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716:
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33:
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3037:. 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, and so on. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for
2072:
433:
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
652:
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
325:
is often used to illustrate it. One bacterium splits itself into two, each of which splits itself resulting in four, then eight, 16, 32, and so on. The amount of increase keeps increasing because it is proportional to the ever-increasing number of bacteria. Growth like this is observed in real-life
539:
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
3067:
French children are offered a riddle, which appears to be an aspect of exponential growth: "the apparent suddenness with which an exponentially growing quantity approaches a fixed limit". The riddle imagines a water lily plant growing in a pond. The plant doubles in size every day and, if left
3068:
alone, it would smother the pond in 30 days killing all the other living things in the water. Day after day, the plant's growth is small, so it is decided that it won't be a concern until it covers half of the pond. Which day will that be? The 29th day, leaving only one day to save the pond.
2491:
1727:
2970:
In reality, initial exponential growth is often not sustained forever. After some period, it will be slowed by external or environmental factors. For example, population growth may reach an upper limit due to resource limitations. In 1845, the
Belgian mathematician
2987:
Exponential growth models of physical phenomena only apply within limited regions, as unbounded growth is not physically realistic. Although growth may initially be exponential, the modelled phenomena will eventually enter a region in which previously ignored
203:
it grows at 30% of its size. If an exponentially growing function grows at a rate that is 3 times is present size, then it always grows at a rate that is 3 times its present size. When it is 10 times as big as it is now, it will grow 10 times as fast.
1166:
1954:
1454:
877:
616:. (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
627:, 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
790:
3018:
Studies show that human beings have difficulty understanding exponential growth. Exponential growth bias is the tendency to underestimate compound growth processes. This bias can have financial implications as well.
1546:
2496:
1347:
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706:
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.
2268:
2711:{\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=kx\\{\frac {dx}{x}}&=k\,dt\\\int _{x_{0}}^{x(t)}{\frac {dx}{x}}&=k\int _{0}^{t}\,dt\\\ln {\frac {x(t)}{x_{0}}}&=kt.\end{aligned}}}
1024:
808:
728:
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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
3653:
341:
Terms like "exponential growth" are sometimes incorrectly interpreted as "rapid growth". Indeed, something that grows exponentially can in fact be growing slowly at first.
2064:
702:, 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. For example, the video
2483:
334:. In real cases, initial exponential growth often does not last forever, instead slowing down eventually due to upper limits caused by external factors and turning into
993:
146:
2942:
676:. 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.
429:
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
201:
175:
486:
electrons also are accelerated, creating larger numbers of free electrons. The resulting exponential growth of electrons and ions may rapidly lead to complete
803:
555:
1258:
2823:
3059:" refers to the time when an exponentially growing influence is having a significant economic impact on an organization's overall business strategy.
723:
2897:
Growth rates may also be faster than exponential. In the most extreme case, when growth increases without bound in finite time, it is called
540:
approximation to think of the first 53 generations as a latency period leading up to the actual explosion, which only takes 3–4 generations."
2186:
1230:
If a species of bacteria doubles every ten minutes, starting out with only one bacterium, how many bacteria would be present after one hour?
3647:
2890:
There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). See
3681:
2025:
in the last formula is not a numerical division either, but converts a dimensionless number to the correct quantity including unit.
1858:, have a one-to-one connection given by the following equation (which can be derived by taking the natural logarithm of the above):
4246:
3233:
2996:
model) or other underlying assumptions of the exponential growth model, such as continuity or instantaneous feedback, break down.
1722:{\displaystyle x(t)=x_{0}\cdot e^{kt}=x_{0}\cdot e^{t/\tau }=x_{0}\cdot 2^{t/T}=x_{0}\cdot \left(1+{\frac {r}{100}}\right)^{t/p},}
3878:
2891:
595:
also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors.
3455:
3428:
3382:
893:
68:
to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now.
247:
2367:
2082:
Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/
2079:
1539:
Thus the law of exponential growth can be written in different but mathematically equivalent forms, by using a different
380:
3789:
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2721:
4152:
3624:
3610:
3596:
3582:
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which can be expressed as the product of a number of units and a unit of time) represent the same growth rate, with
207:
If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing
106:
Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function
17:
3972:
3967:
3962:
3957:
3952:
3947:
3942:
3937:
3052:) on the 41st and there simply was not enough rice in the whole world for the final squares. (From Swirski, 2006)
3049:
2808:
In the long run, exponential growth of any kind will overtake linear growth of any kind (that is the basis of the
3399:
624:
91:
of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as
4261:
4256:
959:
384:
4187:
4067:
3830:
3674:
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80:
3215:
1161:{\displaystyle x(t+\tau )=a\cdot b^{(t+\tau )/\tau }=a\cdot b^{t/\tau }\cdot b^{\tau /\tau }=x(t)\cdot b\,.}
4072:
4062:
4002:
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36:
The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.
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3637:/ Ed. by M. G. Dmitriev and A. P. Petrov, pp. 367–9. Moscow: Russian State Social University, 2004.
2972:
1949:{\displaystyle k={\frac {1}{\tau }}={\frac {\ln 2}{T}}={\frac {\ln \left(1+{\frac {r}{100}}\right)}{p}}}
4162:
3153:
1776:
242:
goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is
3603:
Of
Literature and Knowledge: Explorations in Narrative Thought Experiments, Evolution, and Game Theory
3033:
According to legend, vizier Sissa Ben Dahir presented an Indian King Sharim with a beautiful handmade
2308:
2091:
4147:
4012:
1793:
1514:
613:
4225:
3667:
3257:"Constant Growth Rate Can Be Supported by Decreasing Energy Flux and Increasing Aerobic Glycolysis"
148:
grows at an ever increasing rate, but is very remote from growing exponentially. For example, when
2035:
3168:
3113:
3008:
426:
373:
72:
2446:
2276:. For example, if one wishes to empirically estimate the growth rate from intertemporal data on
1449:{\displaystyle x(1{\text{ hr}})=1\cdot 2^{(60{\text{ min}})/(10{\text{ min}})}=1\cdot 2^{6}=64.}
4266:
4220:
4115:
3864:
3799:
3763:
3004:
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1772:
528:
520:
493:
88:
84:
4251:
4182:
4172:
4110:
3835:
3372:
3143:
3118:
3083:
2901:. In between exponential and hyperbolic growth lie more classes of growth behavior, like the
224:
2912:
2028:
A popular approximated method for calculating the doubling time from the growth rate is the
109:
3573:
3571:
Meadows, Donella H., Dennis L. Meadows, Jørgen
Randers, and William W. Behrens III. (1972)
3556:
2960:
The J-shaped exponential growth (left, blue) and the S-shaped logistic growth (right, red).
