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Rob Eastaway. Maths on the Back of an Envelope: Clever ways to (roughly) calculate anything
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CONTENTS
PROLOGUE. HOW MANY CATS?
1. THE PERILS OF PRECISION. ENVELOPES VERSUS CALCULATORS
TEST YOURSELF
SPURIOUS PRECISION
PRECISION VERSUS ACCURACY
DRESSING UP NUMBERS WITH DRESSAGE
A NUMBER IS ONLY AS STRONG AS ITS WEAKEST LINK
BODY TEMPERATURE A BIT LOW? BLAME IT ON SPURIOUS PRECISION
VARIABILITY
SENSITIVITY
EXPONENTIAL GROWTH
CLIMATE CHANGE AND COMPLEXITY
MAD COWS AND MAD FORECASTS
DOES THIS NUMBER MAKE SENSE?
THE CASE FOR BACK-OF-ENVELOPE THINKING
2. TOOLS OF THE TRADE. THE ESSENTIAL TOOLS OF ESTIMATION
ARE YOU AN ARITHMETICIAN?
TEST YOURSELF
BASIC ARITHMETIC. ADDITION AND SUBTRACTION
MULTIPLICATION AND TIMES TABLES
Tip 1
Tip 2
Tip 3
Tip 4
Tip 5
TEST YOURSELF
DIVISION
DECIMALS AND FRACTIONS. PLACE VALUE AND DECIMAL POINTS
DECIMAL POINTS – A MATTER OF LIFE OR DEATH?
MULTIPLYING FRACTIONS
TEST YOURSELF
PERCENTAGES
Tip 1
Tip 2
Tip 3
Tip 4
TEST YOURSELF
REMOVING VAT
CALCULATING WITH POWERS OF TEN. MULTIPLICATION
TEST YOURSELF
DIVIDING BY LARGE NUMBERS
TEST YOURSELF
USING ‘STANDARD FORM’ FOR LARGE NUMBERS
TEST YOURSELF
STAR WARS POWER
KNOW YOUR MEGAS FROM YOUR TERAS
KEY FACTS
TEST YOURSELF
ZEQUALS
A NEW MATHEMATICAL SYMBOL h
TEST YOURSELF
CALCULATING WITH ZEQUALS
TEST YOURSELF
THE INACCURACY OF ZEQUALS
TEST YOURSELF
WHO WANTS TO BE A MILLIONAIRE? (PART 1)
3. EVERYDAY ESTIMATION. ESTIMATION AND MONEY. SHOPPING BILLS AND SPREADSHEETS
TEST YOURSELF
HOW CAN THAT SHOP STILL BE IN BUSINESS?
SAVING, BORROWING AND PERCENTAGES
DOUBLE YOUR MONEY – THE RULE OF 72
CONVERTING CURRENCY
ESTIMATING SIZE. ESTIMATING DISTANCES
1. The Dropped-Stone Method
2. The Finger Method
3. The Crisp-Packet Method
CIRCLES AND PI
AREAS AND SQUARE ROOTS
TEST YOURSELF
WHO WANTS TO BE A MILLIONAIRE? (PART 2)
METRIC AND IMPERIAL CONVERSIONS. WHO NEEDS IMPERIAL?
THE MARS ORBITER FIASCO
MILES AND KILOMETRES
How To Remember Your Conversions
BACK-OF-ENVELOPE CONVERSIONS
TEST YOURSELF
A QUIRKY METHOD FOR MILE–KILOMETRE CONVERSION
ESTIMATION AND STATISTICS. AVERAGES AND UNCERTAINTY
WORKING OUT PROBABILITY
BACK-OF-ENVELOPE SURVEYS
HOW LONG WILL WE BE IN THIS QUEUE?
THE CHANCE OF MULTIPLE EVENTS
SPOTTING TRENDS
PREMIER LEAGUE GOALS: CERTAINTY WITHIN UNCERTAINTY
4. FIGURING WITH FERMI. THE FERMI APPROACH
COUNTING
HOW MANY WORDS ARE THERE IN YOUR FAVOURITE BOOK?
HOW MANY HAIRS ARE THERE ON AN ADULT HUMAN’S HEAD?
DO MORE PEOPLE GO TO FOOTBALL MATCHES AT THE WEEKEND THAN GO TO CHURCH?
HOW MANY TENNIS BALLS ARE USED AT WIMBLEDON?
