Doubling and halving 1. Abstract Some time ago a message appeared on TV that in the last quarter of 2008 the circulation of daily newspapers has been decreased with 5.5% I was telling this to a friend who is an education consultant and added to it, that, if this continues to be so, the circulation will be halved in 13 quarters of a year. Not a pleasant prospect. The arithmetic is simply to be solved by heart: the "rule of 72" (also known as "rule of 70" or "rule of 69") says, that with a steady change of p% per term the inital quantity will be halved or doubled after 72/p terms, depending on the fact whether there is a decrease or an increase. 2. Description & mathematical reduction A principal, put at 4% annual will be doubled after 72/4 = 18 years. Exchange rates that continuously show a daily increase of 1.2% will have been doubled after 72/1.2 = 60 days. A population that steadily increases annually with 1.5% will double in 72/1.5 = 48 years. The other way around if one wants to cash an amount of € 100.000 after 12 years with a 6% interest rate, he/she has to deposit € 50.000 by now. It is a reliable method with endless applications, also already because the "index"- rate is surprisingly elastic. My friend says: "That's a nice little problem of arithmetic. Why isn't that covered in the basic arithmetic package at the schools?" I said: "Probably the kids are not allowed to know something they don't understand yet, or maybe never, let alone all that master is not understanding." An important application stems from mathematical demography and mathematics of finance, because in those fields there's always the pressing question: at which rate after how many years will a doubling take place? Here's the deduction1. The formula for geometric increment, as mathematicians are calling it, is in which is p the increment percentage per time unit. The unit of time may be a month, a year or longer or shorter, as long as p en t are expressed in the same unit. A doubling takes place when In order to solve n we need to make use of logarithms, which leads to a somewhat more complicated looking image: With help of the Taylor series development (we leave that here behind) it may be shown that the expression in the denominator for the values of x in our subject may be equalled to x. We may therefore write: and simpler: Because the number 72 is somewhat "smoother" than 70 (it's divisible by more different numbers than the number 70), we replace the value in the nominator by 72 - there's not the slightist objection against - the extra 2 is again divided by such and so many, and we have only to do with approximations. 3. Practical applications What then is the practical meaning of this knowledge? Example 1. If a population (or segment of a population) increases by 2%, this population will be doubled in number after 36 years. Example 2. If we have a period of time with an inflation of 3% purchasing-power will be halved after 24 years. Example 3. An exchange rate that increases with 2% per week will be doubled after 36 weeks. Example 4. Halving time radioactivity: halving time of the radiation of e.g. plutonium-239 is 24.400 years. This means that 72/24.400 = ± 0,003% is decayed yearly. Example 5. If you want a steady growth of the economy with 2 or 3% it implies a doubling in 36 or 24 years respectively. And so, there are countless examples from daily practice in which the "rule of 72" is a neat help to establish one's thoughts. Especially, for instance, when politicians threaten us with disaster and evil, in order to let us swallow their weird plans and arguments. 4. By lack of this knowledge The most flagrant example occured during one of the first speeches of Geert Wilders in the Dutch Parliament. He tried to make believe his audience that if things continue as they do, the muslims will take over here within 20 years. In our country there were, when Wilders told that, 1.6 mln muslims, and that population part was increasing then by about 3% per year. In order to "take over" that group must grow to more than 8 million. That's far more than two doublings. By a steady grow of 3% the first doubling will take place after 24 years, and, with that same grow, the second one after again 24 years. In the case of Wilders there could be projected in 2052 6.4 million muslims. But, there's something else going on: the procreative behaviour of the muslims appears to adapt to norms and values of those in The Netherlands. The growth rate is decreasing dramatically and moves already now between two and one and a half percent. Thus we have to adjust the second doubling to a grow percentage of, say, 1.8, which means that the second doubling will take place only after 40 years, thus 64 years in total after the datum date of September 2004. That is 2068. By that time the growing percentages will be leveled and we may be "happy" if the entire population still increases by 1%. A third doubling will then take place after an other 72 years. Apart from the fact that they all have to wait until they'll have reached the requiered age to vote, in the meantime looking away from the given that also the "native" people is not doing nothing. When then? Never! 5. ... even with our politicians Johan de Witt, in fact the first actuary in history, was reading loudly his mathematically complicated, and for us unreadable document, "Waerdije van Lyfrenten" (1671) in Parliament, in order to get his financement for his war against Britain, but is has become clear that none of learned people in this Government or Parliament, neither Balkenende, nor Hirsch Ballin, nor Bos, nor Pechtold, nor Marijnissen nor their advisors had the 72/n-rule at hand. If they had had then we wouldn't have had to listen to the endless untasteful debates of maverick Geert Wilders, and a quarter of an hour explanantion would have eliminated the fear for muslim take over. Jan Karman Middelburg (Neth) January 2010/2017 1 Nathan Keyfitz - Applied Mathematical Demography - 1985 Springer Verlag