2303:
487:
212:
3471:
Stango, Victor; Zinman, Jonathan (2009). "Exponential Growth Bias and
Household Finance".
3045:
th square demanded over a million grains on the 21st square, more than a million million (
8:
4057:
3133:
3093:
3078:
459:
872:{\displaystyle {\begin{aligned}a&=24\\b&={\frac {1}{2}}\\r&=5\end{aligned}}}
326:
activity or phenomena, such as the spread of virus infection, the growth of debt due to
4105:
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3997:
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100:
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92:
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After one hour, or six ten-minute intervals, there would be sixty-four bacteria.
699:
581:
at a constant interest rate provides exponential growth of the capital. See also
512:
501:
4197:
4088:
3987:
3719:
3714:
3690:
3631:
On the
Possible Reasons for the Hyperexponential Growth of the Earth Population
3193:
3183:
3103:
2976:
2965:
2902:
1540:
654:
609:
588:
508:
335:
83:
to the quantity itself. Often the independent variable is time. Described as a
2019:, but for uniformity this has been avoided here. In this case the division by
653:
be physically impossible with an exponential algorithm). Also, the effects of
568:
Economic growth is expressed in percentage terms, implying exponential growth.
4240:
4157:
3932:
3912:
3887:
3280:
3178:
2441:
2005:
for the (dimensionless) number of units of time rather than the time itself,
1802:
1474:
1008:
703:
687:
430:
422:
4192:
3298:
3088:
715:
592:
446:
795:
785:{\displaystyle {\begin{aligned}a&=3\\b&=2\\r&=5\end{aligned}}}
4202:
4142:
3794:
3758:
3108:
691:
524:
331:
3992:
3773:
3034:
2813:
2029:
605:
582:
550:
can result in the exponential growth of the amplified signal, although
463:
387: in this section. Unsourced material may be challenged and removed.
349:
76:
48:
3648:
Growth in a Finite World – Sustainability and the
Exponential Function
1342:{\displaystyle x(t)=a\cdot b^{t/\tau }=1\cdot 2^{t/(10{\text{ min}})}}
4098:
4093:
3927:
3804:
3729:
3724:
3173:
2975:
first proposed a mathematical model of growth like this, called the "
2906:
2793:
2180:
1784:
1762:
551:
547:
2881:{\displaystyle \lim _{t\to \infty }{\frac {t^{\alpha }}{ae^{t}}}=0.}
362:
4032:
3768:
3734:
3709:
3328:
3123:
2956:
2272:
This allows an exponentially growing variable to be modeled with a
1530:
the growth rate is given by the dimensionless positive number
665:, then a processor twice as fast could only solve problems of size
527:
of neutron absorption exceeds the probability of neutron escape (a
523:
by adjacent uranium atoms, causing them to fission in turn. If the
479:
467:
442:
438:
3659:
3617:
Blueprint to a
Billion: 7 Essentials to Achieve Exponential Growth
3356:
3216:"Opinion | Stop Saying 'Exponential.' Sincerely, a Math Nerd"
449:
is available. Each infected person can infect multiple new people.
3922:
3128:
516:
505:
445:) typically will spread exponentially at first, if no artificial
3856:
4177:
3982:
3917:
2892:
Degree of a polynomial § Computed from the function values
27:
Growth of quantities at rate proportional to the current amount
32:
698:" as an analogy to the spread of viruses. With media such as
532:
353:
Bacteria exhibit exponential growth under optimal conditions.
3420:
Functions and Change: A Modeling
Approach to College Algebra
3447:
Population
Ecology: An Introduction to Computer Simulations
2488:
The differential equation is solved by direct integration:
536:
475:
79:) of a quantity with respect to an independent variable is
2001:
is simply the number of units of time. Using the notation
474:
that it frees up additional electrons as it collides with
470:
becomes sufficiently accelerated by an externally applied
3907:
3400:"Gangnam Style vs Call Me Maybe: A Popularity Comparison"
3250:
546:
within the linear range of electrical or electroacoustic
3370:
2094:, hover over a graph to highlight it and its complement.
1748:
Parameters (negative in the case of exponential decay):
3554:
Meadows, Donella. Randers, Jorgen. Meadows, Dennis.
1771:; in finance it is also called the logarithmic return,
1765:(number of times per unit time) of growing by a factor
638:
requires 10 seconds to complete, and a problem of size
3654:
Dr. Albert
Bartlett: Arithmetic, Population and Energy
2263:{\displaystyle \log x(t)=\log x_{0}+t\cdot \log(1+r).}
694:, can spread in an exponential manner, often said to "
215:
of definition with equal intervals, it is also called
183:
154:
112:
3635:
Mathematical Modeling of Social and Economic Dynamics
2915:
2826:
2724:
2494:
2449:
2370:
2311:
2189:
2114:
2099:
2038:
1864:
1549:
1518:
or 2), the growth rate is given by the non-zero time
1354:
1261:
1027:
962:
896:
806:
726:
250:
3349:
3314:"Introduction to Nuclear Weapon Physics and Design"
3251:Slavov, Nikolai; Budnik, Bogdan A.; Schwab, David;
3417:Crauder, Bruce; Evans, Benny; Noell, Alan (2008).
2936:
2880:
2768:
2710:
2477:
2402:
2352:
2262:
2183:of both sides of the exponential growth equation:
2164:
2058:
1948:
1721:
1448:
1341:
1160:
987:
938:
871:
784:
298:
195:
169:
140:
3416:
3334:
3234:"10 Scientific Words You're Probably Using Wrong"
230:The formula for exponential growth of a variable
87:, a quantity undergoing exponential growth is an
4238:
2828:
434:remodeling their metabolism and gene expression.
3496:
3494:
3244:
2410:saying that the change per instant of time of
71:In more technical language, its instantaneous
3872:
3675:
3519:
3491:
3470:
645:requires 20 seconds, then a problem of size
3350:Ariel Cintrón-Arias (2014). "To Go Viral".
1543:. The most common forms are the following:
64:occurs when the a quantity grows at a rate
3879:
3865:
3682:
3668:
3423:. Houghton Mifflin Harcourt. p. 398.
3013:
3443:
3355:
3288:
2992:factors become significant (leading to a
2643:
2561:
2108:exhibits exponential growth according to
1791:is the time it takes to grow by a factor
1154:
981:
403:Learn how and when to remove this message
3530:. Chelsea Green Publishing. p. 21.
