Editing Exponential growth (section)

==Examples==
[[File:e.coli-colony-growth.gif|right|frame|Bacteria exhibit exponential growth under optimal conditions.]]
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===Biology===
* The number of [[microorganism]]s in a [[microbiological culture|culture]] will increase exponentially until an essential nutrient is exhausted, so there is no more of that nutrient for more organisms to grow. Typically the first organism [[cell division|splits]] into two daughter organisms, who then each split to form four, who split to form eight, and so on. Because exponential growth indicates constant growth rate, it is frequently assumed that exponentially growing cells are at a steady-state. However, cells can grow exponentially at a constant rate while remodeling their metabolism and gene expression.<ref name="SlavovBudnik2014">{{cite journal|last1=Slavov|first1=Nikolai| last2=Budnik|first2=Bogdan A.|last3=Schwab|first3=David|last4=Airoldi|author-link4=Edoardo Airoldi|first4=Edoardo M.|last5=van Oudenaarden|first5=Alexander|title=Constant Growth Rate Can Be Supported by Decreasing Energy Flux and Increasing Aerobic Glycolysis | journal=Cell Reports|volume=7|issue=3|year=2014|pages=705–714|issn=2211-1247| doi=10.1016/j.celrep.2014.03.057| pmid=24767987|pmc=4049626}}</ref>    
* A virus (for example [[COVID-19]], or [[smallpox]]) typically will spread exponentially at first, if no artificial [[immunization]] is available. Each infected person can infect multiple new people.

===Physics===
* [[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 [[atom]]s or [[molecule]]s 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]] (the concept behind [[nuclear reactors]] and [[nuclear weapons]]). Each [[uranium]] [[atomic nucleus|nucleus]] that undergoes [[Nuclear fission|fission]] produces multiple [[neutron]]s, each of which can be [[absorption (chemistry)|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 (mathematics)|function]] of the [[shape]] and [[mass]] of the uranium), 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."<ref>{{cite web|url=http://nuclearweaponarchive.org/Nwfaq/Nfaq2.html| title=Introduction to Nuclear Weapon Physics and Design|publisher=Nuclear Weapons Archive|last=Sublette|first=Carey|access-date=26 May 2009}}</ref>
* [[Positive feedback]] within the linear range of electrical or electroacoustic [[Amplifier|amplification]] can result in the exponential growth of the amplified signal, although [[resonance]] effects may favor some [[component frequency|component frequencies]] of the signal over others.

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

===Finance===
* [[Compound interest]] at a constant interest rate provides exponential growth of the capital.{{sfn|Crauder|Evans|Noell|2008|pp=314–315}} See also [[rule of 72]].
* [[Pyramid scheme]]s or [[Ponzi scheme]]s also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors.

===Computer science===
* [[Clock rate|Processing power]] of computers. See also [[Moore's law]] and [[technological singularity]]. (Under exponential growth, there are no singularities. The singularity here is a metaphor, meant to convey an unimaginable future. The link of this hypothetical concept with exponential growth is most vocally made by futurist [[Raymond Kurzweil|Ray Kurzweil]].)
* 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 {{math|2<sup>''x''</sup>}}, if a problem of size {{math|1=''x'' = 10}} requires 10 seconds to complete, and a problem of size {{math|1=''x'' = 11}} requires 20 seconds, then a problem of size {{math|1=''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 increases the feasible problem size by a constant. E.g. if a slow processor can solve problems of size {{mvar|x}} in time {{mvar|t}}, then a processor twice as fast could only solve problems of size {{math|''x'' + constant}} in the same time {{mvar|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 phenomena ===
* Internet contents, such as [[internet meme]]s or [[viral video|video]]s, can spread in an exponential manner, often said to "[[viral phenomenon|go viral]]" as an analogy to the spread of viruses.<ref name=aca>{{cite arXiv|title=To Go Viral|author=Ariel Cintrón-Arias|date=2014|class=physics.soc-ph|eprint=1402.3499}}</ref> With media such as [[social networks]], one person can forward the same content to many people simultaneously, who then spread it to even more people, and so on, causing rapid spread.<ref>{{cite book|author1=Karine Nahon|author2=Jeff Hemsley|title=Going Viral|url=https://books.google.com/books?id=Hjdh8fID3nUC&pg=PA16|date=2013|publisher=Polity|isbn=978-0-7456-7129-1|page=16}}</ref>  For example, the video [[Gangnam Style]] was uploaded to YouTube on 15 July 2012, reaching hundreds of thousands of viewers on the first day, millions on the twentieth day, and was cumulatively viewed by hundreds of millions in less than two months.<ref name=aca/><ref>{{cite web|url=http://youtube-trends.blogspot.com/2012/09/gangnam-style-vs-call-me-maybe.html|title=Gangnam Style vs Call Me Maybe: A Popularity Comparison| work=YouTube Trends|author=YouTube|date=2012}}</ref>