# Exponential growth

**exponentiallyexponentialgrow exponentiallygrows exponentiallyexponentially growingincreases exponentiallyexponentially increasinggrowth rateexponential increaseexponential rate**

Exponential growth is a specific way that a quantity may increase over time.wikipedia

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### Exponential function

**exponentialexponentiallyexp**

Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth).

The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.

### Geometric progression

**geometric sequencegeometricgeometrical**

In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay since the function values form a geometric progression.

### Positive feedback

**positive feedback looppositiveexacerbated**

When the loop gain is positive and above 1, there will typically be exponential growth, increasing oscillations, chaotic behavior or other divergences from equilibrium.

### Nuclear weapon

**atomic bombnuclear weaponsnuclear**

In fission weapons, a mass of fissile material (enriched uranium or plutonium) is forced into supercriticality—allowing an exponential growth of nuclear chain reactions—either by shooting one piece of sub-critical material into another (the "gun" method) or by compression of a sub-critical sphere or cylinder of fissile material using chemically-fueled explosive lenses.

### Pyramid scheme

**pyramid schemespyramidpyramid sales scheme**

The behavior of pyramid schemes follows the mathematics concerning exponential growth quite closely.

### Doubling time

**doublecell doubling timesPopulation doubling time**

When the relative growth rate (not the absolute growth rate) is constant, the quantity undergoes exponential growth and has a constant doubling time or period, which can be calculated directly from the growth rate.

### Moore's law

**Moore’s Lawcomputational powermass-produced**

The observation is named after Gordon Moore, the co-founder of Fairchild Semiconductor and CEO of Intel, whose 1965 paper described a doubling every year in the number of components per integrated circuit, and projected this rate of growth would continue for at least another decade.

### Technological singularity

**Singularityintelligence explosionThe Singularity**

Kurzweil claims that technological progress follows a pattern of exponential growth, following what he calls the "law of accelerating returns".

### E-folding

**e''-foldinge''-foldingse-folding time**

In science, e-folding is the time interval in which an exponentially growing quantity increases by a factor of e; it is the base-e analog of doubling time.

### Rule of 72

**Rule of 70**

A popular approximated method for calculating the doubling time from the growth rate is the rule of 70,

These rules apply to exponential growth and are therefore used for compound interest as opposed to simple interest calculations.

### Quadratic growth

**quadraticallygrows quadraticallyquadratic**

Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth).

* Exponential growth

### Exponential decay

**mean lifetimedecay constantexponentially**

If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead.

### Negative feedback

**negative feedback loopnegative-feedbacknegative**

Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored negative feedback factors become significant (leading to a logistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.

Whereas positive feedback tends to lead to instability via exponential growth, oscillation or chaotic behavior, negative feedback generally promotes stability.

### Logistic function

**logisticlogistic curvelogistic growth**

Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored negative feedback factors become significant (leading to a logistic growth model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down. For a nonlinear variation of this growth model see logistic function.

The logistic function was introduced in a series of three papers by Pierre François Verhulst between 1838 and 1847, who devised it as a model of population growth by adjusting the exponential growth model, under the guidance of Adolphe Quetelet.

### Nuclear chain reaction

**chain reactionpredetonationreactivity**

Chain reactions naturally give rise to reaction rates that grow (or shrink) exponentially, whereas a nuclear power reactor needs to be able to hold the reaction rate reasonably constant.

### Economic growth

**growthGDP growthgrowth rate**

* Economic growth is expressed in percentage terms, implying exponential growth.

Over long periods of time, even small rates of growth, such as a 2% annual increase, have large effects.

### Arthrobacter

**Arthrobacter'' sp.**

Arthrobacter (from the Greek, "jointed small stick”) is a genus of bacteria that is commonly found in soil. All species in this genus are Gram-positive obligate aerobes that are rods during exponential growth and cocci in their stationary phase. Arthrobacter have a distinctive method of cell division called "snapping division" or reversion in which the outer bacterial cell wall ruptures at a joint.

### Hyperbolic growth

**hyperbolic**

In the most extreme case, when growth increases without bound in finite time, it is called hyperbolic growth.

Other models suggest exponential growth, logistic growth, or other functions.

### Bacterial growth

**stationary phaselag phaselog phase**

However, if the number surviving exceeds unity on average, the bacterial population undergoes exponential growth.

### Cell growth

**proliferationcell proliferationgrowth**

Cell populations go through a particular type of exponential growth called dowaiting.

### Accelerating change

**law of accelerating returnsaccelerateaccelerating rate**

In 1910 during the town planning conference of London Daniel Burnham noted that "But it is not merely in the number of facts or sorts of knowledge that progress lies: it is still more in the geometric ratio of sophistication, in the geometric widening of the sphere of knowledge, which every year is taking in a larger percentage of people as time goes on."

### Malthusian growth model

**Malthusian parametergrowthgrowth model**

A Malthusian growth model, sometimes called a ssimple exponential growth model, is essentially exponential growth based on the idea of the function being proportional to the speed to which the function grows.

### Ackermann function

**Ackermann's functioninverse Ackermann functionAckermann number**

In between exponential and hyperbolic growth lie more classes of growth behavior, like the hyperoperations beginning at tetration, and A(n,n), the diagonal of the Ackermann function.

For small values of m like 1, 2, or 3, the Ackermann function grows relatively slowly with respect to n (at most exponentially).

### Logarithmic growth

**logarithmiclogarithmic curvegrow logarithmically**

Logarithmic growth is the inverse of exponential growth and is very slow.

### Wheat and chessboard problem

**Second half of the chessboardaccording to the old storyInvention of chess (story)**

The second half of the chessboard is the time when an exponentially growing influence is having a significant economic impact on an organization's overall business strategy.

The exercise of working through this problem may be used to explain and demonstrate exponents and the quick growth of exponential and geometric sequences.