Exponential growth occurs when a quantity grows as anexponential function of time. The quantity grows at a ratedirectly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now.
In more technical language, its instantaneousrate of change (that is, thederivative) of a quantity with respect to an independent variable isproportional to the quantity itself. Often the independent variable is time. Described as afunction, a quantity undergoing exponential growth is anexponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such asquadratic growth). Exponential growth isthe inverse oflogarithmic growth.
Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function grows at an ever increasing rate, but is much slower than growing exponentially. For example, when it grows at 3 times its size, but when it grows at 30% of its size. If an exponentially growing function grows at a rate that is 3 times is present size, then it always grows at a rate that is 3 times its present size. When it is 10 times as big as it is now, it will grow 10 times as fast.
If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoingexponential decay instead. In the case of a discretedomain of definition with equal intervals, it is also calledgeometric growth orgeometric decay since the function values form ageometric progression.
The formula for exponential growth of a variablex at the growth rater, as timet goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is
wherex0 is the value ofx at time 0. The growth of a bacterialcolony is often used to illustrate it. One bacterium splits itself into two, each of which splits itself resulting in four, then eight, 16, 32, and so on. The amount of increase keeps increasing because it is proportional to the ever-increasing number of bacteria. Growth like this is observed in real-life activity or phenomena, such as the spread of virus infection, the growth of debt due tocompound interest, and the spread ofviral videos. In real cases, initial exponential growth often does not last forever, instead slowing down eventually due to upper limits caused by external factors and turning intologistic growth.
Terms like "exponential growth" are sometimes incorrectly interpreted as "rapid growth". Indeed, something that grows exponentially can in fact be growing slowly at first.[1][2]
The number ofmicroorganisms in aculture will increase exponentially until an essential nutrient is exhausted, so there is no more of that nutrient for more organisms to grow. Typically the first organismsplits into two daughter organisms, who then each split to form four, who split to form eight, and so on. Because exponential growth indicates constant growth rate, it is frequently assumed that exponentially growing cells are at a steady-state. However, cells can grow exponentially at a constant rate while remodeling their metabolism and gene expression.[3]
A virus (for exampleCOVID-19, orsmallpox) typically will spread exponentially at first, if no artificialimmunization is available. Each infected person can infect multiple new people.
Avalanche breakdown within adielectric material. A freeelectron becomes sufficiently accelerated by an externally appliedelectrical field that it frees up additional electrons as it collides withatoms ormolecules of the dielectric media. Thesesecondary electrons also are accelerated, creating larger numbers of free electrons. The resulting exponential growth of electrons and ions may rapidly lead to completedielectric breakdown of the material.
Nuclear chain reaction (the concept behindnuclear reactors andnuclear weapons). Eachuraniumnucleus that undergoesfission produces multipleneutrons, each of which can beabsorbed by adjacent uranium atoms, causing them to fission in turn. If theprobability of neutron absorption exceeds the probability of neutron escape (afunction of theshape andmass of the uranium), the production rate of neutrons and induced uranium fissions increases exponentially, in an uncontrolled reaction. "Due to the exponential rate of increase, at any point in the chain reaction 99% of the energy will have been released in the last 4.6 generations. It is a reasonable approximation to think of the first 53 generations as a latency period leading up to the actual explosion, which only takes 3–4 generations."[4]
Pyramid schemes orPonzi schemes also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors.
Processing power of computers. See alsoMoore's law andtechnological singularity. (Under exponential growth, there are no singularities. The singularity here is a metaphor, meant to convey an unimaginable future. The link of this hypothetical concept with exponential growth is most vocally made by futuristRay Kurzweil.)
Incomputational complexity theory, computer algorithms of exponential complexity require an exponentially increasing amount of resources (e.g. time, computer memory) for only a constant increase in problem size. So for an algorithm of time complexity2x, if a problem of sizex = 10 requires 10 seconds to complete, and a problem of sizex = 11 requires 20 seconds, then a problem of sizex = 12 will require 40 seconds. This kind of algorithm typically becomes unusable at very small problem sizes, often between 30 and 100 items (most computer algorithms need to be able to solve much larger problems, up to tens of thousands or even millions of items in reasonable times, something that would be physically impossible with an exponential algorithm). Also, the effects ofMoore's Law do not help the situation much because doubling processor speed merely increases the feasible problem size by a constant. E.g. if a slow processor can solve problems of sizex in timet, then a processor twice as fast could only solve problems of sizex + constant in the same timet. So exponentially complex algorithms are most often impractical, and the search for more efficient algorithms is one of the central goals of computer science today.
Internet contents, such asinternet memes orvideos, can spread in an exponential manner, often said to "go viral" as an analogy to the spread of viruses.[6] With media such associal networks, one person can forward the same content to many people simultaneously, who then spread it to even more people, and so on, causing rapid spread.[7] For example, the videoGangnam Style was uploaded to YouTube on 15 July 2012, reaching hundreds of thousands of viewers on the first day, millions on the twentieth day, and was cumulatively viewed by hundreds of millions in less than two months.[6][8]
A quantityx depends exponentially on timet ifwhere the constanta is the initial value ofx, the constantb is a positive growth factor, andτ is thetime constant—the time required forx to increase by one factor ofb:
Example:If a species of bacteria doubles every ten minutes, starting out with only one bacterium, how many bacteria would be present after one hour? The question impliesa = 1,b = 2 andτ = 10 min.