3528:The Limits to Growth: The 30-Year Update
2982:
2955:
2297:
794:
714:
348:
31:
3525:
3500:
3397:
3022:
2778:In the above differential equation, if
939:{\displaystyle x(t)=a\cdot b^{t/\tau }}
177:it grows at 3 times its size, but when
14:
4239:
3860:
3663:
3450:. John Wiley & Sons. p. 37.
3345:
3343:
3255:; van Oudenaarden, Alexander (2014).
3046:
2803:
1818:(a dimensionless number) in a period
680:
321:at time 0. The growth of a bacterial
3311:
3213:
2179:over time, as can be seen by taking
385:adding citations to reliable sources
356:
299:{\displaystyle x_{t}=x_{0}(1+r)^{t}}
3689:
3503:Capitalism: as if the world matters
3371:Karine Nahon; Jeff Hemsley (2013).
2796:variation of this growth model see
2403:{\displaystyle {\frac {dx}{dt}}=kx}
2165:{\displaystyle x(t)=x_{0}(1+r)^{t}}
599:
211:instead. In the case of a discrete
24:
3656:— streaming video and audio 58 min
3589:Capitalism as if the world matters
3340:
2951:
2838:
2100:Reformulation as log-linear growth
25:
4278:
4153:Indefinite and fictitious numbers
3886:
3641:
3505:. London: Earthscan. p. 49.
2769:{\displaystyle x(t)=x_{0}e^{kt}.}
1991:is the unit of time the quotient
1001:is a positive growth factor, and
223:since the function values form a
3561:Chelsea Green Publishing, 2004.
3485:10.1111/j.1540-6261.2009.01518.x
2785:, then the quantity experiences
2418:is proportional to the value of
2353:{\displaystyle x(t)=x_{0}e^{kt}}
2172:, then the log (to any base) of
2071:
710:
361:
4247:Ordinary differential equations
3464:
3437:
3335:Crauder, Evans & Noell 2008
1809:is the time it takes to double.
1738:expresses the initial quantity
625:computational complexity theory
482:of the dielectric media. These
372:needs additional citations for
3577:. New York: University Books.
3410:
3391:
3364:
3305:
3226:
3207:
2931:
2919:
2835:
2734:
2728:
2672:
2666:
2598:
2592:
2459:
2453:
2321:
2315:
2254:
2242:
2205:
2199:
2153:
2140:
2124:
2118:
1773:continuously compounded return
1559:
1553:
1416:
1405:
1397:
1386:
1369:
1358:
1334:
1323:
1271:
1265:
1145:
1139:
1072:
1060:
1043:
1031:
972:
966:
906:
900:
886:depends exponentially on time
287:
274:
122:
116:
13:
1:
4068:Conway chained arrow notation
3831:Conway chained arrow notation
3200:
3062:
3057:second half of the chessboard
1015:to increase by one factor of
3273:10.1016/j.celrep.2014.03.057
3214:Suri, Manil (4 March 2019).
3029:Wheat and chessboard problem
2362:linear differential equation
2059:{\displaystyle T\simeq 70/r}
562:
7:
3149:Law of accelerating returns
3071:
1188:has exponential growth. If
686:Internet contents, such as
344:
10:
4283:
4163:Largest known prime number
3783:Inverse for right argument
3548:
3501:Porritt, Jonathan (2005).
3154:List of exponential topics
3026:
2998:
2963:
2478:{\displaystyle x(0)=x_{0}}
572:
558:of the signal over others.
453:
415:
4216:
4148:Extended real number line
4128:
4081:
4063:Knuth's up-arrow notation
4050:
4041:
3894:
3841:Knuth's up-arrow notation
3818:
3782:
3743:
3697:
3526:Meadows, Donella (2004).
3316:. Nuclear Weapons Archive
2816:growth, that is, for all
988:{\displaystyle x(0)=a\,,}
614:technological singularity
95:). Exponential growth is
4073:Steinhaus–Moser notation
3846:Steinhaus–Moser notation
3444:Bernstein, Ruth (2003).
2973:Pierre François Verhulst
1524:. For any non-zero time
952:is the initial value of
56: Exponential growth
3605:. New York: Routledge.
3169:Malthusian growth model
3114:Combinatorial explosion
3014:Exponential growth bias
3009:Apparent infection rate
2090:approximations. In the
1011:—the time required for
631:, if a problem of size
608:of computers. See also
554:effects may favor some
519:, each of which can be
141:{\textstyle f(x)=x^{3}}
4116:Fast-growing hierarchy
3473:The Journal of Finance
3377:. Polity. p. 16.
3005:Malthusian catastrophe
2961:
2944:, the diagonal of the
2938:
2937:{\displaystyle A(n,n)}
2882:
2810:Malthusian catastrophe
2770:
2712:
2479:
2404:
2354:
2264:
2166:
2060:
1950:
1723:
1483:and an amount of time
1450:
1343:
1162:
989:
940:
879:
873:
792:
786:
494:Nuclear chain reaction
354:
300:
197:
171:
142:
58:
4262:Mathematical modeling
4257:Temporal exponentials
4173:Long and short scales
4111:Grzegorczyk hierarchy
3836:Grzegorczyk hierarchy
3559:: The 30-Year Update.
3144:Information explosion
3119:Exponential algorithm
3084:Albert Allen Bartlett
2999:Further information:
2983:Limitations of models
2959:
2939:
2883:
2771:
2713:
2480:
2405:
2355:
2298:Differential equation
2265:
2167:
2061:
1951:
1812:The percent increase
1724:
1512:not equal to 1 (e.g.
1451:
1344:
1232:The question implies
1163:
990:
941:
874:
798:
787:
718:
556:component frequencies
437:A virus (for example
352:
301:
225:geometric progression
198:
172:
143:
66:directly proportional
35:
3629:Tsirel, S. V. 2004.
3574:The Limits to Growth
3557:The Limits to Growth
3023:Rice on a chessboard
2913:
2824:
2722:
2492:
2447:
2368:
2309:
2304:exponential function
2187:
2112:
2036:
1862:
1547:
1477:non-negative number
1352:
1259:
1025:
960:
894:
804:
724:
496:(the concept behind
488:dielectric breakdown
381:improve this article
330:, and the spread of
248:
181:
152:
110:
89:exponential function
4188:Orders of magnitude
4058:Scientific notation
3810:Super-logarithm (4)
3769:Root extraction (3)
3337:, pp. 314–315.
3253:Airoldi, Edoardo M.
3134:Hausdorff dimension
3094:Asymptotic notation
3079:Accelerating change
2642:
2602:
2015:can be replaced by
946:where the constant
719:exponential growth:
460:Avalanche breakdown
234:at the growth rate
41: Linear growth
4106:Ackermann function
3826:Ackermann function
3720:Exponentiation (3)
3715:Multiplication (2)
3619:, Wiley Dec 2005,
3615:Thomson, David G.