HOW MANY ATTENDED DONALD TRUMP’S INAUGURATION?
WHAT ARE THE CHANCES?
WHAT’S THE CHANCE OF WINNING THE LOTTERY TWICE?
THE EXACT ODDS OF WINNING A LOTTERY
WHAT WERE THE CHANCES OF THE ORKNEY BABIES?
WHAT’S THE CHANCE OF TWO HOLES IN ONE?
WHAT’S THE CHANCE OF DEALING OUT FOUR ‘PERFECT’ HANDS?
ENERGY, CLIMATE AND THE ENVIRONMENT
WHAT USES THE MOST ENERGY AROUND THE HOUSE?
HOW MANY KETTLES IN A LIFETIME?
HOW MUCH FRESH WATER IS FLUSHED DOWN LONDON TOILETS EVERY DAY?
COWS VERSUS HUMANS – WHICH EMITS THE MOST METHANE?
HOW MANY PLANES ARE IN THE SKY AT THE MOMENT?
CAN WE PLANT A TRILLION TREES?
FERMI FOR THE FUN OF IT
HOW LONG TO COUNT TO A MILLION?
HOW OFTEN DOES A TEENAGER SAY ‘LIKE’ IN ONE YEAR?
ARE YOU DESCENDED FROM RICHARD III?
HOW MUCH FURTHER CAN YOU THROW A SHOT PUT WHEN YOU ARE IN MEXICO CITY?
WHERE ARE THE ALIENS?
LAST WORD. WHEN THE ROBOTS TAKE OVER. THE COUNTDOWN CONUNDRUM
APPENDIX. SIGNIFICANT FIGURES
WHERE THE RULE OF 72 COMES FROM
WHO WANTS TO BE A MILLIONAIRE? (PART 2)
RANGE OF A SHOT PUT
HOW LONG TO COUNT TO A MILLION?
ANSWERS AND TIPS. ENVELOPES VERSUS CALCULATORS
ARE YOU AN ARITHMETICIAN?
MULTIPLICATION AND TIMES TABLES
MULTIPLYING FRACTIONS
PERCENTAGES
MULTIPLICATION
DIVIDING BY LARGE NUMBERS
USING STANDARD FORM FOR LARGE NUMBERS
KEY FACTS
ZEQUALS
CALCULATING WITH ZEQUALS
INACCURACY OF ZEQUALS
SHOPPING BILLS AND SPREADSHEETS
AREAS AND SQUARE ROOTS
BACK-OF-ENVELOPE CONVERSIONS
Endnotes. 1: THE PERILS OF PRECISION
2: TOOLS OF THE TRADE
3: EVERYDAY ESTIMATION
4: FIGURING WITH FERMI
ACKNOWLEDGEMENTS
ABOUT THE AUTHOR
ABOUT THE PUBLISHER
Отрывок из книги
A few years ago, at a school event, I asked the audience of teenagers to submit some estimation questions that I would then attempt to answer live on stage. One pupil posed this simple question: ‘How many cats are there in the world?’
Cats are always a popular topic, so I took it on.
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The figure 98.6 came from the translation of Wunderlich’s report into English. At the time, Fahrenheit was the commonly used scale in Britain. To convert 37 °C to Fahrenheit, you multiply by 9, divide by 5 and add 32; i.e. 37 °C converts to 98.6 °F. So the English translation – which reached a far bigger audience than the German original, gave the figure 98.6 °F as the human norm. Technically, they were right to do this, but the decimal place created a misleading impression. If Wunderlich had quoted the temperature as 37.0 °C, it would have been reasonable to quote this as 98.6 °F, but Wunderlich deliberately didn’t quote his rough figure to the decimal place. For a figure that can vary by nearly a whole degree between healthy individuals, 98.6 °F was (and is) spurious precision. And in any case, a study in 2015 using modern, more accurate thermometers, found that we’ve been getting it wrong all these years, and that the average human temperature is 98.2 °F, not 98.6 °F.
In the General Election of May 2017, there was a shock result in London’s Kensington constituency. The sitting MP was a Conservative with a healthy majority, but in the small hours of the Friday, news came through that the result was too close to call, and there was going to be a recount. Hours later, it was announced that there needed to be a second recount. And then, when even that failed to resolve the result, the staff were given a few hours to get some sleep, and then returned for a third recount the following day.
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