Read here about the concept of growth, a concept critical to the comprehension of scale. Growth is generally considered as the opposite of decay, though mathematically it can be expressed as negative decay.


Most people have had the experience of nurturing a growing, living being. This experience is common to any living being and is experienced by others such as a puppy, plant, or yeast in a loaf of bread.

Exponents and growth

In growth, the exponent is the number of generations.

Negative Growth

Negative growth is commonly called decay, see decay.

Examples of Positive Growth

Bacterial Growth

One of the most common examples of exponential growth deals with bacteria. Bacteria can multiply at an alarming rate when each bacteria splits into two new cells, thus doubling. Each doubling cycle is called a generation. For example, if we start with only one bacteria and a generation takes an hour, then after one hour we have two bacteria. After two hours we have four, and so, until by the end of one day we will have over 16 million bacteria.
In reality, exponential growth does not continue indefinitely. There would, eventually, come a time when there would no longer be any room for the bacteria, or nutrients to sustain them. Such a pattern of growth is described by a logistics or sigmoid curve.

The number of microorganisms in a culture broth will grow exponentially until an essential nutrient is exhausted. Typically the first organism splits into two daughter organisms, who then each split to form four, who split to form eight, and so on.

Disease Propagation

A virus (for example SARS, or smallpox) typically will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people.

Disease and idea propagation can also be modeled by small world graphs.

Human Population Growth

Human population, if the number of births and deaths per person per year were to remain at current levels (but also see logistic growth).

Number of patents

US patents granted, number from 1870 to 2005, shows an exponential increase.

Educational Spending

USA Education expenditure in constant dollars, 1880—2005, elementary/secondary and post secondary, shows an exponential increase.

Another implication of the law of accelerating returns is exponential growth in education and learning. Over the past 120 years, we have increased our investment in K-12 education (per student and in constant dollars) by a factor of ten. There has been a hundredfold increase in the number of college students. Automation started by amplifying the power of our muscles and in recent times has been amplifying the power of our minds. So for the past two centuries, automation has been eliminating jobs at the bottom of the skill ladder while creating new (and better-paying) jobs at the top of the skill ladder. The ladder has been moving up, and thus we have been exponentially increasing investments in education at all levels.

Dielectric Material Avalanche Breakdown

Avalanche breakdown within a dielectric material. A free electron becomes sufficiently accelerated by an externally applied electrical field that it frees up additional electrons as it collides with atoms or molecules 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

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. "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."

Telescopic Depth

Scope of the SETI project by plotting the capability of the varied scanning efforts against three major parameters: distance from Earth, frequency of transmission, and the fraction of the sky.

Medical Scanning

Resolution of noninvasive brain scanning in mm, 1970—2000; brain scanning image reconstruction time in seconds, 1970—2005;
As opposed to Moore’s law and miniaturization curves, in medical practice more, larger and more expensive machines chase ever-smaller problems, such as the growth of breast tumor detection regimes from hand check, to ordinary X-ray, mammogram, and now today MRI scan. See Deborah Rhodes talk.

Feedback In Amplification

Positive feedback within the linear range of electrical or electroacoustic amplification can result in the exponential growth of the amplified signal, although resonance effects may favor some component frequencies of the signal over others.

Heat Transfer

Heat transfer experiments yield results whose best fit line are exponential decay curves.

Economic growth

Economic growth is expressed in percentage terms, implying exponential growth. For example, U.S. GDP per capita has grown at an exponential rate of approximately two percent per year for two centuries.

See for example the growth in
  • Number of telephones.
  • Number of mobile devices.
  • Mass use of inventions.
  • Private manufacturing output per hour in constant dollars 1945—2005

The growth in the money supply: see Financial.

Compound Interest

Compound interest at a constant interest rate provides exponential growth of the capital. See also rule of 72.

Pyramid or Ponzi scheme

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.

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.


There has been an exponential increase in the use of petroleum products in our society, over the course of this century. Peak oil is the name of the point where the resources have been hal exploited. It is a classic example of a progress curve.

Moore's Law

Growth in processing power of computers over time. See also Moore's law.

Relationship Of Computational Algorithm To Resources

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 2x, 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 today.

Internet Traffic Growth

Internet traffic growth has been exponential. See also network effects.

