Exponential growth
In
mathematics, a quantity that grows exponentially is one whose growth rate is always proportional to its current size. Such growth is said to follow an exponential law . 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.
Encyclopedia
In
mathematics, a quantity that
grows exponentially is one whose growth rate is always proportional to its current size. Such growth is said to follow an
exponential law . 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.
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 , 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 . 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 2
n cents; thus, at week 15 the payout is 2
15 = 32768c = $327.68. All formulas of the form
kn, where
k is an unchanging number greater than 1 , 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:
The red line represents 50
x, 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 stays constant while the base changes. This function is larger than the other two when
x is between about 7 and 9. Then the exponential function 2
x 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 is growing exponentially. For example, if the average number of offspring of each individual 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 . So, in an account starting with $1 and earning 5% annually, the account will have after 1 year, after 10 years, and $131.50 after 100 years. Since the starting balance and rate don't change, the quantity can work as the value
k in the formula
kn given earlier.
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:
where
k > 0 is the
constant of proportionality . . The solution to this equation is the
exponential function -- hence the name
exponential growth . The constant 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 as well as any
polynomial growth, i.e., for all α:
There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear . 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.
Examples of exponential growth
- Biology.
- Microorganisms in a culture dish will grow exponentially, at first, after the first microorganism appears .
- A virus of sufficient infectivity 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 .
- 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 smooth 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 .
- In computational complexity theory, computer algorithms of exponential complexity require an exponentially increasing amount of resources 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, then a problem of size x=11 will require 20 seconds, and x=12 will require 40 seconds. This kind of algorithm typically becomes unusable at very small problem sizes, often between 30 and 100 items . 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 one. 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+1 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.
- an analysis of Wikipedia's growth rates based on article growth rates and other statistics
- 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
...
produces multiple
neutrons, each of which can be absorbed by adjacent uranium atoms, causing them to fission in turn. If the probability of neutron absorption exceeds the probability of neutron escape ,
k > 0 and so the production rate of neutrons and induced uranium fissions increases exponentially, in an uncontrolled reaction.
- Exponential increases appear in each 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 grains on the 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.
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.
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.
See also
...
...
- exponential algorithm
- asymptotic notation
- EXPSPACE
- EXPTIME
- rule of 72/rule of 70
- list of exponential topics
- Malthusian growth model
External links
- - 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.
References
- Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III. 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 0-471-74747-5
- 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.