3591:, Earthscan 2005.
3220:The New York Times
3159:Logarithmic growth
2962:
2946:Ackermann function
2934:
2878:
2842:
2804:Other growth rates
2766:
2708:
2706:
2628:
2572:
2475:
2400:
2350:
2260:
2162:
2056:
1946:
1846:, and for a given
1719:
1446:
1339:
1158:
985:
936:
880:
869:
867:
799:exponential decay:
793:
782:
780:
681:Internet phenomena
515:produces multiple
355:
296:
193:
167:
138:
101:logarithmic growth
62:Exponential growth
59:
4234:
4233:
4124:
4123:
3854:
3853:
3747:for left argument
3457:978-0-470-85148-7
3430:978-1-111-78502-4
3384:978-0-7456-7129-1
3312:Sublette, Carey.
3164:Logistic function
3139:Hyperbolic growth
2990:negative feedback
2899:hyperbolic growth
2870:
2827:
2812:) as well as any
2798:logistic function
2787:exponential decay
2686:
2616:
2549:
2517:
2389:
1944:
1933:
1900:
1879:
1777:force of interest
1695:
1491:physical quantity
1414:
1395:
1367:
1332:
1223:exponential decay
846:
672:in the same time
579:Compound interest
544:Positive feedback
466:material. A free
413:
412:
405:
328:compound interest
209:exponential decay
196:{\textstyle x=10}
170:{\textstyle x=1,}
16:(Redirected from
4274:
4048:
4047:
3978:Eddington number
3923:Hundred thousand
3881:
3874:
3867:
3858:
3857:
3819:Related articles
3684:
3677:
3670:
3661:
3660:
3601:Swirski, Peter.
3542:
3541:
3523:
3517:
3516:
3498:
3489:
3488:
3479:(6): 2807–2849.
3468:
3462:
3461:
3441:
3435:
3434:
3414:
3408:
3407:
3398:YouTube (2012).
3395:
3389:
3388:
3368:
3362:
3361:
3359:
3347:
3338:
3332:
3326:
3325:
3323:
3321:
3309:
3303:
3302:
3292:
3248:
3242:
3241:
3230:
3224:
3223:
3211:
3189:Quadratic growth
3099:Bacterial growth
3048:
3044:
3040:
3001:Limits to Growth
2943:
2941:
2940:
2935:
2887:
2885:
2884:
2879:
2871:
2869:
2868:
2867:
2854:
2853:
2844:
2841:
2819:
2784:
2775:
2773:
2772:
2767:
2762:
2761:
2749:
2748:
2717:
2715:
2714:
2709:
2707:
2687:
2685:
2684:
2675:
2661:
2641:
2636:
2617:
2612:
2604:
2601:
2587:
2586:
2585:
2550:
2545:
2537:
2518:
2516:
2508:
2500:
2484:
2482:
2481:
2476:
2474:
2473:
2439:
2428:
2417:
2413:
2409:
2407:
2406:
2401:
2390:
2388:
2380:
2372:
2359:
2357:
2356:
2351:
2349:
2348:
2336:
2335:
2293:
2289:
2282:linearly regress
2279:
2274:log-linear model
2269:
2267:
2266:
2261:
2226:
2225:
2175:
2171:
2169:
2168:
2163:
2161:
2160:
2139:
2138:
2107:
2075:
2065:
2063:
2062:
2057:
2052:
2024:
2018:
2014:
2004:
2000:
1990:
1982:being infinite.
1981:
1975:
1969:
1962:
1955:
1953:
1952:
1947:
1945:
1940:
1939:
1935:
1934:
1926:
1906:
1901:
1896:
1885:
1880:
1872:
1857:
1851:
1845:
1839:
1833:
1823:
1817:
1770:
1760:
1744:
1737:
1728:
1726:
1725:
1720:
1715:
1714:
1710:
1701:
1697:
1696:
1688:
1671:
1670:
1658:
1657:
1653:
1637:
1636:
1624:
1623:
1619:
1603:
1602:
1590:
1589:
1574:
1573:
1535:
1529:
1523:
1511:
1506:. For any fixed
1505:
1499:proportional to
1498:
1488:
1482:
1472:
1455:
1453:
1452:
1447:
1439:
1438:
1420:
1419:
1415:
1412:
1404:
1396:
1393:
1368:
1365:
1348:
1346:
1345:
1340:
1338:
1337:
1333:
1330:
1322:
1300:
1299:
1295:
1252:
1245:
1238:
1220:
1216:
1208:
1201:
1194:
1187:
1183:
1176:
1167:
1165:
1164:
1159:
1132:
1131:
1127:
1111:
1110:
1106:
1084:
1083:
1079:
1020:
1014:
1006:
1000:
994:
992:
991:
986:
955:
951:
945:
943:
942:
937:
935:
934:
930:
889:
885:
878:
876:
875:
870:
868:
847:
839:
791:
789:
788:
783:
781:
675:
671:
664:
660:
651:
644:
637:
630:
606:Processing power
600:Computer science
498:nuclear reactors
490:of the material.
472:electrical field
408:
401:
397:
394:
388:
365:
357:
320:
317:is the value of
316:
305:
303:
302:
297:
295:
294:
273:
272:
260:
259:
241:
237:
233:
217:geometric growth
202:
200:
199:
194:
176:
174:
173:
168:
147:
145:
144:
139:
137:
136:
93:quadratic growth
55:
46:
40:
21:
18:Geometric growth
4282:
4281:
4277:
4276:
4275:
4273:
4272:
4271:
4237:
4236:
4235:
4230:
4212:
4168:List of numbers
4136:
4134:
4132:
4130:
4120:
4077:
4043:
4037:
4008:Graham's number
3998:Skewes's number
3900:
3898:
3896:
3890:
3885:
3855:
3850:
3814:
3795:Subtraction (1)
3790:Predecessor (0)
3778:
3759:Subtraction (1)
3754:Predecessor (0)
3739:
3693:
3691:Hyperoperations
3688:
3644:
3551:
3546:
3545:
3538:
3524:
3520:
3513:
3499:
3492:
3469:
3465:
3458:
3442:
3438:
3431:
3415:
3411:
3396:
3392:
3385:
3369:
3365:
3348:
3341:
3333:
3329:
3319:
3317:
3310:
3306:
3249:
3245:
3240:. 11 July 2014.
