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Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. In the case of a discrete
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
of definition with equal intervals, it is also called geometric growth or geometric decay since the function values form a geometric progression. The formula for exponential growth of a variable at the growth rate , as time goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is x_t = x_0(1+r)^t where is the value of at time 0. The growth of a
bacteria Bacteria (; singular: bacterium) are ubiquitous, mostly free-living organisms often consisting of one biological cell. They constitute a large domain of prokaryotic microorganisms. Typically a few micrometres in length, bacteria were am ...
l
colony In modern parlance, a colony is a territory subject to a form of foreign rule. Though dominated by the foreign colonizers, colonies remain separate from the administration of the original country of the colonizers, the '' metropolitan state' ...
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 to compound interest, and the spread of viral 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 into logistic 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.


Examples


Biology

* The number of
microorganism A microorganism, or microbe,, ''mikros'', "small") and ''organism'' from the el, ὀργανισμός, ''organismós'', "organism"). It is usually written as a single word but is sometimes hyphenated (''micro-organism''), especially in old ...
s in a
culture Culture () is an umbrella term which encompasses the social behavior, institutions, and norms found in human societies, as well as the knowledge, beliefs, arts, laws, customs, capabilities, and habits of the individuals in these groups ...
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 organism splits 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. * A virus (for example
COVID-19 Coronavirus disease 2019 (COVID-19) is a contagious disease caused by a virus, the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The first known case was identified in Wuhan, China, in December 2019. The disease quick ...
, or
smallpox Smallpox was an infectious disease caused by variola virus (often called smallpox virus) which belongs to the genus Orthopoxvirus. The last naturally occurring case was diagnosed in October 1977, and the World Health Organization (WHO) c ...
) typically will spread exponentially at first, if no artificial
immunization Immunization, or immunisation, is the process by which an individual's immune system becomes fortified against an infectious agent (known as the immunogen). When this system is exposed to molecules that are foreign to the body, called ''non-s ...
is available. Each infected person can infect multiple new people.


Physics

* Avalanche breakdown within a
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
material. A free
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
becomes sufficiently accelerated by an externally applied electrical field that it frees up additional electrons as it collides with
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
s or
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and b ...
s of the dielectric media. These ''secondary'' electrons also are accelerated, creating larger numbers of free electrons. The resulting exponential growth of electrons and ions may rapidly lead to complete dielectric breakdown of the material. * Nuclear chain reaction (the concept behind nuclear reactors and nuclear weapons). Each
uranium Uranium is a chemical element with the symbol U and atomic number 92. It is a silvery-grey metal in the actinide series of the periodic table. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons. Uranium is weak ...
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: * Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
that undergoes fission produces multiple
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the atomic nucleus, nuclei of atoms. Since protons and ...
s, each of which can be absorbed by adjacent uranium atoms, causing them to fission in turn. If the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
of neutron absorption exceeds the probability of neutron escape (a function of the shape and
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
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." * Positive feedback within the linear range of electrical or electroacoustic amplification can result in the exponential growth of the amplified signal, although
resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscil ...
effects may favor some component frequencies of the signal over others.


Economics

* Economic growth is expressed in percentage terms, implying exponential growth.


Finance

* Compound interest at a constant interest rate provides exponential growth of the capital. See also rule of 72. * Pyramid schemes or Ponzi schemes also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors.


Computer science

* Processing power of computers. See also Moore's law and
technological singularity The technological singularity—or simply the singularity—is a hypothetical future point in time at which technological growth becomes uncontrollable and irreversible, resulting in unforeseeable changes to human civilization. According to the m ...
. (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 futurist Ray Kurzweil.) * In
computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...
, 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 complexity , if a problem of size requires 10 seconds to complete, and a problem of size requires 20 seconds, then a problem of size 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 of Moore's Law do not help the situation much because doubling processor speed merely allows you to increase the problem size by a constant. E.g. if a slow processor can solve problems of size in time , then a processor twice as fast could only solve problems of size in the same time . 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 phenomena

* Internet contents, such as internet memes or
video Video is an electronic medium for the recording, copying, playback, broadcasting, and display of moving visual media. Video was first developed for mechanical television systems, which were quickly replaced by cathode-ray tube (CRT) sy ...
s, can spread in an exponential manner, often said to " go viral" as an analogy to the spread of viruses. With media such as social 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. For example, the video Gangnam Style was uploaded to
YouTube YouTube is a global online video sharing and social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by Google, and is the second mo ...
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.


