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the report, the correct statistic was quoted, that 54.32 out of every 1,000 women aged 15 to 17 in the 10 most deprived areas had fallen pregnant. Fifty-four out of 1,000 is 5.4%, not 54%. Perhaps it was the spurious precision of the 54.32’ figure that had confused the report writers.

      Other questionable numbers require a little more thought. The National Survey of Sexual Attitudes has been published every 10 years since 1990. It gives an overview of sexual behaviour across Britain.

      One statistic that often draws attention when the report is published is the number of sexual partners that the average man and woman has had in their lifetime.

      The figures in the first three reports were as follows:

Average (mean) number of opposite-sex partners in lifetime (ages 16–44)
	Men	Women
1990–91	8.6	3.7
1999–2001	12.6	6.5
2010–2012	11.7	7.7

The figures appear quite revealing, with a surge in the number of partners during the 1990s, while the early 2000s saw a slight decline for men and an increase for women.

      But there is something odd about these numbers. When sexual activity happens between two opposite-sex people, the overall ‘tally’ for all men and women increases by one. Some people will be far more promiscuous than others, but across the whole population, it is an incontravertible fact of life that the total number of male partners for women will be the same as the number of women partners for men. In other words, the two averages ought to be the same.

      There are ways you can attempt to explain the difference. For example, perhaps the survey is not truly representative – maybe there is a large group of men who have sex with a small group of women that are not covered in the survey.

      However, there is a more likely explanation, which is that somebody is lying. The researchers are relying on individuals’ honesty – and memory – to get these statistics, with no way of checking if the numbers are right.

      What appears to be happening is that either men are exaggerating, or women are understating, their experience. Possibly both. Or it might just be that the experience was more memorable for the men than for the women. But whatever the explanation, we have some authentic-looking numbers here that under scrutiny don’t add up.

THE CASE FOR BACK-OF-ENVELOPE THINKING

      I hope this opening section has demonstrated why, in many situations, quoting a number to more than one or two significant figures is misleading, and can even lull us into a false sense of certainty. Why? Because a number quoted to that precision implies that it is accurate; in other words, that the ‘true’ answer will be very close to that. Calculators and spreadsheets have taken much of the pain out of calculation, but they have also created the illusion that any numerical problem has an answer that can be quoted to several decimal places.

      There are, of course, situations where it is important to know a number to more than three significant figures. Here are a few of them:

       In financial accounts and reports. If a company has made a profit of £2,407,884, there will be some people for whom that £884 at the end is important.

       When trying to detect small changes. Astronomers looking to see if a remote object in the sky has shifted in orbit might find useful information in the tenth significant figure, or even more.

       Similarly in the high end of physics there are quantities linked to the atom that are known to at least 10 significant figures.

       For precision measurements such as those involved in GPS, which is identifying the location of your car or your destination, and where the fifth significant figure might mean the difference between pulling up outside your friend’s house and driving into a pond.

      But take a look at the numbers quoted in the news – they might be in a government announcement, a sports report or a business forecast – and you’ll find remarkably few numbers where there is any value in knowing them to four or more significant figures.

      And if we’re mainly dealing with numbers with so few significant figures, the calculations we need to make to find those numbers are going to be simpler. So simple, indeed, that we ought to be able to do most of them on the back of an envelope or even, with practice, in our heads.

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      TOOLS OF THE TRADE

      THE ESSENTIAL TOOLS OF ESTIMATION

      For most back-of-envelope calculations, the tools of the trade are quite basic.

      The first vital tool is the ability to round numbers to one or more significant figures.

      The next three tools are ones that require exact answers:

       Basic arithmetic (which is built around mental addition, subtraction and a reasonable fluency with times tables up to 10).

       The ability to work with percentages and fractions.

       Calculating using powers of 10 (10, 100, 1,000 and so on) and hence being able to work out ‘orders of magnitude’; in other words, knowing if the answer is going to be in the hundreds, thousands or millions, for example.

      And finally, it is handy to have at your fingertips a few key number facts, such as distances and populations, that crop up in many common calculations.

This section will arm you with a few tips that will help you with your back-of-envelope calculations later on – including a technique you may not have come across that I call Zequals, and how to use it.

      ARE YOU AN ARITHMETICIAN?

      In the opening section there was a quick arithmetic warm-up. It was a chance to find out to what extent you are an Arithmetician.

      Arithmetician is not a word you hear very often.

      In past centuries it was a much more familiar term. Here, for example, is a line from Shakespeare’s Othello: ‘Forsooth, a great arithmetician, one Michael Cassio, a Florentine.’ That line is spoken by Iago, the villain of the play, who is angry that he has been passed over for the job of lieutenant by a man called Cassio. It is an amusing coincidence that Shakespeare’s arithmetician Cassio has a name very similar to Casio, the UK’s leading brand of electronic calculator.

      Iago scoffs that Cassio might be good with numbers, but he has no practical understanding of the real world. (This rather harsh stereotype of mathematical people as being abstract thinkers who are out of touch with reality is one that lives on today.)

      Shakespeare never used the word ‘mathematician’ in any of his plays, though in Tudor times the two words were often used interchangeably, just as ‘maths’ and ‘arithmetic’ are today – much to the annoyance of many mathematicians.

So what is the difference between maths and arithmetic? If you ask mathematicians this question, they come up with many different answers. Things like ‘being able to logically prove what is true’ and ‘seeing patterns and connections’. What they never say is: ‘knowing your times tables’ or ‘adding up the bill’.

      Arithmetic, on the other hand, is entirely about calculations.

      Here’s an example to show what I mean:

      Pick any whole number

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