing vs not replacing selections
9.3 Calculating the probability of multiple events
9.4 The binomial distribution
9.5 Coincidences
10 Inferential Statistics
10.1 Statistical inference
10.2 Procedure for hypothesis testing
10.3 Standard scores (z-scores)
10.4 Student’s t-test (t-test)
10.5 Analysis of variance (ANOVA)
10.6 χ2-test
10.7 Fisher’s exact test
11 Correlation and Regression
11.1 Regression or correlation?
11.2 Correlation
11.3 Regression
Appendix 1 Answers to Problems
Appendix 2 Software for Biologists
Word processors
Presentation and graphics
Internet resources
Statistics software
Appendix 3 Statistical Formulae and Tables
Critical values of the χ2 distribution
Critical values of student’s t-Test
Table of critical values of the F-statistic
Table of critical values of the correlation coefficient, r
Table of binomial probabilities
Appendix 4 Glossary
Index
Copyright © 2003 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England
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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 2002570′.1′51 — dc212002028068
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 0 471 49834 3 Hardback0 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. CannUniversity of Leicester, [email protected]
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?
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.
WARNING!
Using mixed units is dangerous (see Chapter 3).
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
1.2. Estimating
Whenever you have calculated an answer, always make a rough estimate to see if your answer is sensible and to avoid mistakes.
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
LEARNING OBJECTIVES:
On completing this chapter, you should be able to:
understand the basic rules of algebra;perform simple algebraic manipulations;identify and manipulate fractions.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:
In algebra, there are two sorts of statement which you need to be able to recognize:
2.2. Solving equations
To ‘solve’ an equation, you must find the value(s) of the variable(s) which make the equation ‘true’, i.e. makes the terms on either side of the equal sign equal. There are seven steps to follow in order to solve an equation (‘BICORS’):
You will not always have to perform all of these steps, depending on the equation. For example, if an equation does not contain any brackets, just move on to the next step, but do go through all the steps in order. Solving equations often involves simplifying the expressions they contain, which means getting all similar terms (e.g. x) on the same side of the equal sign. All of this sounds more complicated than it actually is and is best illustrated by some examples.
Examples
Solve for x (i.e. find the value of x that makes the equation true):
Expand Brackets (Bicors):
Simplify to Isolate (bIcors) the variable:
Combine (biCors) like terms:
Carry out the Opposite (bicOrs) process:
Simplify the equation by Reducing (bicoRs) fractions to their lowest terms:
Check the answer by Substituting (bicorS) it back into the original equation:
Solve for x (find the value of x):
Solve for z (find the value of z):
Note that equations do not always have a numerical answer – sometimes the value of the variable can only be expressed in terms of its relationship with another variable:
Solve for x (find the true value of x):
Solve for t (find the true value of t):
There are two main sorts of equation:
Although non-linear equations are common in biology, this chapter is primarily concerned with linear equations. Many people find the idea of solving equations difficult. The answer to this is to practise. For this, you can use the problems at the end of this chapter. When you become more confident, you can move on to non-linear equations. Word problems are particularly useful to help you think through what you are being asked, but can be surprisingly difficult for some people. In real life, information is frequently presented in this form rather than as an equation. The trick is to start by converting words into numbers. Again, this is a skill that you can acquire by practice – use the problems at the end of this chapter.
2.3. Why do you need to know all this?
You need to know all this because you cannot go very far in biology without encountering topics like enzyme kinetics.
Example
In an enzyme-catalysed reaction, the reactant (S) combines reversibly with a catalyst (E) to form a complex (ES) with forward and reverse rate constants of k1 and k–1, respectively. The complex then dissociates into product (P) with a reaction rate constant of k2 and the catalyst is regenerated:
From this can be derived the Michaelis–Menten equation:
Km measures enzyme/substrate affinity – a low Km indicates a strong enzyme – substrate affinity and vice versa. However, Km is not just a binding constant that measures the strength of binding between the enzyme and substrate. Its value includes the affinity of substrate for enzyme, but also the rate at which the substrate bound to the enzyme is converted to product (see Table 2.1).
Table 2.1Km for various enzyme reactions
Simplifying this by dividing the top and bottom of this equation by 7.9 * 10−3,
so
2.4. Fractions
When you perform algebraic manipulations, you soon encounter fractions, which means parts of numbers. We all learned to manipulate fractions in school, but in these days of computers and calculators, many people have forgotten how to do this. Remembering how to multiply and divide fractions causes particular problems.
All fractions have three components – a numerator, a denominator and a division symbol:
The division symbol in a simple fraction indicates that the entire expression above the division symbol is the numerator and must be treated as if it were one number, and the entire expression below the division symbol is the denominator and must be treated as if it were one number. The same order of operations (BEDMAS) applies to fractions as to other mathematical terms. Brackets instruct you to simplify the expression within the bracket before doing anything else. The division symbol in a fraction has the same role as a bracket. It instructs you to treat the quantity above (the numerator) as if it were enclosed in a bracket, and to treat the quantity below (the denominator) as if it were enclosed in another bracket:
In a simple fraction, the numerator and the denominator are both integers (whole numbers), e.g.
A complex fraction is a fraction where the numerator, denominator or both contain a fraction, e.g.
To manipulate (e.g. add, subtract, divide or multiply) complex fractions, you must first convert them to simple fractions.
A compound fraction, also called a mixed number, contains integers and fractions, e.g.
As with complex fractions, to manipulate compound fractions, you must first convert them to simple fractions.
No fraction (simple, complex or compound) can have a denominator with an overall value of zero. This is because, if the denominator of a fraction is zero, the overall value of the fraction is not defined, since you cannot divide by zero. A numerator is allowed to take on the value of zero in a fraction, although any legitimate fraction (denominator not equal to zero) with a numerator equal to zero has an overall value of zero. If there is a single minus sign in a simple fraction, the overall value of the fraction will be negative. If there is an even number of minus signs in a simple fraction, the value of the fraction is positive. If there is an odd number of minus signs in a simple fraction, the value of the fraction is negative, e.g.