Basic formula

A quantity depends exponentially on time if x(t)=a\cdot b^ where the constant is the initial value of , x(0) = a \, , the constant is a positive growth factor, and is the time constant—the time required for to increase by one factor of : x(t+\tau) = a \cdot b^ = a \cdot b^ \cdot b^ = x(t) \cdot b\, . If and , then has exponential growth. If and , or and , then has exponential decay. 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 implies , and . x(t)=a\cdot b^ = 1 \cdot 2^ x(1\text) = 1\cdot 2^ = 1 \cdot 2^6 =64. After one hour, or six ten-minute intervals, there would be sixty-four bacteria. Many pairs of a dimensionless non-negative number and an amount of time (a physical quantity which can be expressed as the product of a number of units and a unit of time) represent the same growth rate, with proportional to . For any fixed not equal to 1 (e.g. '' e'' or 2), the growth rate is given by the non-zero time . For any non-zero time the growth rate is given by the dimensionless positive number . Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a different base. The most common forms are the following: x(t) = x_0\cdot e^ = x_0\cdot e^ = x_0 \cdot 2^ = x_0\cdot \left( 1 + \frac \right)^, where expresses the initial quantity . Parameters (negative in the case of exponential decay): * The ''growth constant'' is the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
(number of times per unit time) of growing by a factor ; in finance it is also called the logarithmic return, continuously compounded return, or force of interest. * The '' e-folding time'' ''τ'' is the time it takes to grow by a factor '' e''. * The '' doubling time'' ''T'' is the time it takes to double. * The percent increase (a dimensionless number) in a period . The quantities , , and , and for a given also , have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above): k = \frac = \frac = \frac where corresponds to and to and being infinite. If is the unit of time the quotient is simply the number of units of time. Using the notation for the (dimensionless) number of units of time rather than the time itself, can be replaced by , but for uniformity this has been avoided here. In this case the division by 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 the rule of 70, that is, T \simeq 70 / r.


Reformulation as log-linear growth

If a variable exhibits exponential growth according to x(t) = x_0 (1+r)^t, then the log (to any base) of grows linearly over time, as can be seen by taking
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
s of both sides of the exponential growth equation: \log x(t) = \log x_0 + t \cdot \log (1+r). This allows an exponentially growing variable to be modeled with a
log-linear model A log-linear model is a mathematical model that takes the form of a function whose logarithm equals a linear combination of the parameters of the model, which makes it possible to apply (possibly multivariate) linear regression. That is, it h ...
. For example, if one wishes to empirically estimate the growth rate from intertemporal data on , one can linearly regress on .


Differential equation

The
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
x(t) = x_0 e^ satisfies the linear differential equation: \frac = kx saying that the change per instant of time of at time is proportional to the value of , and has the initial value x(0) = x_0. The differential equation is solved by direct integration: \begin \frac & = kx \\ pt\frac x & = k\, dt \\ pt\int_^ \frac & = k \int_0^t \, dt \\ pt\ln \frac & = kt. \end so that x(t) = x_0 e^. In the above differential equation, if , then the quantity experiences exponential decay. For a
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
variation of this growth model see logistic function.


Other growth rates

In the long run, exponential growth of any kind will overtake linear growth of any kind (that is the basis of the
Malthusian catastrophe Malthusianism is the idea that population growth is potentially exponential while the growth of the food supply or other resources is linear, which eventually reduces living standards to the point of triggering a population die off. This event, c ...
) as well as any
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
growth, that is, for all : \lim_ \frac = 0. There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). See . Growth rates may also be faster than exponential. In the most extreme case, when growth increases without bound in finite time, it is called hyperbolic growth. In between exponential and hyperbolic growth lie more classes of growth behavior, like the hyperoperations beginning at tetration, and A(n,n), the diagonal of the Ackermann function.


