Is That a Big Number?
Mathematics is much more than numbers, but in an historical as well as in an educational context numbers are certainly essential for the development of mathematical skills. Counting is one of the primitive mathematical activities, but as societies became more complex, numbers became much more essential for the socio-economical fabric. Sometimes the balance tips the wrong way by reducing something or even someone to just a number expressing an amount of a particular something that can be associated with them. It has become more and more important to have an idea of what a number means or what it stands for. Whether it is small or big in the context it is used. Think of the kcal of the food we eat, the GNP or the debt per capita or the happiness index of your country, the meaning of Olympic records, or the area of forest destroyed by wildfire. This book is about the numeracy that people should gain to be able to understand the meaning of numbers in our daily life so that they can discuss about them knowing what they are talking about. This understanding of numbers has been promoted before by J.A. Paulos in his book Innumeracy (Hill and Wang, 2001) and his semi-autobiographical sequel A Numerate Life (Prometheus, 2015).
To get an idea of what is small and what is big, one should be able to get a mental picture of what a number means. So the book starts with enumerating five ways to attach a meaning to the adjectives "big" or "small".
- Landmark numbers are numbers that one should memorise so that they are readily available for comparison
- Visualisation means to use your imagination to get a mental picture of the amount
- Divide and conquer: break a larger number up into smaller parts (e.g. x rows of y columns having stacks that are z high)
- Rates and ratios are usually smaller and often more relevant for what the numbers mean
- Log scales will bring numbers varying over a wide range to within reasonable bounds and separates them better in dense parts of the range
These ideas are illustrated abundantly with an incredible number of facts. Some are marked in frames containing possible landmark numbers, others are enumerated in lists of random alignments or number ladders. Random alignments are lists of (unrelated) numbers that happen to appear in almost perfect ratios (like for example the height of St. Paul's in London which is 100 times the height of R2D2 from Star Wars). Such lists are usually found at the end of the chapters. Number ladders are lists of increasing or decreasing numbers that run through a whole scale (for example the weight of animals ranging from an Indian flying fox (1 kg) to a blue whale (110 tons)). Each chapter also starts with a multiple choice question for the reader to test his/her numeracy (like which is largest in a list of four populations). The answers and references for the latter are given at the end of the book.
The chapters are grouped into four parts that reflect the context in which the numbers appear. They are also somewhat arranged in historical order. There is in fact also a lot of history and etymology about the origin of names that have been invented to indicate units by which quantities are measured, or the origin of terms used in for example our monetary system. The metric system has simplified a lot, but before that, there were many different words for units to measure a quantity, and it may be a surprise to read how many are still in use that have escaped metric standardisation.
Part 1 is about counting, which is our oldest use of numbers. When it gets to really big numbers (how many stars in the sky?) or in other instances (what percentage of alcohol in your beer?) it involves more than "just counting 1,2,3,...". The reader is instructed how to "visualise" numbers mainly by the first two methods of the list above. Methods 3 and 4 of the list are better illustrated in the second part.
In that second part, we learn about measures. A spatial dimension was historically often measured using body parts. This still resounds in our names for units of length like inches, feet, fathoms, and other less known units. Time has for obvious astronomical reasons escaped the decimal subdivision that is now used in the naming of space dimensions. It is the reason why twelve or sixty (which happen to have many divisors) have played a basic role in early mathematical cultures, and we still use the terms "dozen" and "gross" today (strange that Elliott didn't mention these two terms). The hierarchical subdivision of (pre)history in time spans like ages < epochs < periods < eras < eons (where x < y means that y consists of several x's) is an illustration of the fact that a divide and conquer technique is a way to get an idea of our geological time scale. To discuss history, it is important to get at least a rough idea about the rise and decline of the different civilisations that directly or indirectly had an influence on the age we are currently living in. So both historical and prehistorical periods of time get much attention in the book. In this part it is also illustrated that to measure areas, size needs to be squared and for volumes (and hence also for mass and weight) size must be cubed. It explains why the legs of an elephant are relatively much thicker than the legs of a mosquito. The strength depends on the cross section (size squared) and the weight on the volume (size cubed). That is why a giant Godzilla cannot be real. Scales are used to quantify wind speed and hurricanes or earth quakes. These scales are actually logarithmic. So log-scales are first introduced in a (lightly) technical intermezzo (which also mentions Benford's law, the slider rule and mortality rates).
Numbers get much bigger in the third part dealing with numbers in science. For example naming astronomical distances, measuring energy or capacity of a digital memory, or quantifying the information content of a text. Also measuring the complexity of solving combinatorial problems dives into the large numbers very fast. The fourth and final part is probably the most important for the general public since it treats numbers in a political and socio-economical context. This means among other things money (exchange rates) and economy (GDP), population (quantities and dynamics), quality of life (Millennium Development Goals and happiness index), etc.
The book illustrates well that knowledge (being numerate) is power. So we should be able to unveil the true value of a number used in an argument and know whether it has to be considered alarmingly big or small, and hence defend ourselves against deceit, fake news, and false arguments which has become an essential skill in our current society. This book is an essential tool if you want to work on this skill for personal use.
However, what the book does not discuss is that one should be careful with just ranking the numbers. Numbers do represent something, but it may not always be clear what that "something" really is. GNP is considered a measure for the wealth of a nation, but is it? Should military expenses, included in the GNP, be taken into account to measure wealth? Is an IQ really a measure of intelligence? A number will represent the result of a test or a poll, but what the test or poll is supposed to measure is not always clear because it may not correspond to what is actually measured. Moreover, these usually refer to just an sample while more general statements about a much wider population are concluded. In a world where everything is being managed, numbers are used to manipulate and define strategies, but often reality can not be caught in just a number. Believing that the number stands for some effect may be a generally accepted hoax. So the numeracy discussed in this book is just one aspect of being knowledgeable about a topic. Knowing whether a number is small or big is only one element to be taken into account for the interpretation of numbers. Even more important is to know what exactly these numbers measure and how they were obtained. This is not so important for the first three pars of this book since most of the numbers there concern quantities that can be objectively measured. Only in the fourth part when numbers are discussed in a social context, one should be careful with their interpretation and drawing conclusions.