Mathematics Explained for Healthcare Practitioners

Derek Haylock, Paul Warburton, SAGE Publications Ltd

• Publisher: SAGE Publications Ltd
• Year: 2013
• Print ISBN: 9781446211199, 9781446211182
• eBook ISBN: 9781446289914
• Edition: 1
• Topic: Medicine
• Subtopic: Nursing Skills

If you find mathematics a struggle, or if mathematics makes you panic - whether you?re a student preparing for a calculations test, or a qualified healthcare professional - this is the book for you.

Its focus is first and foremost on helping you develop a proper and lasting understanding of mathematical concepts and processes in a healthcare context, with an emphasis on developing mental strategies to reinforce your feeling for number and numerical relationships. Clear, accessible chapters take you step-by-step through specific sets of objectives, and once you feel you have fully mastered a particular mathematical process, you?re encouraged to practise your skills through a range of practical examples. Each chapter also contains a ?Spot the Errors? feature designed to reflect the busy, pressurised healthcare environment.

Written by Derek Haylock, a best-selling mathematics educator, and Paul Warburton, co-ordinator of the Non-Medical Prescribing programme at Edge Hill University, the book is directly linked to the NMC?s competencies and will give you the confidence to perform the safe, error-free calculations required of all healthcare professionals.

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UNDERSTANDING THE NUMBER SYSTEM

1

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OBJECTIVES

The practitioner should be able to:

• understand the place-value principle in our number system for both whole numbers and decimal numbers
• use and interpret powers of 10
• recognize the positions of whole numbers or decimal numbers on a section of a number line
• appreciate where zeros are needed in decimal numbers
• express a decimal number as tenths, hundredths or thousandths

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Our mission throughout this book is to increase your understanding of mathematics, because we know that this will help to make you a more confident and competent healthcare practitioner. So, we begin with refreshing your understanding of numbers! The purpose of this chapter is to remind you of some fundamental ideas about whole numbers and decimals and to ensure that you have a clearer grasp of how our base-ten place-value number system works. The practitioner will need to be able to apply this understanding of the number system in the context of their healthcare practice, especially when the numbers are used in measurement. Various aspects of measurement are not explained until Chapters 4–7 of this book, so it is from those chapters on that we will illustrate and apply mathematical ideas more specifically in healthcare contexts.

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SPOT THE ERRORS

Identify any obvious mathematical errors in the following ten statements.

1	The digit 7 in the number 87 654 represents seven thousand.
2	Counting in ones, the next number after three thousand and ninety-nine is four thousand.
3	On a number line 8050 lies halfway between 8000 and 8100.
4	To calculate the cost per day of a 28-day course of medicine costing £75.60, a pharmacist enters 75.60 ÷ 28 on a calculator and gets the result 2.7; this means the cost per day is two pounds and seven pence.
5	The number 0.0008 is ‘eight ten-thousandths’.
6	The number halfway between 0.007 and 0.008 is 0.075.
7	The number 4567 is equal to (4 × 10^3) + (5 × 10^2) + (6 × 10^1) + (7 × 10^0).
8	The number 0.067 is equal to (6 × 10^–2) + (7 × 10^–3).
9	In this set {0.09, 0.8, 0.084, 0.18, 0.48}, the greatest number is 0.8 and the smallest number is 0.084.
10	1.275 is equal to 1 unit and 275 thousandths, which is 1275 thousandths.

(errors identified on page 4)

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How does place value work?

Over history there have been many systems for representing numbers. The system used internationally today is essentially a Hindu-Arabic system that has ten as its base and uses the principle of place value. This principle means that, for example, the symbol ‘5’ occurring in a numeral might sometimes mean ‘five’ (as in 65), but it could mean ‘fifty’ (as in 56) or ‘five thousand’, as in (5678), or ‘five hundredths’ (as in 2.75), and so on, depending on the place in which it is written. Most ancient number systems did not use this principle. For example, in Roman numerals the symbol ‘V’ wherever it is written, for example in XV or in CLXVIII, means ‘five’.

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Numeral and number

A numeral is a symbol used to represent a number. For example, the number of teeth that an adult should have is represented by the numeral 32.

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The place-value system for numbers is based on powers of ten. These are: one, ten, a hundred, a thousand, ten thousand, a hundred thousand, a million, and so on, getting ten times bigger each time. These powers of ten can be expressed symbolically and in words in a number of ways as shown in the table in Figure 1.1.

Figure 1.1

A table showing increasing powers of 10

There are a few things to note here. In general, ten to the power n (10ⁿ) means 1 multiplied by 10, n times, and this is equal to a number written as 1 followed by n zeros. For example, 10⁹ would be 1 multiplied by 9 tens: 1 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10; written out in full as a numeral this number (a billion) is 1 followed by 9 zeros.

The expression 10² (ten to the power 2) is also read as ‘ten squared’; and 10³ (ten to the power 3) is also read as ‘ten cubed’.

We have included ‘ten to the power zero’ at the top of the table above, as a way of writing the number 1. Although this might seem a bit weird, you should at least be able to see how it fits into the pattern of the table. This use of ‘to the power zero’ is a mathematical convention adopted for the sake of completeness; it also provides a bridge for extending the place-value system from whole numbers to decimal numbers (see below). Think of it as meaning ‘1 not multiplied by any tens’ or ‘1 followed by no zeros’.

This table could go on for ever, continuing with ten million, a hundred million, a billion, ten billion, and so on. But the larger the power of ten the less useful becomes the actual name. For example, ‘a trillion’ might not convey much to us, until someone tells us that it actually stands for 10¹² (1 followed by 12 zeros). An important (and very large) number in science that we will mention in Chapter 14 involves 10²³; written like this, as a power of ten, this is much easier to grasp than referring to it as ‘a hundred sextillion’!

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ERRORS IDENTIFIED

The obvious errors are in Statements 2, 4 and 6.

Statement 2

The next number after three thousand and ninety-nine (3099) is three thousand one hundred (3100). We hope this is obvious now you see the numbers written as numerals.

Statement 4

The cost is two pounds and seventy pence (£2.70) per day. The numbers 2.7, 2.70, 2.700, 2.7000, and so on, all represent the same quantity, since the zeros simply tell us that there are no tenths, no hundredths, no thousandths, and so on. Calculators usually d...

Table of contents

• Cover Page
• Title Page
• Copyright
• Contents
• About the Authors
• Acknowledgements
• Introduction
• A Disclaimer
• 1 Understanding the Number System
• 2 Addition and Subtraction Skills
• 3 Multiplication and Division Skills
• 4 Units of Length
• 5 Units of Liquid Volume
• 6 Units of Weight
• 7 Using a Calculator
• 8 Rounding
• 9 Calculations with Decimal Numbers
• 10 Imperial Units Still in Use
• 11 Understanding Fractions and Ratios
• 12 Proportionality and ‘Per’
• 13 Percentages
• 14 Miscellaneous Mathematics
• 15 Application
• Answers to Check-ups
• Index