THE greatest shortcoming of the human race is our inability to understand the exponential function.鈥� These are the words of the late Albert Bartlett, a physicist at the University of Colorado, Boulder, whose . Arguably, he鈥檚 right.
Take saving for retirement. 鈥淪tart early鈥� is the mantra, but it is easy to overlook just how much difference a few years can make. It all comes down to exponential growth 鈥� an often abused term that refers to anything that grows in proportion to its current value. It dictates that a forward-thinking 18-year-old can retire as a millionaire at 65 by investing around 拢250 a month with an average annual return of 7 per cent.
That figure might sound high by today鈥檚 standards, but it鈥檚 a rough average of the stock market return since 1960. The surprise is that when our saver reaches 55, the savings will amount to a little under 拢500,000. Thanks to the power of compound interest, however 鈥� exponential growth by another name 鈥� it will double to 拢1 million just 10 years later. Wait until you鈥檙e 30 to start saving that 拢250 and you鈥檒l only reach about half as much (see diagram). Starting at 30, you鈥檇 actually need to save more than 拢600 a month to make it to 拢1 million by 65.FIG-mg30510501.jpg
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Exponential growth鈥檚 stealth factor is nicely illustrated by the story of the man who invented chaturanga, an Indian precursor to chess. He presented his king with a beautifully laid out board divided into 64 squares and when asked to name his reward, requested a grain of wheat to be placed on the first square, two on the next, four on the third, and so on. It sounded a modest reward, but had the king obliged across the board, he would have given away more than 18 billion billion grains. Fail to understand exponential growth, and our debts can rapidly spiral out of control too. This is an engine for creation and destruction wrapped up in deceptively simple maths.
In reference to the chaturanga legend, US futurist Ray Kurzweil refers to the sudden changes that spring from exponential growth as the 鈥渟econd half of the chessboard鈥�. The number of transistors that can fit on a electronic chip provides an example. Over the past few decades, it has roughly doubled every 18 months, a phenomenon known as Moore鈥檚 law. The accelerating effect of exponential growth explains why we spent 25 years with bulky desktop computers before rapidly switching over to sleek smartphones. Kurzweil is famed for believing that this sort of technological growth will lead to an event called the singularity, when computers will become powerful and smart enough to improve themselves and outpace us all.
The spread of viruses often works in a similar way: one ill person infects a few others, who in turn each infect a few more, until we鈥檝e got an epidemic on our hands. Immunisation acts as the limiting factor, which is why the world scrambled to treat last year鈥檚 Ebola outbreak, which at one point saw the number of known cases doubling every few weeks.
When it comes to exponential growth, you can鈥檛 trust your short-term instincts. Whether it鈥檚 finance or technology, the largest changes won鈥檛 happen for some time. But when they do, your whole world can be turned upside down in an instant.
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This article appeared in print under the headline 鈥淣eed to know: Exponential growth鈥�
