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Encyclopedia > Exponential growth

The phrase exponential growth is often used in nontechnical contexts to mean merely surprisingly fast growth. In a strictly mathematical sense, though, exponential growth has a precise meaning and does not necessarily mean that growth will happen quickly. In fact, a population can grow exponentially but at a very slow absolute rate (as when money in a bank account earns a very low interest rate, for instance), and can grow surprisingly fast without growing exponentially. And some functions, such as the logistic function, approximate exponential growth over only part of their range. The "technical details" section below explains exactly what is required for a function to exhibit true exponential growth. A bank account is a monetary account with a banking institution recording the balance of money for a customer. ... Logistic curve, specifically the sigmoid function A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. ...

But the general principle behind exponential growth is that the larger a number gets, the faster it grows. Any exponentially growing number will eventually grow larger than any other number which grows at only a constant rate for the same amount of time (and will also grow larger than any function which grows only subexponentially). This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at \$1 and increases by \$1 each week. Although the second option, growing at a constant rate of \$1/week, pays more in the short run, the first option eventually grows much larger:

Week 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Option 1 \$0.01 \$0.02 \$0.04 \$0.08 \$0.16 \$0.32 \$0.64 \$1.28 \$2.56 \$5.12 \$10.24 \$20.48 \$40.96 \$81.92 \$163.84 \$327.68 \$655.36 \$1310.72 \$2621.44
Option 2 \$1 \$2 \$3 \$4 \$5 \$6 \$7 \$8 \$9 \$10 \$11 \$12 \$13 \$14 \$15 \$16 \$17 \$18 \$19
The graph illustrates how an exponential growth surpasses both linear and cubic growths

We can describe these cases mathematically. In the first case, the allowance at week n is 2n cents; thus, at week 15 the payout is 215 = 32768c = \$327.68. All formulas of the form kn, where k is an unchanging number greater than 1 (e.g., 2), and n is the amount of time elapsed, grow exponentially. In the second case, the payout at week n is simply n + 1 dollars. The payout grows at a constant rate of \$1 per week. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ...

This image shows a slightly more complicated example of an exponential function overtaking subexponential functions:

The red line represents 50x, similar to option 2 in the above example, except increasing by 50 a week instead of 1. Its value is largest until x gets around 7. The blue line represents the polynomial x3. Polynomials grow subexponentially, since the exponent (3 in this case) stays constant while the base (x) changes. This function is larger than the other two when x is between about 7 and 9. Then the exponential function 2x (in green) takes over and becomes larger than the other two functions for all x greater than about 10.

Anything that grows by the same percentage every year (or every month, day, hour etc.) is growing exponentially. For example, if the average number of offspring of each individual (or couple) in a population remains constant, the rate of growth is proportional to the number of individuals. Such an exponentially growing population grows three times as fast when there are six million individuals as it does when there are two million. Bank accounts with fixed-rate compound interest grow exponentially provided there are no deposits, withdrawals or service charges. Mathematically, the bank account balance for an account starting with s dollars, earning an annual interest rate r and left untouched for n years can be calculated as s(1 + r)n. So, in an account starting with \$1 and earning 5% annually, the account will have $1times(1+0.05)^1=1.05$ after 1 year, $1times(1+0.05)^{10}=1.62$ after 10 years, and \$131.50 after 100 years. Since the starting balance and rate do not change, the quantity $1times(1+0.05)=1.05$ can work as the value k in the formula kn given earlier. Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ...

## Technical details

Let x be a quantity growing exponentially with respect to time t. By definition, the rate of change dx/dt obeys the differential equation: Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ...

$!, frac{dx}{dt} = k x$

where k ≠ 0 is the constant of proportionality (related to the average number of offspring per individual in the case of the population). (See logistic function for a simple correction of this growth model where k is not constant). The solution to this equation is the exponential function $!, x(t)=x_0 e^{kt}$ -- hence the name exponential growth ('e' being a mathematical constant). The constant $!, x_0$ is the initial size of the population. Logistic curve, specifically the sigmoid function A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. ... The exponential function is one of the most important functions in mathematics. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ...