3232:
3231:
3227:
3212:
3208:
3203:
3198:
3074:
3065:
3042:
3038:
3031:
3025:
3016:
3011:
2994:logistic growth
2985:
2977:logistic growth
2968:
2954:
2952:Logistic growth
2914:
2911:
2910:
2903:hyperoperations
2863:
2859:
2855:
2849:
2845:
2843:
2831:
2825:
2822:
2821:
2817:
2806:
2779:
2754:
2750:
2744:
2740:
2723:
2720:
2719:
2705:
2704:
2688:
2680:
2676:
2662:
2660:
2651:
2650:
2637:
2632:
2618:
2605:
2603:
2588:
2581:
2577:
2576:
2569:
2568:
2551:
2538:
2536:
2533:
2532:
2519:
2509:
2501:
2499:
2495:
2493:
2490:
2489:
2469:
2465:
2448:
2445:
2444:
2430:
2419:
2415:
2411:
2381:
2373:
2371:
2369:
2366:
2365:
2341:
2337:
2331:
2327:
2310:
2307:
2306:
2300:
2291:
2284:
2277:
2221:
2217:
2188:
2185:
2184:
2173:
2156:
2152:
2134:
2130:
2113:
2110:
2109:
2105:
2102:
2097:
2096:
2095:
2081:
2076:
2048:
2037:
2034:
2033:
2020:
2016:
2006:
2002:
1992:
1986:
1977:
1971:
1964:
1963:corresponds to
1957:
1925:
1918:
1914:
1907:
1905:
1886:
1884:
1871:
1863:
1860:
1859:
1853:
1847:
1841:
1835:
1829:
1828:The quantities
1819:
1813:
1766:
1756:
1754:growth constant
1739:
1736:
1730:
1706:
1702:
1687:
1680:
1676:
1675:
1666:
1662:
1649:
1645:
1641:
1632:
1628:
1615:
1611:
1607:
1598:
1594:
1582:
1578:
1569:
1565:
1548:
1545:
1544:
1531:
1525:
1519:
1507:
1500:
1494:
1484:
1478:
1462:
1434:
1430:
1411:
1400:
1392:
1385:
1381:
1364:
1353:
1350:
1349:
1329:
1318:
1314:
1310:
1291:
1287:
1283:
1260:
1257:
1256:
1247:
1240:
1233:
1218:
1210:
1203:
1196:
1189:
1185:
1178:
1171:
1123:
1119:
1115:
1102:
1098:
1094:
1075:
1059:
1055:
1026:
1023:
1022:
1016:
1012:
1002:
996:
961:
958:
957:
953:
947:
926:
922:
918:
895:
892:
891:
887:
883:
866:
865:
855:
849:
848:
838:
831:
825:
824:
814:
807:
805:
802:
801:
800:
779:
778:
768:
762:
761:
751:
745:
744:
734:
727:
725:
722:
721:
720:
713:
700:social networks
683:
673:
666:
662:
658:
646:
639:
632:
628:
602:
589:Pyramid schemes
575:
565:
511:that undergoes
502:nuclear weapons
456:
418:
409:
398:
392:
389:
378:
366:
347:
336:logistic growth
318:
315:
309:
290:
286:
268:
264:
255:
251:
249:
246:
245:
239:
235:
231:
221:geometric decay
182:
179:
178:
153:
150:
149:
132:
128:
111:
108:
107:
57:
53:
51:
44:
42:
38:
28:
23:
22:
15:
12:
11:
5:
4280:
4270:
4269:
4264:
4259:
4254:
4249:
4232:
4231:
4229:
4228:
4223:
4217:
4214:
4213:
4211:
4210:
4205:
4200:
4198:Power of three
4195:
4190:
4185:
4180:
4178:Number systems
4175:
4170:
4165:
4160:
4155:
4150:
4145:
4139:
4137:
4133:(alphabetical
4126:
4125:
4122:
4121:
4119:
4118:
4113:
4108:
4103:
4102:
4101:
4096:
4089:Hyperoperation
4085:
4083:
4079:
4078:
4076:
4075:
4070:
4065:
4060:
4054:
4052:
4045:
4039:
4038:
4036:
4035:
4030:
4025:
4020:
4015:
4010:
4005:
4003:Moser's number
4000:
3995:
3990:
3988:Shannon number
3985:
3980:
3975:
3970:
3965:
3960:
3955:
3950:
3945:
3940:
3935:
3930:
3925:
3920:
3915:
3910:
3904:
3902:
3892:
3891:
3884:
3883:
3876:
3869:
3861:
3852:
3851:
3849:
3848:
3843:
3838:
3833:
3828:
3822:
3820:
3816:
3815:
3813:
3812:
3807:
3802:
3797:
3792:
3786:
3784:
3780:
3779:
3777:
3776:
3774:Super-root (4)
3771:
3766:
3761:
3756:
3750:
3748:
3741:
3740:
3738:
3737:
3732:
3727:
3722:
3717:
3712:
3707:
3701:
3699:
3695:
3694:
3687:
3686:
3679:
3672:
3664:
3658:
3657:
3651:
3650:— Presentation
3643:
3642:External links
3640:
3639:
3638:
3627:
3613:
3599:
3585:
3569:
3550:
3547:
3544:
3543:
3536:
3518:
3511:
3490:
3463:
3456:
3436:
3429:
3409:
3404:YouTube Trends
3390:
3383:
3363:
3339:
3327:
3304:
3267:(3): 705–714.
