We live in a world characterized by change. This is the 21st Century and the world is (as it has always been) in a flux where so many things are undergoing change: the values and beliefs of people; the world climate; the prices of all commodities and goods; amount of money required to send your children to college; the prices of houses before and after the sub-prime crisis, etc. Someone even once remarked that “the only constant is change” – a seemingly contradictory statement. On reflection, the truth of this statement will become self-evident: the only phenomenon that fully describes the modern world, the common denominator that ascribes all human and earthly events and activities is change, and that fact is something that itself remains constant or unchanging.
It is therefore important for humankind to be able to measure changes in all phenomena, be they quantitative or qualitative. At the early stage of human civilization, people learned to measure changes by pure observation. For example, prehistoric tribes noticed that meat when placed over fire changed its colour, texture and taste, and this change happened over time. Similarly, a keen observer in early societies might have noticed that the height of a shadow cast upon a vertical pole by the sun varied according to the time of the day – whether it was the morning sunlight, the strong mid-day sun or the fading lights of the late evening.
What we could usefully discern from the two above illustrations is that to observe change alone is not enough. One needs to measure the rate of change or the amount of change effected over a period of time. In other words, one needs to quantify change and that is where mathematics comes in. In common parlance, the rate of change simply measures the amount of change brought about over time; but in mathematical jargon the rate of change measure the rate at which one variable changes with respect to another variable. This second variable is often the time, but not always. In general algebraic notation, the rate of change of y with respect to x measures the rate at which the amount of variable y changes as the amount of variable x changes.
If this seems like playing with words or just a matter of semantics to you, I am not surprised because to a non-mathematician, the last sentence of the paragraph above doesn’t seem to explain anything. However, to a person who has learned Math (specifically Calculus, an important branch of Math) up to High School or undergraduate level, this is but one way used to describe how Math measures change.
Before the advent of Calculus ( most Math historians attribute the discovery or the ‘invention’ of Calculus to Leibniz and Newton, two Giants of Math ) simple notions of rates of change could be described by making use of relatively simple formulae and equations involving few terms. A specific example is in order: to measure speed of a cyclist, for instance, one measures the time taken by the cyclist to travel a certain distance of say 500 m. If the cyclist took 5 minutes to cover a distance of 500 m, then by the simple Math formula, his speed is simply 500m divided by 5 minutes, which equals 100 m per minute. So we say, on average, the cyclist could cover 100 m in 1 minute of time. Other slightly more complicated formulae had been devised by mathematicians to compute a great variety of rates such as the rate of flow of a viscous liquid, the rate at which an amount of money grew when put in a savings account in a bank, the rate at which sea-birds died in a remote island, etc.
Another way Math is used to describe change is by graphing. If you have done some Math I am sure you have learned the use of graphs to illustrate the relation between two quantities (the variables’ ). Say you want to study the relation between the price of commodities like palm oil over a period of 10 years. You could plot the prices of palm oil ( the y-variable) against the years ( the x-variable) and suppose you obtain a straight line graph, in which case the rate of change of the price of palm oil per year can be computed by finding the gradient or slope of the line. Finding the slope of the line is a relatively simple matter, akin to finding the speed of the cyclist discussed above. Now should one obtain not a straight line, but a curve instead how is one going to compute the rate of change of the price of oil?
Math provides a powerful tool in the form of derivatives (or differential coefficient ) of a mathematical function which describes the curve in an algebraic equation y = f(x) where y represents the price of oil and x the years. And Math introduces the symbol dy/dx, which is the derivative of y with respect to x and one learns various ways of obtaining the value of dy/dx, from simple to complex, algebraic to trigonometric, functions. Engineers, economist, biologists, etc all make use of this powerful mathematical tool in their work to compute many things (‘variables’ or ‘parameters’) like the cost of materials used in building dams, bridges; the price of commodities, the exchange rate between currencies; the mortality rate of certain species of fish, birds, mammals, etc.
There are many other ways Math is used to describe change. Commensurate with the varieties of ways in which change or variation is taking place, Math has come out with many novel ways to measure the change. Branches of Math have evolved over the decades – examples are stochastic processes, matrix algebra, game theories – these are but some of the more complex ways Math is used to describe change in an ever-changing world.