Exponential growth

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In mathematics, a quantity that grows exponentially (or geometrically) is one that grows at a rate proportional to its size. Such growth is said to follow an exponential law (but see also Malthusian growth model). This implies that for any exponentially growing quantity, the larger the quantity gets, the faster it grows. But it also implies that the relationship between the size of the dependent variable and its rate of growth is governed by a strict law, of the simplest kind: direct proportion. It is proved in calculus that this law requires that the quantity is given by the exponential function, if we use the correct time scale. This explains the name.

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Intuition

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.

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
Option 1: 1c, 2c, 4c, 8c, 16c, 32c, 64c, $1.28, $2.56, $5.12, $10.24, $20.48, $40.96, $81.92, $163.84, $327.68
Option 2:$1, $2, $3, $4,  $5,  $6,  $7,  $8,    $9,   $10,    $11,    $12,    $13,    $14,     $15,     $16

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.

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

Image:Exponential.png

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 green 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 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 <math>s (1+r)^n</math>. So, in an account starting with $1 and earning 5% annually, the account will have <math>\$1\times(1+0.05)^1=\$1.05</math> after 1 year, <math>\$1\times(1+0.05)^{10}=\$1.62</math> after 10 years, and $131.50 after 100 years. Since the starting balance and rate don't change, the quantity <math>\$1\times(1+0.05)=\$1.05</math> can work as the value k in the formula kn given earlier.

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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:

<math> \!\, \frac{dx}{dt} = k x</math>

where k > 0 is the constant of proportionality (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 <math> \!\, x(t)=x_0 e^{kt}</math> -- hence the name exponential growth ('e' being a mathematical constant). The constant <math>\!\, x_0</math> is determined by the initial size of the population.

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:

<math>\lim_{x\rightarrow\infty} {x^\alpha \over Ce^x} =0</math>

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 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.

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Examples of exponential growth

  • Multi-level marketing
Exponential increases appear in each level of a starting member's downline as each subsequent member recruits more people.
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Exponential stories

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

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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 <math>2^{n-1}</math> grains on the <math>n</math>th square demanded over a million grains on the 21st square, more than a million million 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)

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

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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 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 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)

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See also

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External links

  • Exponent calculator - One of the best ways to see how exponents work is to simply try different examples. This calculator enables you to enter an exponent and a base number and see the result.
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References

Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III. (1972) The Limits to Growth. New York: University Books. ISBN 0-87663-165-0

Porritt, J. Capitalism as if the world matters, Earthscan 2005. ISBN 1-84407-192-8

Thomson, David G. Blueprint to a Billion: 7 Essentials to Achieve Exponential Growth, Wiley Dec 2005, ISBN 0-471-74747-5es:Crecimiento exponencial fr:Croissance exponentielle hu:Exponenciális növekedés nl:Exponentiële groei no:Eksponentiell vekst pl:Wzrost wykładniczy ru:Экспоненциальный рост

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