3243:
3225:
3205:
3204:
3202:
3199:
3197:
3196:
3191:
3186:
3181:
3176:
3171:
3166:
3161:
3156:
3151:
3146:
3141:
3136:
3131:
3126:
3121:
3116:
3111:
3106:
3104:Bounded growth
3101:
3096:
3091:
3086:
3081:
3075:
3073:
3070:
3064:
3061:
3041:grains on the
3024:
3021:
3015:
3012:
2984:
2981:
2966:Logistic curve
2964:Main article:
2953:
2950:
2933:
2930:
2927:
2924:
2921:
2918:
2877:
2874:
2866:
2862:
2858:
2852:
2848:
2840:
2837:
2834:
2830:
2805:
2802:
2765:
2760:
2757:
2753:
2747:
2743:
2739:
2736:
2733:
2730:
2727:
2703:
2700:
2697:
2694:
2691:
2689:
2683:
2679:
2674:
2671:
2668:
2665:
2659:
2656:
2653:
2652:
2649:
2646:
2640:
2635:
2631:
2627:
2624:
2621:
2619:
2615:
2611:
2608:
2600:
2597:
2594:
2591:
2584:
2580:
2575:
2571:
2570:
2567:
2564:
2560:
2557:
2554:
2552:
2548:
2544:
2541:
2535:
2534:
2531:
2528:
2525:
2522:
2520:
2515:
2512:
2507:
2504:
2498:
2497:
2472:
2468:
2464:
2461:
2458:
2455:
2452:
2399:
2396:
2393:
2387:
2384:
2379:
2376:
2360:satisfies the
2347:
2344:
2340:
2334:
2330:
2326:
2323:
2320:
2317:
2314:
2299:
2296:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2224:
2220:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2177:grows linearly
2159:
2155:
2151:
2148:
2145:
2142:
2137:
2133:
2129:
2126:
2123:
2120:
2117:
2104:If a variable
2101:
2098:
2078:
2077:
2070:
2069:
2068:
2055:
2051:
2047:
2044:
2041:
1943:
1938:
1932:
1929:
1924:
1921:
1917:
1913:
1910:
1904:
1899:
1895:
1892:
1889:
1883:
1878:
1875:
1870:
1867:
1826:
1825:
1810:
1798:
1785:e-folding time
1780:
1734:
1718:
1713:
1709:
1705:
1700:
1694:
1691:
1686:
1683:
1679:
1674:
1669:
1665:
1661:
1656:
1652:
1648:
1644:
1640:
1635:
1631:
1627:
1622:
1618:
1614:
1610:
1606:
1601:
1597:
1593:
1588:
1585:
1581:
1577:
1572:
1568:
1564:
1561:
1558:
1555:
1552:
1445:
1442:
1437:
1433:
1429:
1426:
1423:
1418:
1410:
1407:
1403:
1399:
1391:
1388:
1384:
1380:
1377:
1374:
1371:
1363:
1360:
1357:
1336:
1328:
1325:
1321:
1317:
1313:
1309:
1306:
1303:
1298:
1294:
1290:
1286:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1157:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1130:
1126:
1122:
1118:
1114:
1109:
1105:
1101:
1097:
1093:
1090:
1087:
1082:
1078:
1074:
1071:
1068:
1065:
1062:
1058:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
984:
980:
977:
974:
971:
968:
965:
933:
929:
925:
921:
917:
914:
911:
908:
905:
902:
899:
864:
861:
858:
856:
854:
851:
850:
845:
842:
837:
834:
832:
830:
827:
826:
823:
820:
817:
815:
813:
810:
809:
777:
774:
771:
769:
767:
764:
763:
760:
757:
754:
752:
750:
747:
746:
743:
740:
737:
735:
733:
730:
729:
712:
709:
708:
707:
688:internet memes
682:
679:
678:
677:
621:
601:
598:
597:
596:
586:
574:
571:
570:
569:
564:
561:
560:
559:
541:
491:
455:
452:
451:
450:
435:
423:microorganisms
421:The number of
417:
414:
411:
410:
369:
367:
360:
346:
343:
313:
293:
289:
285:
282:
279:
276:
271:
267:
263:
258:
254:
192:
189:
186:
166:
163:
160:
157:
135:
131:
127:
124:
121:
118:
115:
75:(that is, the
73:rate of change
52:
43:
37:
26:
9:
6:
4:
3:
2:
4279:
4268:
4267:Growth curves
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4244:
4242:
4227:
4224:
4222:
4219:
4218:
4215:
4209:
4206:
4204:
4201:
4199:
4196:
4194:
4191:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4158:Infinitesimal
4156:
4154:
4151:
4149:
4146:
4144:
4141:
4140:
4138:
4127:
4117:
4114:
4112:
4109:
4107:
4104:
4100:
4097:
4095:
4092:
4091:
4090:
4087:
4086:
4084:
4080:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4055:
4053:
4049:
4046:
4040:
4034:
4031:
4029:
4028:Rayo's number
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3959:
3956:
3954:
3951:
3949:
3946:
3944:
3941:
3939:
3936:
3934:
3931:
3929:
3926:
3924:
3921:
3919:
3916:
3914:
3911:
3909:
3906:
3905:
3903:
3893:
3889:
3888:Large numbers
3882:
3877:
3875:
3870:
3868:
3863:
3862:
3859:
3847:
3844:
3842:
3839:
3837:
3834:
3832:
3829:
3827:
3824:
3823:
3821:
3817:
3811:
3808:
3806:
3805:Logarithm (3)
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3787:
3785:
3781:
3775:
3772:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3751:
3749:
3746:
3742:
3736:
3733:
3731:
3730:Pentation (5)
3728:
3726:
3725:Tetration (4)
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3705:Successor (0)
3703:
3702:
3700:
3696:
3692:
3685:
3680:
3678:
3673:
3671:
3666:
3665:
3662:
3655:
3652:
3649:
3646:
3645:
3636:
3632:
3628:
3626:
3625:0-471-74747-5
3622:
3618:
3614:
3612:
3611:0-415-42060-1
3608:
3604:
3600:
3598:
3597:1-84407-192-8
3594:
3590:
3586:
3584:
3583:0-87663-165-0
3580:
3576:
3575:
3570:
3568:
3567:9781603581554
3564:
3560:
3558:
3553:
3552:
3539:
3537:9781603581554
3533:
3529:
3522:
3514:
3512:1-84407-192-8
3508:
3504:
3497:
3495:
3486:
3482:
3478:
3474:
3467:
3459:
3453:
3449:
3448:
3440:
