Talk:Exponential growth

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Contents 1 SARS 2 Nuclear reactor example 3 increasing rate 4 Does science grow exponentially? 5 human population 6 Correcting for the ordinal numbering system: 7 Exponential Law and Malthusian Law 8 Nothing grows exponentially 9 alternate general formula for growth 10 geometric growth 11 New graph 11.1 What about a graph 11.2 graph 12 What's the opposite? 13 little more maths 14 Wikipedia? 15 to be recast 15.1 Intuitition

[edit] SARS

It looks like SARS didn't exactly grow exponentially as expected. Maybe a different example would be better? ;) Revolver

[edit] Nuclear reactor example

The nuclear reactor example doesn't really work. The damping of the control rods (hopefully!) prevents exponential growth in decay rate. Needs work.--Rwinkel 15:56, 8 Aug 2004 (UTC)

[edit] increasing rate

The opening paragraph still needs work. I seem to keep missing Michael Hardy's points, so maybe I'm not the best person to write it. My understanding is that the usual case for an exponentially growing (as opposed to decaying) function is that it grows faster the larger it gets. Of course, other functions can also do this, but exponentially growing ones always will. Can someone supply a counterexample if that's not the case? Lunkwill 22:17, 13 Apr 2005 (UTC)

I'll return to this soon and do some work that's not as hasty as the terse edits I've done lately. Michael Hardy 02:22, 14 Apr 2005 (UTC)

[edit] Does science grow exponentially?

I removed "science" from the list of things which grow exponentially, and another user wanted to know my reasoning behind it, so here it is:

Saying "science grows exponentially" has little meaning to begin with. Do you mean scientific knowledge? Scientific practice? Scientific ambitions? Scientific institutions? Scientific practitioners? Scientific method? Are you referring to all branches of science, or is it a statement based on idealized versions of physics or biology? (Does immunology grow exponentially? How about botany?) So first you'd have to be more specific about your metric before claiming anything about it, there is no monolithic science when you get down to it, the term refers to a bundle of things, many of which are historically contingent (i.e. the role and demographic of scientific practitioners changes radically between 1880 and 1930).

Second, you'd have to have the data to measure it. Can we measure "scientific knowledge"? How would we do so? Is science about quantity or quality? Would a thousand papers about phologiston theory count as scientific growth? Even something as mundane as a list of the number of scientists in the world at any given time -- when do you start the list? Do we stretch science all the way back to the Greeks? The Middle Ages? Or are we talking about science as an independent profession, one which doesn't really get under way until the late 18th century?

Lastly, are we assuming this is some inherent property of science itself? Because science does not compel itself to grow (or even work) -- it requires a number of outside variables. Scientific funding seems to be the most important one (no money, no science), and there's no reason at all to expect that to be exponential. In the 20th century, scientific funding (both in amount and its sources) hugs very closely to political and economic trends of the times (i.e. science in Russia grew tremendously between the 19th century and the 1980s, and then it collapsed almost completely after the collapse of its funding source, the USSR).

So anyway, I don't think it is a very meaningful statement. I modified it a bit on the history of science page to be descriptive rather than prescriptive ("science has grown" vs. "science grows"), and changed it to something more specific ("scientific practice" vs. "science"), but even then it ought to be questioned whether it is anything more than hyperbole. And if it is in doubt whether it is hyperbole, then it should only be hyperbole on a relevant page: we don't need hyperbole about science on a page about exponential growth, but could tolerate it on a page about the history of science.

Do I make sense? That's why I think it is a meaningless statement, and likely incorrect at best. I also don't know why scientific growth would be exponential and not, say, linear. Does each scientific discovery produce two or three more? I don't see any general rule to it which would make me think it could be reduced to mathematics. I'm just not sure it makes sense. --Fastfission 14:04, 16 Apr 2005 (UTC)

Your explanation is very interesting to me, as it reminds me how oversimplistic and flawed are such vague statements. It also offers me many new insights. If you agree, I will include the information given in your post, especially from the section about outside variables, in the article "history of science". What was that about growth of science in Russia?! ;)

One more question and sorry for bothering you - What could be said about the growth of different aspects of science in time (scientific knowledge, number of scientists...)? Thanks for taking the time to explain this to me and other Wikipedians. Happy wiki-ing! --Eleassar777 14:53, 16 Apr 2005 (UTC)

Loren Graham has a book (What have we learned about science and technology from the Russian experience?) which poses a number of fun questions about science using Russia as an example; the best one (and most relevant to this discussion) is "What is more important to science, money or freedom?" where he basically concludes that while freedom might be nice in an idealistic sense, without money, science grinds to a halt, while without freedom, science finds ways around the difficulties (it works with the system). Very enjoyable.

On the number of scientists, a lot of that depends on what one defines as a "scientist." One metric often used is number of PhDs granted in a given field. In physics, for example, from 1900-1940 there is a fairly linear growth; after WWII the US government encouraged more physicist and it jumped up to a huge amount during the Korean war and after Sputnik. But the market started to slow up and by 1970 the number had peaked and dropped. See figure 1 and 2 in: David Kaiser, "Scientific Manpower, Cold War Requisitions, and the Production of American Physicists after World War II," Historical Studies in the Physical and Biological Sciences 33 (Fall 2002): 131-159 (available online here).

