Millions, Billions, Zillions

Journalists reporting on somebody else's results may easily be mistaken in citing the numbers that are not theirs, Perhaps authors change numbers on purpose to bias their arguments. In such cases unprepared and naive readers are easily deceived. In this book you can learn how to defend yourself from mistakes as an author and from deceptions as a reader.

Kernighan is the guide on a tour where he shows all the number traps that people can easily fall into. There are of course the big numbers like millions and billions mentioned in the title. It is hard to get a mental idea of what they actually mean. As a consequence an interchange of millions and billions is a mistake easily made without being noticed. One should also be aware of what the really big numbers actually stand for when they are indicated by prefixes like mega, giga, tera, and peta. Even exa, zetta, and maybe yotta can come into the picture. In modern texts these terms regularly occur referring to the large amounts of digital data stored in for example the Deep Web. At the other end of the spectrum we should know something about micro, nano, pico, femto, atto etc. when reading about the tiny parts of the hardware on which these data are stored, or even further down the scale when reading about the Theory of Everything where the natural playground is at a subatomic level. Like in mathematics the super large and the super small often match to keep everything within finite boundaries.

To avoid errors of course the units have to be right. Mistaking barrels for gallons, years for months, or hours for seconds may give quite unexpected and hard to believe interpretations of the numbers that are on display. Besides mistakes in scales, there is the extra complication that there are different systems of units like American miles, gallons, or degrees Fahrenheit that should somehow match with European kilometres, litres and degrees Celsius. Even a mile can have many different meanings in different contexts. All this requires carefully introducing the proper conversions. Just picking up a number from a website may easily lead to such mistakes if the proper conversion is not made. Another typical mistake is to confuse a square mile and a mile squared. This means that you should be aware that if you double the length of the sides of a square you get a surface four times as large. For a volume it is even more dramatic because that will give a volume or mass that is eight times as large.

The previous examples are possible sources of mistakes. How should we detect them and how could we protect ourselves against malicious attempts to deceive us? We could compare different sources or different ways of computing. When the results are approximately the same we can probably trust the numbers. It is of course useful to check with some numbers from your own experience or numbers that you know like for example the population of your country. Approximate rounded values for these numbers can be used in crude and simple computations to get at least an idea about the magnitude of the number that should be expected. If the number presented to you deviates considerably, either it is fake or your computations are wrong. For example Little's Law is applicable for a simple check in many situations. Given the population of your country and an average human life span, Little's Law allows you to estimate for example the number of people that will turn 64 twenty years from now.

Usually numbers appear in non-scientific texts with approximate rounded values. If one spots a specious number that is given with many digits, then it is probably the result of a computation or conversion and only the most significant digits or rounded values are actually meaningful. They are probably the result of some conversion, like a mountain over 4.000 meter high should not be referred to as over 13.123 feet. Statistics is another possible source of deception: the median is not the average, a correlation does not imply causation. It might also be interesting to know who did the statistics and the sampling or polling. Results may be consistently biassed towards the results of which some lobbyist or pressure group wants to convince you. Another well known trick is to fiddle with the scales used in the graphical representation of the numbers in pie or bar charts. Numbers representing a percent are again possible pitfalls that can put you on the wrong foot. A percent is definitely different from a percent point and you should also be aware that a percent increase is computed in terms of the lower number, while a percent decrease is referring to a percent of the larger number: a 50% decrease can only be compensated by a 100% increase.

Kernighan gives many examples of all these issues, mostly from newspapers and websites. He also keeps his readers alert by continuously pushing them to do some mental calculation to estimate some results for themselves. As some kind of a test at the end of the book he gives many such problems that one should be able to approximately solve (he also gives his own estimates): How many miles did Google drive to get the pictures for Street View (for your country)? How long did that take? How much did it cost? Or, if you have a garden with some trees in it, how many leaves do you have to rake every autumn? And there are many of these so called Fermi problems throughout the book. Kernighan gives some tricks to solve them, hence the "test" at the end. However it certainly requires a lot of practising and training which the reader has to do for him or herself to acquire some routine in this,

In this time of "fake news" and in a society that is more and more spammed by numbers, it seems like problems of numeracy among a general public is gaining interest and public awareness. More books devoted to different aspects of this issue seem to appear lately. Among the earlier examples are the books by John Allen Paulos Innumeracy (1988) and A mathematician reads the newspaper (1996). Kernighan mentions them in his survey of "books for further reading". However several more books appeared since 2010. Almost simultaneously with this one I received Is This a Big Number? (2018) by Andrew Elliott which is also reviewed here. Elliott has a more positive approach to the problem: how should I interpret the numbers presented to me (assuming they are correct) while Kernighan is more defensive: how to to stay away from mistakes or deceiving numbers.

It should also be noted that Kernighan uses American units most of the time, and his examples are mostly related to the American situation, or from the American newspapers. His advise is of course generally applicable, but it might be an extra hurdle to take for European readers who are used to the metric system of metre-litre-gram. It is surprising that Kernighan does not discuss the difference between the short scale (a billion is $10^9$) and long scale (a billion is $10^{12}$) hence also missing the milliard ($10^9$) and billiard ($10^{15}$) and giving different meaning to trillions, and nomenclature higher up. These are also obvious ways to get the wrong numbers cited.

This book does not need mathematics to read and it is actually not about mathematics at all. There is not a formula made explicit, even though the rule of 72 is explained (it takes 72/x units of time to double your capital when it has a compounding interest of x percent) and an idea is given about what it means to grow exponentially or by powers of 10 or powers of 2. The style of Kernighan is fluent and casual, but not particularly funny. The charm sits in his continuous teasing to make you think of these Fermi problems, and of course in his ample illustrations of how often authors are mistaken in citing numbers and how easily a reader can be deceived.

Adhemar Bultheel
Book details

This book about numeracy learns you how to defend yourself against making mistakes with numbers or to recognize incorrectly cited numbers by explaining what the possible sources of these errors are. On the positive side, one learns how to solve Fermi problems, that is to make rough numerical estimates of certain quantities, using little available data and where the few computations can me made "on the back of an envelope".



978-0-691-18277-3 (hbk), 978-0-691-19013-6 (ebk)
£ 17.99 (hbk)

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