3432:
3426:
3422:
3421:
3413:
3405:
3401:
3394:
3386:
3380:
3376:
3375:
3367:
3358:
3353:
3346:
3344:
3336:
3331:
3315:
3308:
3300:
3296:
3291:
3286:
3282:
3278:
3274:
3270:
3266:
3262:
3258:
3254:
3247:
3239:
3238:HowStuffWorks
3235:
3229:
3221:
3217:
3210:
3206:
3195:
3192:
3190:
3187:
3185:
3182:
3180:
3179:Menger sponge
3177:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3076:
3069:
3060:
3058:
3053:
3051:
3036:
3030:
3020:
3010:
3006:
3002:
2997:
2995:
2991:
2980:
2978:
2974:
2967:
2958:
2949:
2947:
2928:
2925:
2922:
2916:
2908:
2905:beginning at
2904:
2900:
2895:
2893:
2888:
2875:
2872:
2864:
2860:
2856:
2850:
2846:
2832:
2815:
2811:
2801:
2799:
2795:
2790:
2788:
2782:
2776:
2763:
2758:
2755:
2751:
2745:
2741:
2737:
2731:
2725:
2701:
2698:
2695:
2692:
2690:
2681:
2677:
2669:
2663:
2657:
2654:
2647:
2644:
2638:
2633:
2629:
2625:
2622:
2620:
2613:
2609:
2606:
2595:
2589:
2582:
2578:
2573:
2565:
2562:
2558:
2555:
2553:
2546:
2542:
2539:
2529:
2526:
2523:
2521:
2513:
2510:
2505:
2502:
2486:
2470:
2466:
2462:
2456:
2450:
2443:
2442:initial value
2437:
2433:
2426:
2422:
2397:
2394:
2391:
2385:
2382:
2377:
2374:
2363:
2345:
2342:
2338:
2332:
2328:
2324:
2318:
2312:
2305:
2295:
2288:
2283:
2275:
2270:
2257:
2251:
2248:
2245:
2239:
2236:
2233:
2230:
2227:
2222:
2218:
2214:
2211:
2208:
2202:
2196:
2193:
2190:
2182:
2178:
2157:
2149:
2146:
2143:
2135:
2131:
2127:
2121:
2115:
2093:
2089:
2085:
2080:
2074:
2067:
2053:
2049:
2045:
2042:
2039:
2031:
2026:
2023:
2013:
2009:
1999:
1995:
1989:
1983:
1980:
1974:
1967:
1960:
1941:
1936:
1930:
1927:
1922:
1919:
1915:
1911:
1908:
1902:
1897:
1893:
1890:
1887:
1881:
1876:
1873:
1868:
1865:
1856:
1850:
1844:
1838:
1832:
1822:
1816:
1811:
1808:
1805:
1804:
1803:doubling time
1799:
1796:
1795:
1790:
1787:
1786:
1781:
1778:
1774:
1769:
1764:
1759:
1755:
1751:
1750:
1749:
1746:
1742:
1733:
1716:
1711:
1707:
1703:
1698:
1692:
1689:
1684:
1681:
1677:
1672:
1667:
1663:
1659:
1654:
1650:
1646:
1642:
1638:
1633:
1629:
1625:
1620:
1616:
1612:
1608:
1604:
1599:
1595:
1591:
1586:
1583:
1579:
1575:
1570:
1566:
1562:
1556:
1550:
1542:
1537:
1534:
1528:
1522:
1517:
1516:
1510:
1504:
1497:
1492:
1487:
1481:
1476:
1475:dimensionless
1470:
1466:
1459:
1456:
1443:
1440:
1435:
1431:
1427:
1424:
1421:
1408:
1401:
1389:
1382:
1378:
1375:
1372:
1361:
1355:
1326:
1319:
1315:
1311:
1307:
1304:
1301:
1296:
1292:
1288:
1284:
1280:
1277:
1274:
1268:
1262:
1254:
1250:
1243:
1236:
1231:
1226:
1224:
1214:
1206:
1199:
1192:
1181:
1174:
1168:
1155:
1151:
1148:
1142:
1136:
1133:
1128:
1124:
1120:
1116:
1112:
1107:
1103:
1099:
1095:
1091:
1088:
1085:
1080:
1076:
1069:
1066:
1063:
1056:
1052:
1049:
1046:
1040:
1037:
1034:
1028:
1019:
1010:
1009:time constant
1005:
999:
995:the constant
982:
978:
975:
969:
963:
950:
931:
927:
923:
919:
915:
912:
909:
903:
897:
862:
859:
857:
852:
843:
840:
835:
833:
828:
821:
818:
816:
811:
797:
775:
772:
770:
765:
758:
755:
753:
748:
741:
738:
736:
731:
717:
711:Basic formula
705:
704:Gangnam Style
701:
697:
693:
689:
685:
684:
669:
656:
649:
642:
635:
626:
622:
619:
615:
611:
607:
604:
603:
594:
593:Ponzi schemes
590:
587:
584:
580:
577:
576:
567:
566:
557:
553:
549:
548:amplification
545:
542:
538:
534:
530:
526:
522:
518:
514:
510:
507:
503:
499:
495:
492:
489:
485:
481:
477:
473:
469:
465:
461:
458:
457:
448:
444:
440:
436:
432:
428:
424:
420:
419:
407:
404:
396:
386:
382:
376:
375:
370:This section
368:
364:
359:
358:
351:
342:
339:
337:
333:
329:
324:
312:
306:
291:
283:
280:
277:
269:
265:
261:
256:
252:
243:
228:
226:
222:
218:
214:
210:
205:
190:
187:
184:
164:
161:
158:
155:
133:
129:
125:
119:
113:
104:
102:
98:
94:
90:
86:
82:
78:
74:
69:
67:
63:
50:
34:
30:
19:
4252:Exponentials
4193:Power of two
4183:Number names
3918:Ten thousand
3800:Division (2)
3764:Division (2)
3735:Hexation (6)
3710:Addition (1)
3634:
3616:
3602:
3588:
3587:Porritt, J.
3572:
3555:
3527:
3521:
3502:
3476:
3472:
3466:
3446:
3439:
3419:
3412:
3403:
3393:
3373:
3366:
3330:
3318:. Retrieved
3307:
3264:
3261:Cell Reports
3260:
3246:
3237:
3228:
3219:
3209:
3089:Arthrobacter
3066:
3054:
3032:
3017:
2986:
2969:
2896:
2889:
2807:
2791:
2780:
2777:
2487:
2435:
2431:
2424:
2420:
2301:
2286:
2271:
2103:
2087:
2083:
2027:
2021:
2011:
2007:
1997:
1993:
1987:
1984:
1978:
1972:
1965:
1958:
1854:
1848:
1842:
1836:
1830:
1827:
1820:
1814:
1806:
1801:
1792:
1788:
1783:
1767:
1757:
1753:
1747:
1740:
1731:
1538:
1532:
1526:
1520:
1513:
1508:
1502:
1495:
1485:
1479:
1468:
1464:
1460:
1457:
1255:
1248:
1241:
1234:
1229:
1227:
1212:
1204:
1197:
1190:
1179:
1172:
1169:
1017:
1003:
997:
948:
881:
667:
647:
640:
633:
618:Ray Kurzweil
483:
447:immunization
399:
390:
379:Please help
374:verification
371:
340:
332:viral videos
310:
307:
244:
229:
220:
216:
206:
