Numbers have something that mere words seem to lack. Let’s call it an aura of absolute truth, of incontestability.
This is even more the case if the are figures not round. Sir George Everest realised this characteristic very well when measuring the height of the mountain which was then still known as 'Peak 15'. He measured it at exactly 29,000 feet, but afraid that people would not believe that such a perfectly round figure was the result of an exact measurement process, he added two feet and told the world the mountain was 29,002 feet high. Subsequent measurements using various more advanced technologies recorded heights of 29,029 feet (8,848 m), 29,035 feet (8,850 m), and 29,017 feet (8,844 m). Even today, the exact height of Mount Everest is a point of discussion among the cognoscenti of such minutiae.
This example shows the high esteem figures actually have, as well as how relative true exactness can be. We often fall prey to viewing figures as if they are 'absolute facts' because they are produced in the 'truth factories' of expert consultants, university laboratories, research companies, and study centres. But any so-called 'exact figure' is rarely completely unambiguous (see article 'Studies can prove anything you want them to'). In most cases, somewhere down the chain of investigation and calculation, certain suppositions, estimations, extrapolations, or just plain simple choices are made that significantly influence the result.
Consequently, it is wise not to automatically buy into all figures simply because they appear to have the underpinning of scientific or mathematical method supporting them. And in fact, you can often come close to the actual physical reality by doing a quick and dirty, back-of-the-envelope calculation yourself.
One of the most famous of the adherents of the 'back-of-the-envelope' idea was the Italian nuclear physicist Enrico Fermi (1901 – 1954). Often regarded as the twentieth century’s most accomplished theorist as well as experimentalist, Fermi was well known for emphasising that most complex scientific equations can be approximated within order of magnitude by making simple calculations 'on the back of an envelope'. As part of the Manhattan Project, he was present as an observer at the first nuclear bomb explosion on 16 July 1945 in Los Alamos. At the specific moment of detonation, he implemented the envelope concept in a very original way. He had torn an envelope into small snippets and as the shock wave hit the Base Camp, he dribbled them into the air. After watching the shock wave moving the confetti, he calculated the yield of the explosion to be around 10 kilotons TNT. The actual yield was 18.6 kiloton TNT; Fermi’s estimate however was less than a factor of two from the actual result.
The power of back-of-the-envelope calculations is often underestimated. Now, I do not recommend that everyone should use envelopes in the same way and under similar conditions as Fermi exposed himself to at Los Alamos that day; he died of stomach cancer at the age of 53. But nobody ever died from using a piece of paper and a pen for making a rough estimate.
Back-of-the-envelope calculations have the advantage that they are transparent and easy to follow, don’t pretend at exactness, and give a quick idea of the order of magnitude of things. This can be useful, for example, when evaluating the feasibility of a project at an early developmental phase, or when comparing various policy options.
Let’s consider, for example, the following question: do we have enough arable land to make all our cars run on biodiesel? Well, let’s compare land use for biodiesel with land use for food. It seems difficult to make this comparison on the level of land use itself, but we could make it quite easily on the level of energy. What’s the energy use of a car compared to the energy use of a person? Assume the average person in the EU or US consumes 2,500 food kilo calories per day (Wikipedia), or about one million per year. A calorie is about 4,000 joules, so we eat approximately 4 billion joules per year. Now suppose the average car owner drives 15,000 kilometres per year and the car uses 6 litres of conventional gasoline per 100 kilometres. That totals to a fuel consumption of 900 litres per year. Knowing that gasoline has a density of approximately 30 million joules per litre (Wikipedia), it follows that each car consumes an average of 27 billion joules per year – about 7 times the energy that we eat. If every joule requires the same surface of arable land, making all cars drive on biodiesel would take about 7 times the land surface that we use for food production.
Now suppose that in making that last assumption, the above calculation is a factor 10 from reality. That would still mean that biodiesel would need about three quarters of the land that is currently used for food production, which still seems unfeasable. Conclusion: it is unlikely that we have enough arable land to produce enough biodiesel for all cars that are currently driving around in the US and EU. However, more detailed information on the relation between land surface and energy yield is required to confirm this result.
The book 'Sustainable Energy — without the hot air' by David Mac Kay — already mentioned several times previously on this blog — is built upon the use of such rough calculations. It figures out which options for a sustainable energy future are technically feasible, and what price we have to pay for them in terms of land use, waste, efficiency, geopolitical consequences, and so on. It is a textbook example of how far you can go with back-of-the-envelope calculations.
In the introduction to his book, MacKay strongly emphasises that such calculations are only useful if making use of unambiguous, traceable, and comparable figures. He always gives the source of the figures that he uses, and prefers making a rough calculation himself to copying questionable figures from external sources. Moreover, he calculates all figures for energy production and consumption back to kWh per person per day, making them much more easily comparable.
If these simple rules are not applied with a degree of rigour, rough estimate calculations quickly degenerate into an 'anything goes' situation. The Office of Metropolitan Architecture (OMA) demonstrated this in its master plan for a huge ring of wind farms in the North Sea (see article 'Energy master plan by OMA: North Sea super ring of wind farms'). According to OMA’s calculations, such a North Sea ring could supply the whole of Europe with energy. However, OMA failed elucidate all of its data sources and calculation steps. In fact, it presented a back-of-the-envelope calculation as if it were the result of a thorough scientific calculation. Calculating with David MacKay’s figures, such a plan would yield only about one tenth of what OMA suggests.
Considering the example above, the question has to be asked: How many journalists and newspaper readers actually make their own back-of-the-envelope calculations? Quick, rough estimates are not only useful to check your own propositions; they are also an efficient way to verify the credibility of the calculations or figures that are communicated in the media and elsewhere.
Recently there was a pertinent example of this with an article on the behaviour of bears in The Guardian newspaper in the UK. A man studying bears claimed that it’s more dangerous to encounter another human being than a grizzly bear, since one human out of 18,000 kills somebody, and only one bear out of 50,000 does the same. A smart reader corrected this reasoning with the following back-of-the-envelope calculation:
'If the average human meets another human 2,000 times a year (6 people per day), and the average bear meets hikers 20 times a year (guesstimate). Then bears may kill people in 1 / (20 x 50,000) meetings. For humans it would be 1 / (2,000 x 18,000) meetings. (2,000 x 18,000) / (20 x 50,000) = 36. In other words, each time a bear meets a human, it is about 40 times more likely that it will kill him/her than if it were a person.'
Finally, along with all the advantages mentioned above, back-of-the-envelope calculations are also quite didactical. They will, in most cases, increase one’s insight into the matter in question and certainly give you a feeling of the order of magnitude of things.
And if nothing else, these kinds of calculations can at least improve the prospects of the envelope manufacturing industry, which have been under pressure since the advent of e-mail.
Some interesting reading material on the subject: