to work. And commentators on statistics use it to help make sense of the myriad numbers that are thrown out by politicians, ‘expert’ pundits and marketers.
On a more mundane level, it’s the maths you might use every day to ensure you aren’t getting ripped off by a so-called ‘deal’ that turns out to be anything but.
It is also maths and arithmetic that can be done without needing to resort to a calculator.
But wait a minute. Maths without a calculator? To many people, this notion seems quaintly old-fashioned, or even masochistic. Why grapple with manual or mental calculations when most of us have a phone (with a calculator) readily to hand almost all of the time?
This is not an anti-calculator book. Calculators are indispensable tools that have enabled us to do in seconds what used to take minutes, hours or even days. If you need to know exactly what £31.40 × 96 is, then unless you are a savant or somebody with plenty of time on your hands, a calculator is the only sensible option for working it out. And I’m probably typical in usually having a calculator – or a spreadsheet – to hand if I’m doing my tax return, or totting up expenses after a work-related trip.
But much of the time we don’t need to know the exact answer. It’s an approximate figure that matters. The point of back-of-envelope maths is to help see the bigger picture behind numbers.
Suppose a sales team has a target of £10,000. If they report that they have sold 96 units at £31.40 each – that’s roughly:
100 × £30 = £3,000 revenue.
That’s massively short of the £10,000 target, even if the estimate is out by a few per cent.
When the government announces a £1 billion increase in health spending, is that significant? Spread between 50 million people? It won’t be exactly one billion pounds of course, nor will it be spread evenly between 50 million people, but with back-of-envelope maths, we can work out it will represent an average of something nearer to £20 (i.e. hardly anything) than £200 per person.
Of course, even these simplified calculations can be done on a calculator. But the reality is that they rarely are.
The argument: ‘Who needs to do arithmetic when we all have calculators?’ is usually a red herring. In situations where a calculation is not essential, most of us do it in our heads or on the back of an envelope, or don’t do it at all.
And there are some who use their ability to figure things mentally to their advantage. I have a friend who made his fortune as a wheeler-dealer in finance. I asked him to share some advice.
‘I have two tips for succeeding when negotiating a deal with somebody,’ he said. ‘The first is: learn how to be able to read upside down, so that you can decipher the documents of the person opposite you. And my second is: be able to do the calculations faster than they can.’
TEST YOURSELF
How is your arithmetic without the aid of a calculator? Try these 10 questions. There is no time pressure, and you’re allowed to use pencil and paper if you want. As you do these questions, you might want to think about how you do them. Are you recalling facts you’ve memorised? Do you use a pencil-and-paper method?
(a) 17 + 8
(b) 62 – 13
(c) 2,020 – 1,998
(d) 9 × 4
(e) 8 × 7
(f) 40 × 30
(g) 3.2 × 5
(h) One-quarter of 120
(i) What is 75% as a fraction?
(j) What is 10% of 94?
Solutions
I can still remember the thrill when I first got a calculator of my own. It was made by Commodore, and had red LED digits and buttons that made a satisfying click when you pressed them. It was a Christmas present, and I was 16 years old. I was captivated. Just being able to enter a number like 123456 and press the square root button was enough to send a tingle of excitement down my spine, as I gazed at all those digits after the decimal point. I’d never seen numbers to such precision before.
There were two things that came out of the arrival of cheap calculators.
The first was that we could all now do calculations that we would never have conceived of doing before. It was empowering, liberating and gave us a chance to see the bigger picture of mathematics without getting bogged down in the nitty-gritty of calculation.2
The second thing that happened was that we could now quote answers to several decimal places. The square root of 83? Certainly, sir, just give me one second – and how many digits would you like after the decimal point?
What could possibly go wrong?
SPURIOUS PRECISION
A tourist in a natural history museum was very impressed by the skeleton of a Tyrannosaurus Rex.
‘How old is that fossil?’ she asked one of the guides.
‘It’s 69 million years and 22 days,’ said the guide.
‘That’s incredible, how do you know the age so precisely?’ asked the tourist.
‘Well, it was 69 million years old when I started working at the museum, and that was 22 days ago,’ replied the guide.
The thoughtless precision of the museum guide in this old joke nicely illustrates why there is no point in stating a number to several figures if the overall measurement is only a rough estimate. Yet it is a mistake that is made time and again when presenting and interpreting numbers in everyday life.
Quoting a number to more precision than is justified is often called spurious accuracy, though it should really be called spurious precision and we will encounter it several times in this book. It is one of the strongest arguments against the unthinking overuse of calculators. The fact that you can work out numbers to several decimal places at the touch of a button doesn’t mean that you should.
PRECISION VERSUS ACCURACY
The words precision and accuracy are often used interchangeably, to indicate how ‘right’ a measurement or number is. It is certainly possible for a number to be accurate and precise; for example: 74 × 23.2 = 1,716.8.
But used mathematically, precision and accuracy mean different things.
‘Accuracy’ is an indication of how close you are to the right answer. Suppose we are playing darts. I throw a dart at a dartboard and just miss the bullseye. My throw was quite accurate, but if you then throw and hit the bullseye, your throw was more accurate than mine. Likewise … if I tot up the items in my shopping basket and estimate that the total will be £65 while you reckon it will be £70, and the bill turns out to be £69.43, then you were more accurate than I was.
Precision, on the other hand, is an indication of how confident you are in a number to a particular level of detail, so that you or somebody else would come up with the same figure if you did a measurement or calculation again. If you think the shopping basket will add up to £69, you are confident that you are right to the nearest pound; but if you suggest the bill will be £69.40, you are being more precise, and are confident your figure is right to the nearest 10 pence. Even more precise is £69.41. In maths terms, precision is about how many significant figures (this is an important concept – please see here) you can quote a number to.
As a society, we put a lot of faith in precision. If we see a number such as 84.36, we tend to believe that the person who produced the number is confident of that figure to the second decimal place. We might