105:
81:proportional
70:
61:
60:
49:Cubic growth
29:
4203:Power of 10
4143:Busy beaver
3948:Quintillion
3943:Quadrillion
3374:Going Viral
3194:Stein's law
3184:Moore's law
3109:Cell growth
2092:SVG version
2032:, that is,
1461:Many pairs
882:A quantity
655:Moore's Law
610:Moore's law
525:probability
393:August 2013
97:the inverse
4241:Categories
4208:Sagan Unit
4042:Expression
3993:Googolplex
3958:Septillion
3953:Sextillion
3899:numerical
3201:References
3063:Water lily
3035:chessboard
3027:See also:
2814:polynomial
2280:, one can
2181:logarithms
2030:rule of 70
670:+ constant
583:rule of 72
464:dielectric
238:, as time
77:derivative
4099:Pentation
4094:Tetration
4082:Operators
4051:Notations
3973:Decillion
3968:Nonillion
3963:Octillion
3895:Examples
3357:1402.3499
3281:2211-1247
3174:Power law
2907:tetration
2851:α
2839:∞
2836:→
2794:nonlinear
2658:
2630:∫
2574:∫
2240:
2234:⋅
2215:
2194:
2043:≃
1912:
1891:
1877:τ
1763:frequency
1673:⋅
1639:⋅
1621:τ
1605:⋅
1576:⋅
1428:⋅
1413: min
1394: min
1379:⋅
1331: min
1308:⋅
1297:τ
1281:⋅
1228:Example:
1149:⋅
1129:τ
1121:τ
1113:⋅
1108:τ
1092:⋅
1081:τ
1070:τ
1053:⋅
1041:τ
932:τ
916:⋅
563:Economics
552:resonance
484:secondary
480:molecules
462:within a
4131:articles
4129:Related
4033:Infinity
3938:Trillion
3913:Thousand
3299:24767987
3124:EXPSPACE
3072:See also
3050:trillion
2718:so that
2440:has the
2414:at time
1366: hr
1251:= 10 min
696:go viral
661:in time
529:function
521:absorbed
517:neutrons
504:). Each
468:electron
443:smallpox
439:COVID-19
345:Examples
85:function
4226:History
4044:methods
4018:SSCG(3)
4013:TREE(3)
3933:Billion
3928:Million
3908:Hundred
3745:Inverse
3698:Primary
3549:Sources
3290:4049626
3129:EXPTIME
2086:and 72/
1970:and to
1761:is the
1217:, then
1211:0 <
1184:, then
1007:is the
573:Finance
531:of the
513:fission
509:nucleus
506:uranium
454:Physics
427:culture
416:Biology
4135:order)
3983:Googol
3623:
3609:
3595:
3581:
3565:
3534:
3509:
3454:
3427:
3381:
3320:26 May
3297:
3287:
3279:
3047:a.k.a.
3007:, and
2909:, and
2792:For a
2783:< 0
2429:, and
1956:where
1840:, and
1729:where
1215:< 1
1207:> 0
1200:> 1
1193:< 0
1182:> 1
1175:> 0
692:videos
431:splits
323:colony
308:where
213:domain
54:
47:
45:
39:
4221:Names
4023:BH(3)
3901:order
3352:arXiv
3055:The "
1852:also
1775:, or
1473:of a
1202:, or
533:shape
476:atoms
441:, or
425:in a
3621:ISBN
3607:ISBN
3593:ISBN
3579:ISBN
3563:ISBN
3532:ISBN
3507:ISBN
3452:ISBN
3425:ISBN
3379:ISBN
3322:2009
3295:PMID
3277:ISSN
2302:The
2285:log
1976:and
1800:The
1782:The
1752:The
1541:base
1501:log
1246:and
1221:has
1209:and
1195:and
1177:and
650:= 12
643:= 11
636:= 10
612:and
537:mass
535:and
500:and
3481:doi
3285:PMC
3269:doi
2979:".
2829:lim
2290:on
2237:log
2212:log
2191:log
1985:If
1968:= 0
1961:= 0
1931:100
1743:(0)
1693:100
1489:(a
1444:64.
1244:= 2
1237:= 1
1170:If
890:if
690:or
623:In
591:or
478:or
383:by
219:or
99:of
4243::
3897:in
3633:.
3493:^
3477:64
3475:.
3402:.
3342:^
3293:.
3283:.
3275:.
3263:.
3259:.
3236:.
3218:.
3003:,
2948:.
2894:.
2876:0.
2820::
2800:.
2789:.
2655:ln
2485:.
2364::
2294:.
2066:.
2046:70
1909:ln
1888:ln
1834:,
1745:.
1536:.
1467:,
1409:10
1390:60
1327:10
1253:.
1239:,
1225:.
1021::
956:,
822:24
620:.)
338:.
227:.
191:10
103:.
3880:e
3873:t
3866:v
3683:e
3676:t
3669:v
3540:.
3515:.
3487:.
3483::
3460:.
3433:.
3406:.
3387:.
3360:.
3354::
3324:.
3301:.
3271::
3265:7
3222:.
3043:n
3039:2
2932:)
2929:n
2926:,
2923:n
2920:(
2917:A
2873:=
2865:t
2861:e
2857:a
2847:t
2833:t
2818:α
2781:k
2764:.
2759:t
2756:k
2752:e
2746:0
2742:x
2738:=
2735:)
2732:t
2729:(
2726:x
2702:.
2699:t
2696:k
2693:=
2682:0
2678:x
2673:)
2670:t
2667:(
2664:x
2648:t
2645:d
2639:t
2634:0
2626:k
2623:=
2614:x
2610:x
2607:d
2599:)
2596:t
2593:(
2590:x
2583:0
2579:x
2566:t
2563:d
2559:k
2556:=
2547:x
2543:x
2540:d
2530:x
2527:k
2524:=
2514:t
2511:d
2506:x
2503:d
2471:0
2467:x
2463:=
2460:)
2457:0
2454:(
2451:x
2438:)
2436:t
2434:(
2432:x
2427:)
2425:t
2423:(
2421:x
2416:t
2412:x
2398:x
2395:k
2392:=
2386:t
2383:d
2378:x
2375:d
2346:t
2343:k
2339:e
2333:0
2329:x
2325:=
2322:)
2319:t
2316:(
2313:x
2292:t
2287:x
2278:x
2258:.
2255:)
2252:r
2249:+
2246:1
2243:(
2231:t
2228:+
2223:0
2219:x
2209:=
2206:)
2203:t
2200:(
2197:x
2174:x
2158:t
2154:)
2150:r
2147:+
2144:1
2141:(
2136:0
2132:x
2128:=
2125:)
2122:t
2119:(
2116:x
2106:x
2088:t
2084:t
2054:r
2050:/
2040:T
2022:p
2017:t
2012:p
2010:/
2008:t
2003:t
1998:p
1996:/
1994:t
1988:p
1979:T
1973:τ
1966:r
1959:k
1942:p
1937:)
1928:r
1923:+
1920:1
1916:(
1903:=
1898:T
1894:2
1882:=
1874:1
1869:=
1866:k
1855:r
1849:p
1843:T
1837:τ
1831:k
1824:.
1821:p
1815:r
1807:T
1797:.
1794:e
1789:τ
1779:.
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120:x
117:(
114:f
20:)
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