BGP, the core routing protocol on the Internet, has to maintain a routing table in order to remember the paths a packet can be deviated to. When one of these paths repeatedly changes its state from available to not available (and vice-versa), the BGP router controlling that path has to repeatedly add and remove the path record from its routing table (flaps the path), thus spending local resources such as CPU and RAM and, even more, broadcasting useless information to peer routers. To prevent this undesired behavior, an algorithm named route flapping damping assigns each route a weight that gets bigger each time the route changes its state and decays exponentially with time. When the weight reaches a certain limit, no more flapping is done, thus suppressing the route.

Network Effects

The number of networked paths grows or decays exponentially in relationship to the change in the number of nodes. See network effects.

Increase in Key Human Events

Kurzweil claims a review of the literature shows an exponential pattern for any list of key events. His paradigm shift chart of 15 lists of key events shows strong correlation between lists from sources as varied as Encyclopaedia Britannica, American Museum of Natural History, Carl Sagan's "cosmic calendar," and others.

Examples Of Negative Growth

Tournament Play

Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. How many players remain after 5 rounds?

Half Life

In a sample of a radionuclide that undergoes radioactive decay to a different state, the number of atoms in the original state follows exponential decay as long as the remaining number of atoms is large. The decay product is termed a radiogenic nuclide.

The pesticide DDT was widely used in the United States until its ban in 1972. DDT is toxic to a wide range of animals and aquatic life, and is suspected to cause cancer in humans. The half-life of DDT can be 15 or more years. Half-life is the amount of time it takes for half of the amount of a substance to decay. Scientists and environmentalists worry about such substances because these hazardous materials continue to be dangerous for many years after their disposal. For this example, we will set the half-life of the pesticide DDT to be 15 years.

In history of science, some believe that the body of knowledge of any particular science is gradually disproved according to an exponential decay pattern.

Heat Flow

If an object at one temperature is exposed to a medium of another temperature, the temperature difference between the object and the medium follows exponential decay (in the limit of slow processes; equivalent to "good" heat conduction inside the object, so that its temperature remains relatively uniform through its volume). See also Newton's law of cooling.

Chemical Reaction Rate

The rates of certain types of chemical reactions depend on the concentration of one or another reactant. Reactions whose rate depends only on the concentration of one reactant (known as first-order reactions) consequently follow exponential decay. For instance, many enzyme-catalyzed reactions behave this way.

Atmospheric Pressure

Atmospheric pressure decreases approximately exponentially with increasing height above sea level, at a rate of about 12% per 1000m.

Capacitor Capability

The electric charge (or, equivalently, the potential) stored on a capacitor (capacitance C) decays exponentially, if the capacitor experiences a constant external load (resistance R). The exponential time-constant τ for the process is R C, and the half-life is therefore R C ln2. (Furthermore, the particular case of a capacitor discharging through several parallel resistors makes an interesting example of multiple decay processes, with each resistor representing a separate process. In fact, the expression for the equivalent resistance of two resistors in parallel mirrors the equation for the half-life with two decay processes.)

Vibration Decay

Some vibrations may decay exponentially; this characteristic is often used in creating ADSR envelopes in synthesizers.

Pharmacological Decay

In pharmacology and toxicology, it is found that many administered substances are distributed and metabolized (see clearance) according to exponential decay patterns. The biological half-lives "alpha half-life" and "beta half-life" of a substance measure how quickly a substance is distributed and eliminated.

Absorption of EM Radiation

The intensity of electromagnetic radiation such as light or X-rays or gamma rays in an absorbent medium, follows an exponential decrease with distance into the absorbing medium.

Electrical Resistance In Relation To Temperature

The decline in resistance of a Negative Temperature Coefficient Thermistor as temperature is increased.

Divergence of Langauge

The field of glottochronology attempts to determine the time elapsed since the divergence of two langauges from a common root, using the assumption that linguistic changes are introduced at a steady rate; given this assumption, we expect the similarity between them (the number of properties of the langauge that are still identical) to decrease exponentially.

Water Use

Water Use is an example of negative growth, or decay. See water use by the expanding human population.

Species Extinction

Species Extinction is an example of negative growth, or decay. See species extinction as a result of activites undertaken by the human population.

Fisheries Exploitation

Fisheries Exploitation is an example of negative growth, or decay. See fish stock depletion as a result of activites undertaken by the human population, including increased need, increased performance of fish capture and increased waste of such enhanced systems.

Forest Loss

Forest Loss is an example of negative growth, or decay. See forest loss as a result of activites undertaken by the human population.

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