In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. This means 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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In experimental design, a dependent variable is a variable whose values in different treatment conditions are compared. ... This article is about proportionality, the mathematical relation. ... Jump to: navigation, search Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... Jump to: navigation, search The exponential function is one of the most important functions in mathematics. ...

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. The 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 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 (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:

Wikipedia does not have an article with this exact name. ...

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 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 (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 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: In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

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 -- hence the name exponential growth. The constant is determined by the initial size of the population. The 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. ... Jump to: navigation, search The exponential function is one of the most important functions in mathematics. ...

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: Jump to: navigation, search A Malthusian catastrophe, sometimes known as a Malthusian check, 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. ... Jump to: navigation, search In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

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. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

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 constant (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 smooth (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.
• Electroengineering
  • Charging and discharging of capacitors and changes in current in inductors are also exponential growth and decay phenomena. Engineers use a rule of five time constants to estimate when a steady state has been reached.
• Computer technology
  • Processing power of computers. See also Moore's law and technological singularity (under exponential growth, there are no such singularities).
  • 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
  • Atmospheric pressure decreases exponentially with increasing height above sea level, at a rate of about 12% per 1000m.
  • Nuclear chain reaction (the concept behind nuclear weapons). Each uranium nucleus that undergoes fission 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 (a function of the shape and mass of the uranium), k > 0 and so the production rate of neutrons and induced uranium fissions increases exponentially, in an uncontrolled reaction.
  • Newton's law of cooling where T is temperature, t is time, and, A, D, and k > 0 are constants, is an example of exponential decay.

Main articles: Life The most salient example of biological universality is that all living things share a common carbon-based biochemistry and in particular pass on their characteristics via genetic material, which is based on nucleic acids such as DNA and which uses a common genetic code with only minor... Jump to: navigation, search A microorganism or microbe is an organism that is so small that it is microscopic (invisible to the naked eye). ... A microbiological culture is a way to determine the cause of infectious disease by letting the agent multiply (reproduce) in predetermined media. ... The 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. ... Severe Acute Respiratory Syndrome (SARS) is an atypical form of pneumonia. ... West Nile virus is a newly emergent virus of the family Flaviviridae, found in both tropical and temperate regions. ... Jump to: navigation, search Smallpox (also known by the Latin names Variola or Variola vera) is a highly contagious disease unique to humans. ... Immunization (AmE) or Immunisation (BE) has a number of meanings: In medicine immunization is the process by which an individual is exposed to a material that is designed to prime his or her immune system against that material. ... Jump to: navigation, search The world population is the total number of humans alive on the planet Earth at a given time. ... The 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. ... In physiology, a stimulus is a detectable change in the internal or external environment. ... PSYCHOLOGY In psychology and the cognitive sciences, perception is the process of acquiring, interpreting, selecting, and organizing sensory information. ... Jump to: navigation, search Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1. ... Loudness is the quality of a sound which is high in volume (amplitude, or sound pressure). ... Jump to: navigation, search Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... Jump to: navigation, search A schematic representation of hearing. ... Brightness is an attribute of visual perception in which a source appears to emit a given amount of light. ... Jump to: navigation, search Visual perception is one of the senses, consisting of the ability to detect light and interpret (see) it as the perception known as sight or naked eye vision. ... A concept in evolution linking survival of the fittest to natural selection. ... Jump to: navigation, search Accuracy, in science, engineering, industry and statistics, is the degree of conformity of a measured/calculated quantity to its actual (true) value. ... Estimation is generally the calculation of an approximate or uncertain result, often based on approximate, uncertain, incomplete, or noisy data. ... Electrical engineering is an engineering discipline that deals with the study and application of electricity and electromagnetism. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... The term discharge can have several meanings in different contexts: To discharge a weapon is to fire it. ... Jump to: navigation, search Various types of capacitors SMD capacitors: electrolytic at the bottom line, ceramic above them; classic ceramic and electrolytic capacitors at the right side for comparison A capacitor is a device that stores energy in the electric field created between a pair of conductors on which equal... Jump to: navigation, search In electricity, current refers to electric current, which is the flow of electric charge. ... Jump to: navigation, search An inductor is a passive electrical device that stores energy in a magnetic field, typically by combining the effects of many loops of electric current. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... The RC time constant, usually denoted by the Greek letter τ (tau), is a parameter that characterizes the frequency response of a resistance-capacitance (RC) circuit. ... Steady state has a number of meanings: In biochemistry, steady state is a central term in enzyme kinetics. ... Jump to: navigation, search A computer is a device or machine for processing information from data according to a program — a compiled list of instruction. ... Jump to: navigation, search The clock rate is the fundamental rate in cycles per second, measured in hertz, at which a [[computer]] performs its most basic operations such as adding two numbers or transferring a value from one processor register to another. ... Jump to: navigation, search Moores law is a myth in computer folklore about the growth of computing power over time. ... Jump to: navigation, search When plotted on a logarithmic graph, 15 separate lists of paradigm shifts for key events in human history show an exponential trend. ... ... Jump to: navigation, search In finance, interest has three general definitions. ... In finance, the rule of 72 is a simple method of calculating the approximate number of periods over which a quantity will double. ... diurnal (daily) rhythm of air pressure in northern Germany (black curve is air pressure) Atmospheric pressure is the pressure above any area in the Earths atmosphere caused by the weight of air. ... Albert Einsteins letter to President Roosevelt in 1939 about his concern, about (Nuclear chain reactions) Click for closeup of letter A nuclear chain reaction occurs when on average more than one nuclear reaction is caused by another nuclear reaction, thus leading to an exponential increase in the number of... The mushroom cloud of the atomic bombing of Nagasaki, Japan, 1945, rose some 18 km (11 mi) above the epicenter. ... Jump to: navigation, search General Name, Symbol, Number uranium, U, 92 Chemical series actinides Group, Period, Block ?, 7, f Appearance silvery gray metallic Atomic mass 238. ... Jump to: navigation, search A stylized representation of a lithium atom. ... Jump to: navigation, search Sketch of induced nuclear fission, a neutron (n) strikes a uranium nucleus which splits into similar products (F. P.), and releases more neutrons to continue the process, and energy in the form of gamma and other radiation. ... Properties In physics, the neutron is a subatomic particle with no net electric charge and a mass of 939. ... Absorption has a number of meanings: In physics, absorption is a process in which particles of some sort encounter another material and are taken up by or even disappear in it. ... The word probability derives from the Latin probare (to prove, or to test). ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In geometry, two objects are of the same shape if one can be transformed to another (ignoring color) by dilating (that is, by multiplying all distances by the same factor) and then, if necessary, rotating and translating. ... Jump to: navigation, search Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

See also

• exponential decay
• exponential function,
• bacterial growth,arthrobacter
• logistic curve,
• exponential algorithm
• asymptotic notation
• EXPSPACE
• EXPTIME
• Rule of 72/Rule of 70
• list of exponential topics