In the long run, exponential growth of any kind will overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α: Malthusian catastrophe, sometimes known as a Malthusian check, Malthusian crisis, Malthusian dilemma, Malthusian disaster, Malthusian trap, or Malthusian limit is a return to subsistence-level conditions as a result of agricultural (or, in later formulations, economic) production being eventually outstripped by growth in population. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

$lim_{xrightarrowinfty} {x^alpha over Ce^x} =0$

There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). Growth rates may also be faster than exponential. The linear and exponential models are not merely simple candidates but are those of greatest occurrence in nature.

In the above differential equation, if k < 0, then the quantity experiences exponential decay. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

## Characteristic quantities of exponential growth

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: â€œExponentâ€ redirects here. ...

$x(t) = x_0 e^{kt} = x_0 e^{t/tau} = x_0 times 2^{t/T} = x_0 left( 1 + frac{r}{100} right)^t,$

where as in the example above x0 expresses the initial quantity (i.e. x(t) for t = 0).

The quantity k is called the growth constant; the quantity r is known as the growth rate (percent increase per unit time); τ is the e-folding time; and T is the doubling time. Indicating one of these four equivalent quantities automatically permits calculating the three others, which are connected by the following equation (which can be derived by taking the natural logarithm of the above): In science, e-folding is the time interval in which an exponentially growing quantity increases by a factor of e. ... The doubling time is the period of time required for a quantity to double in size or value. ...

$k = frac{1}{tau} = frac{ln 2}{T} = ln left( 1 + frac{r}{100} right).,$

A popular approximated method for calculating the doubling time from the growth rate is the rule of 70, i.e. $T simeq 70 / r$ (or better: $T simeq 70 / r + 0.03$). The Rule of 70 is a financial term derived to determine the time it takes for the value of money to halve due to a given inflation rate. ...

## Limitations of exponential models

As discussed above, an important point about exponential growth is that even when it seems slow on the short run, it becomes impressively fast on the long run, with the initial quantity doubling at the doubling time, then doubling again and again. For instance, a population growth rate of 2% per year may seem small, but it actually implies doubling after 35 years, doubling again after another 35 years (i.e. becoming 4 times the initial population), etc. This implies that both the observed quantity and its time derivative will become several orders of magnitude larger than what was initially meant by the person who conceived the growth model. Because of this, some effects not initially taken into account will distort the growth law, usually moderating it as for instance in the logistic law. Exponential growth of a quantity placed in the real world (i.e. not in the abstract world of mathematics) is a model valid for a temporary period of time only. Logistic curve, specifically the sigmoid function A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. ...

For this reason, some people challenge the exponential growth model on the ground that it is valid for the short term only, i.e. nothing can grow indefinitely. For instance, a population in a closed environment cannot continue growing if it eats up all the available food and resources; industry cannot continue pumping carbon from the underground into the atmosphere beyond the limits connected with oil reservoirs and the consequences of climate change; etc. Problems of this kind exist for every mathematical representation of the real world, but are specially felt for exponential growth, since with this model growth accelerates as variables increase in a positive feedback, to a point where human response time to inconvenience can be insufficient. On these points, see also the Exponential stories below. For the thermonuclear reaction involving carbon that helps power stars, see CNO cycle. ... An oil reservoir, petroleum system or petroleum reservoir is often thought of as being an underground lake of oil, but it is actually composed of hydrocarbons contained in porous rock formations. ... Variations in CO2, temperature and dust from the Vostok ice core over the last 450,000 years For current global climate change, see Global warming. ... Positive feedback is a feedback system in which the system responds to the perturbation in the same direction as the perturbation (It is sometimes referred to as cumulative causation). ...

## Examples of exponential growth

• Biology.
• Microorganisms in a culture dish will grow exponentially, at first, after the first microorganism appears (but then logistically until the available food is exhausted, when growth stops).
• A virus (SARS, West Nile, smallpox) of sufficient infectivity (k > 0) will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people.
• Human population, if the number of births and deaths per person per year were to remain at current levels (but also see logistic growth).
• Many responses of living beings to stimuli, including human perception, are logarithmic responses, which are the inverse of exponential responses; the loudness and frequency of sound are perceived logarithmically, even with very faint stimulus, within the limits of perception. This is the reason that exponentially increasing the brightness of visual stimuli is perceived by humans as a linear increase, rather than an exponential increase. This has survival value. Generally it is important for the organisms to respond to stimuli in a wide range of levels, from very low levels, to very high levels, while the accuracy of the estimation of differences at high levels of stimulus is much less important for survival.
• Computer technology
• Processing power of computers. See also Moore's law and technological singularity (under exponential growth, there are no singularities. The singularity here is a metaphor.).
• In computational 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 complexity 2^x, if a problem of size x=10 requires 10 seconds to complete, and a problem of size x=11 requires 20 seconds, then a problem of size x=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 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 x in time t, then a processor twice as fast could only solve problems of size x+constant in the same time t. So exponentially complex algorithms are most often impractical, and the search for more efficient algorithms is one of the central goals of computer science.
• Internet traffic growth.
• Investment. The effect of compound interest over many years has a substantial effect on savings and a person's ability to retire. See also rule of 72
• Physics
• Multi-level marketing
Exponential increases are promised to appear in each new level of a starting member's downline as each subsequent member recruits more people.

