A group of n people is infected with a disease. Each of these n persons daily infects p other persons coming in close contact. How many people will be infected by this disease after x days?
This scenario is all too familiar nowadays, of course, thanks to the COVID-19 pandemic. (Incidentally, today marks one year since the first COVID-19 case in the Maltese islands.) Although it is a much-simplified model of what happens in real life – most of these infected persons will eventually be unable to infect others, for example – answering this question sheds an important light to the handling of a pandemic. The answer to the question, by the way, is n times (1+p) to the power of x.
Let us now consider a different problem. Suppose we have n euros in a fixed bank account. Every year, the bank increases this amount by the interest rate p%. How much money will we have in the account after x years? The answer is the same: n times (1+p%) to the power of x.
In both scenarios, and in many others in real life, the quantity in the following time period is increasing by multiplying the current quantity by a fixed number, in this case (1+p). The quantity, then, increases in relation to its current size; the bigger it is, the bigger it becomes.
On other occasions, we have a quantity that increases relative to its current size continuously instead of every constant time period, say every day or every year. In mathematics, we would usually model such scenarios as the differential equation y’(t) = k y(t), with the initial condition y(0) = n, for some appropriate value k, where t is the time in seconds. The solution of this differential equation is n times e to the power of kt, where e is approximately equal to 2.718. The function e to the power of t is called the exponential function.
This function is at the heart of every situation, arising from a natural phenomenon or otherwise, having a quantity that grows or diminishes at a rate related to its current size. Other phenomena modelled by the exponential function, besides the two examples already mentioned, include the charge in a discharging capacitor, the number of bacteria growing in a Petri dish, the amount of memory and time required for intractable computer algorithms, and – believe it or not – the way internet memes spread across a social network!
Dr Alexander Farrugia is the mathematics subject coordinator at the University of Malta Ġ.F. Abela Junior College.
Sound bites
• From the local scene: Roughly speaking, a network system is controllable from some of its nodes if the output of the network only depends on the input linked to those nodes. In secondary school, we learn how to factorise polynomials like x2 + 2x into the two factors x and x + 2. In the paper Alexander Farrugia, The rank of Pseudo walk matrices: controllable and recalcitrant pairs. Open Journal of Discrete Applied Mathematics, 3(3) (2020), 41-52, it was discovered that the number of factors of a specific polynomial suffices to roughly decide whether or not a network is controllable from select nodes.
• From the international scene: Very recently, Andrew Booker and Andrew Sutherland, in the paper On a question of Mordell. Proceedings of the National Academy of Sciences of the United States of America (in press), discovered a representation of the numbers 42, 165 and 906 as the sum of three cubes, which representation had eluded mathematicians for centuries.
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Did you know?
• The constant e is an irrational number, or a number which cannot be expressed as a fraction. However, the fraction 19/7 is a reasonable approximation to e. Euler was first to prove that e is irrational, in 1737. Indeed, this number is denoted by ‘e’ in Euler’s honour.
• A derangement is a permutation of distinct objects such that no object appears in its starting position. For example, a derangement of ‘ABCDEF’ is ‘FCEBAD’, but ‘AFEDCB’ is not a derangement of ‘ABCDEF’. The number of all possible permutations divided by the number of all possible derangements is approximately e.
• The exponential function may be extended to complex numbers, which include square roots of negative numbers. Remarkably, all results and equations involving trigonometry may be proved or solved using the complex exponential function.
• Walter Rudin starts his textbook Real and Complex Analysis with the bold claim: “The exponential function is the most important function in mathematics.”