Logistic growth

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. In 1845, the Belgian mathematician Pierre François Verhulst first proposed a mathematical model of growth like this, called the " logistic growth".


Limitations of models

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 ignored negative feedback factors become significant (leading to a logistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.


Exponential growth bias

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. Below are some stories that emphasize this bias.


Rice on a chessboard

According to an old legend, vizier Sissa Ben Dahir presented an Indian King Sharim with a beautiful handmade chessboard. 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, etc. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for grains on the th square demanded over a million grains on the 21st square, more than a million million (
trillion ''Trillion'' is a number with two distinct definitions: *1,000,000,000,000, i.e. one million million, or (ten to the twelfth power), as defined on the short scale. This is now the meaning in both American and British English. * 1,000,000,000,00 ...
) on the 41st and there simply was not enough rice in the whole world for the final squares. (From Swirski, 2006) The
second half of the chessboard The wheat and chessboard problem (sometimes expressed in terms of rice grains) is a mathematical problem expressed in textual form as: The problem may be solved using simple addition. With 64 squares on a chessboard, if the number of grains d ...
is the time when an exponentially growing influence is having a significant economic impact on an organization's overall business strategy.


Water lily

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.


See also

*
Accelerating change In futures studies and the history of technology, accelerating change is the observed exponential nature of the rate of technological change in recent history, which may suggest faster and more profound change in the future and may or may not be ...
*
Albert Allen Bartlett Albert Allen Bartlett (March 21, 1923 – September 7, 2013) was an emeritus professor of physics at the University of Colorado at Boulder, US. Professor Bartlett had lectured over 1,742 times since September, 1969 on ''Arithmetic, Population, ...
* Arthrobacter * Asymptotic notation *
Bacterial growth 250px, Growth is shown as ''L'' = log(numbers) where numbers is the number of colony forming units per ml, versus ''T'' (time.) Bacterial growth is proliferation of bacterium into two daughter cells, in a process called binary fission. Providing ...
*
Bounded growth Bounded growth occurs when the growth rate of a mathematical function is constantly increasing at a decreasing rate. Asymptotically, bounded growth approaches a fixed value. This contrasts with exponential growth, which is constantly increasing at ...
* Cell growth * Combinatorial explosion *
Exponential algorithm In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
* EXPSPACE * EXPTIME *
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
* Hyperbolic growth * Information explosion * Law of accelerating returns * List of exponential topics * Logarithmic growth * Logistic function * Malthusian growth model * Menger sponge * Moore's law * Quadratic growth * Stein's law


References


Sources

* Meadows, Donella. Randers, Jorgen. Meadows, Dennis. '' The Limits to Growth: The 30-Year Update.'' Chelsea Green Publishing, 2004. * Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III. (1972) '' The Limits to Growth''. New York: University Books. * Porritt, J. ''Capitalism as if the world matters'', Earthscan 2005. * Swirski, Peter. ''Of Literature and Knowledge: Explorations in Narrative Thought Experiments, Evolution, and Game Theory''. New York: Routledge. * Thomson, David G. ''Blueprint to a Billion: 7 Essentials to Achieve Exponential Growth'', Wiley Dec 2005, * Tsirel, S. V. 2004
On the Possible Reasons for the Hyperexponential Growth of the Earth Population
''Mathematical Modeling of Social and Economic Dynamics'' / Ed. by M. G. Dmitriev and A. P. Petrov, pp. 367–9. Moscow: Russian State Social University, 2004.


External links


Growth in a Finite World – Sustainability and the Exponential Function
— Presentation
Dr. Albert Bartlett: Arithmetic, Population and Energy
— streaming video and audio 58 min {{Large numbers Ordinary differential equations Temporal exponentials Mathematical modeling Growth curves