## Exponential stories

The surprising characteristics of exponential growth have fascinated people through the ages.

### Rice on a chessboard

A courtier presented the Persian king with a beautiful, hand-made 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 2n − 1 grains on the nth square demanded over a million grains on the 21st square, more than a quadrillion on the 41st and there simply was not enough rice in the whole world for the final squares. (From Meadows et al. 1972, p.29 via Porritt 2005) Chessboard Chessboard with Staunton chess pieces A chessboard is often painted or engraved on a chess table. ...

For variation of this see Second Half of the Chessboard in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization's overall business strategy. An illustration of the principle. ...

### The water lily

French children are told a story in which they imagine having a pond with water lily leaves floating on the surface. The lily population doubles in size every day and if left unchecked will smother the pond in 30 days, killing all the other living things in the water. Day after day the plant seems small and so it is decided to leave it to grow until it half-covers the pond, before cutting it back. They are then asked, on what day that will occur. This is revealed to be the 29th day, and then there will be just one day to save the pond. (From Meadows et al. 1972, p.29 via Porritt 2005) Genera Barclaya Wall. ...

Albert A. Bartlett is a retired Emeritus Professor of Physics University of Colorado, Boulder, USA. Professor Bartlett has lectured over 1,500 times on Arithmetic, Population, and Energy. He has famously stated that The greatest shortcoming of the human race is our inability to understand the exponential function. ... Species Arthrobacter is a genus of bacteria that is commonly found in soil. ... Bacterial growth is process in which two clone daughter cells are produced by the cell division of one bacterium. ... The term cell growth is used in two different ways in biology. ... Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... The exponential function is one of the most important functions in mathematics. ... In Library and Information Science, information explosion is a term used for the ever increasing rate of publication. ... When plotted on a logarithmic graph, 15 separate lists of paradigm shifts for key events in human history show an exponential trend. ... The logistic function or logistic curve is defined by the mathematical formula: for real parameters a, m, n, and . ... In complexity theory, exponential time is the computation time of a problem where the time to complete the computation, m(n), is bounded by an exponential function of the problem size, n (i. ... In mathematical analysis, and in particular in the analysis of algorithms, to classify the growth of functions one has recourse to asymptotic notations. ... In complexity theory, EXPSPACE is the set of all decision problems solvable by a deterministic Turing machine in O(2p(n)) space, where p(n) is a polynomial function of n. ... In computational complexity theory, the complexity class EXPTIME (sometimes called EXP) is the set of all decision problems solvable by a deterministic Turing machine in O(2p(n)) time, where p(n) is a polynomial function of n. ... In finance, the rule of 72, the rule of 71, the rule of 70 and the rule of 69. ... The Rule of 70 is a financial term derived to determine the time it takes for the value of money to halve due to a given inflation rate. ... This is a list of exponential topics, by Wikipedia page. ... The Malthusian growth model, sometimes called the simple exponential growth model, is essentially exponential growth based on a constant rate of compound interest. ...

Results from FactBites:

 Exponential growth - Wikipedia, the free encyclopedia (1829 words) In mathematics, a quantity that grows exponentially is one whose growth rate is always proportional to its current size. This is the reason that exponentially increasing the brightness of visual stimuli is perceived by humans as a smooth (linear) increase, rather than an exponential increase. The resulting exponential growth of electrons and ions may rapidly lead to complete dielectric breakdown of the material.
 Exponential function - Wikipedia, the free encyclopedia (1112 words) If a variable's growth or decay rate is proportional to its size — as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay — then the variable can be written as a constant times an exponential function of time. The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. The definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices (in which case the function is called the matrix exponential).
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