After one hour, or six ten-minute intervals, there would be sixty-four bacteria.
Many pairs(b,τ) of adimensionless non-negative numberb and an amount of timeτ (aphysical quantity which can be expressed as the product of a number of units and a unit of time) represent the same growth rate, withτ proportional tologb. For any fixedb not equal to 1 (e.g.e or 2), the growth rate is given by the non-zero timeτ. For any non-zero timeτ the growth rate is given by the dimensionless positive number b.
Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a differentbase. The most common forms are the following:wherex0 expresses the initial quantityx(0).
Parameters (negative in the case of exponential decay):
The percent increaser (a dimensionless number) in a periodp.
The quantitiesk,τ, andT, and for a givenp alsor, have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above):wherek = 0 corresponds tor = 0 and toτ andT being infinite.
Ifp is the unit of time the quotientt/p is simply the number of units of time. Using the notationt for the (dimensionless) number of units of time rather than the time itself,t/p can be replaced byt, but for uniformity this has been avoided here. In this case the division byp in the last formula is not a numerical division either, but converts a dimensionless number to the correct quantity including unit.
A popular approximated method for calculating the doubling time from the growth rate is therule of 70,that is,.
Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/t and 72/t approximations. In theSVG version, hover over a graph to highlight it and its complement.
If a variablex exhibits exponential growth according to, then the log (to any base) ofxgrows linearly over time, as can be seen by takinglogarithms of both sides of the exponential growth equation:
This allows an exponentially growing variable to be modeled with alog-linear model. For example, if one wishes to empirically estimate the growth rate from intertemporal data onx, one canlinearly regresslogx ont.
In the long run, exponential growth of any kind will overtake linear growth of any kind (that is the basis of theMalthusian catastrophe) as well as anypolynomial growth, that is, for allα:
Growth rates may also be faster than exponential. In the most extreme case, when growth increases without bound in finite time, it is calledhyperbolic growth. In between exponential and hyperbolic growth lie more classes of growth behavior, like thehyperoperations beginning attetration, and, the diagonal of theAckermann function.
In reality, initial exponential growth is often not sustained forever. After some period, it will be slowed by external or environmental factors. For example, population growth may reach an upper limit due to resource limitations.[9] In 1845, the Belgian mathematicianPierre François Verhulst first proposed a mathematical model of growth like this, called the "logistic growth".[10]
Exponential growth models of physical phenomena only apply within limited regions, as unbounded growth is not physically realistic. Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignorednegative feedback factors become significant (leading to alogistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.
Studies show that human beings have difficulty understanding exponential growth. Exponential growth bias is the tendency to underestimate compound growth processes. This bias can have financial implications as well.[11]
According to legend, vizier Sissa Ben Dahir presented an Indian King Sharim with a beautiful handmadechessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third, and so on. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for2n−1 grains on thenth square demanded over a million grains on the 21st square, more than a million million (a.k.a.trillion) on the 41st and there simply was not enough rice in the whole world for the final squares. (From Swirski, 2006)[12]
The "second half of the chessboard" refers to the time when an exponentially growing influence is having a significant economic impact on an organization's overall business strategy.
French children are offered a riddle, which appears to be an aspect of exponential growth: "the apparent suddenness with which an exponentially growing quantity approaches a fixed limit". The riddle imagines a water lily plant growing in a pond. The plant doubles in size every day and, if left alone, it would smother the pond in 30 days killing all the other living things in the water. Day after day, the plant's growth is small, so it is decided that it won't be a concern until it covers half of the pond. Which day will that be? The 29th day, leaving only one day to save the pond.[13][12]
^Stango, Victor; Zinman, Jonathan (2009). "Exponential Growth Bias and Household Finance".The Journal of Finance.64 (6):2807–2849.doi:10.1111/j.1540-6261.2009.01518.x.
^abPorritt, Jonathan (2005).Capitalism: as if the world matters. London: Earthscan. p. 49.ISBN1-84407-192-8.
^Meadows, Donella (2004).The Limits to Growth: The 30-Year Update. Chelsea Green Publishing. p. 21.ISBN9781603581554.
Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III. (1972)The Limits to Growth. New York: University Books.ISBN0-87663-165-0
Porritt, J.Capitalism as if the world matters, Earthscan 2005.ISBN1-84407-192-8
Swirski, Peter.Of Literature and Knowledge: Explorations in Narrative Thought Experiments, Evolution, and Game Theory. New York: Routledge.ISBN0-415-42060-1
Thomson, David G.Blueprint to a Billion: 7 Essentials to Achieve Exponential Growth, Wiley Dec 2005,ISBN0-471-74747-5