A great book which tries to spend a lot of time picking out many different questions about what we mean by "science" and "scientific practice" is Bruno Latour's Laboratory Life -- very recommended if you are interested in thinking about this sort of thing. --Fastfission 17:25, 16 Apr 2005 (UTC)

An old classic which approaches this problem is Derek Price's Little Scìence, Big Science. Price noted that if the number of scientists grew exponentially at then current rates, every person on Earth would soon be a scientist. He argued that many parameters of science (more precisely of technology) don't grow exponentially but grow in accordance with a logistic curve (or more precisely with multiple logistic curves as design principles change). Examples I recall are the energy of particle accelerators and the length of bridges (I don't think the latter is in Price). --SteveMcCluskey 17:29, 1 June 2006 (UTC)

[edit] human population

"Human population, if it is not hindered by predation or environmental problems" Wrong, unless refraining from procreating (voluntarily or by government decree) is considered an environmental problem.

Yeah, I don't know about this. It seems to be directly linked also to the number of children in a family, which seems linked to all sorts of circumstances. In the end, I feel like saying, "Human population when the number of children born is at least X per family," which basically is, "Human population, when the population increases at an exponential rate" which is somewhat circular! --Fastfission 17:30, 4 Jun 2005 (UTC)

Several weeks ago I responded to the objection above by editing the article so that it says the following:

Human population, if the number of births and deaths per person per year remains constant.

Michael Hardy 02:22, 5 Jun 2005 (UTC)

But is that true? I mean, if the number of deaths per year is more than the number of births, even if they remain constant it won't be exponential growth, will it? Again, doesn't this just mean, "if the number of births and deaths per year is a function of exponential growth, and they remain constant, then human birth rate is exponential growth"? --Fastfission 17:31, 5 Jun 2005 (UTC)

[edit] Correcting for the ordinal numbering system:

Week: 1 2
Option 1: 1c, 2c,

We can describe these cases mathematically. In the first case, the allowance at week n is 2n cents; thus, at week 16 the payout is 216 = 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 dollars. The payout grows at a constant rate of $1 per week.

 

Please correct me if I'm wrong, but we need to correct for the ordinal numbering system here. Value at week n = 2^n-1.

p.

Right you are. I fixed it, but you would have been most welcome to fix it, too. Incidentally, MediaWiki will almost always figure out what you mean without HTML tags, and you can sign your messages on talk pages if you want by putting 4 ~ characters in a row. Thanks for your contribution! Lunkwill 20:57, 4 August 2005 (UTC)

[edit] Exponential Law and Malthusian Law

This is an approximate law of population based on Malthusian Growth Model, discovered by Malthus 1798 (Refer Darwin, Dawkins, and Drexler. Refer Exponential Growth law, "Cassell's Laws Of Nature", Trefil 2002. Also, "e: The Story Of A Number" by Eli Maor, 1994 , "What Evolution Is" by Ernst Maor, 2001 , "How Many People Can The Earth Support" by Joel E. Cohen, 1995, "Complex Population Dynamics" by Peter Turchin, 2003, and "The Galilean turn in population ecology" Mark Colyvan and Lev R. Ginzburg, 2003). The Exponential Law is also sometimes referred to as the Malthusian Law (refer "Laws Of Population Ecology" by Dr. Paul Haemig, 2005).

I believe we may need a disambiguation page to ensure that users can get to "Malthusian Law" version of the Exponential Law. DAC

[edit] Nothing grows exponentially

Strictly speaking, if exponential growth requires a constant rate, then nothing grows exponentially. Why? Take the 3 logical cases:

1) positive growth rate. Nothing can grow indefinitely at a constant positive rate due to limits to growth. Any constant positive rate of exponential growth inevitably causes all available resources to be consumed, which then halts such growth!

2) negative growth rate. Anything which sustains a constant negative rate will cease to exist, which then halts all growth as whatever the "thing" was no longer exists!.

3) zero growth rate. It's not growing!

All quoted examples of exponential growth in this article are, in fact, temporary periods of constant rate exponential growth. Another way of saying that things grow via temporary periods of constant rate exponential growth is to say that they grow at variable rates of exponential growth. This is, of course, mathematical heresy because the definiton of exponential growth requires a constant rate!

But you can think of it this way:

• fixed rate compound interest = constant rate exponential growth
• variable rate compound interest = successive periods of constant rate exponential growth (at a potentially different rate per period) = variable rate exponential growth

Regards, --Couttsie 01:17, 24 March 2006 (UTC)

Removed several examples that were actually exponential decay. Added avalanche breakdown as another axample of exponential growth. Bert 14:29, 12 June 2006 (UTC)

Dear Couttsie, I hope you appreciate the newly added paragraph on Limitations of exponential models. Sincerely, Marenco 14:29, 29 September 2006 (UTC)

[edit] alternate general formula for growth

i see it mention but the formula itself is not mentioned, so: Q(t)=a(1+r)^t where

• the coefficient a is the initial value of Q (at t = 0)
• the base b is the growth factor where + b = 1 + r is growth (b > 1) where r is the rate (as a decimal)
• the exponent t is the independent variable
• the dependent variable is the quantity Q

—The preceding unsigned comment was added by Mokaiba (talk • contribs) 26 September 2006.

This formular only changes the letters, so its not really an alternative formula. --Salix alba (talk) 07:59, 26 September 2006 (UTC)

[edit] geometric growth

Why the hell does geometric growth redirect here? Fresheneesz 05:57, 26 October 2006 (UTC)

Because the two terms mean the same thing. -- Dominus 17:35, 26 October 2006 (UTC)

Well.. i'm not quite so sure about that, but since I have no other information to go on (couldn't find anything quickly), i'll put it up as a synonym, which should have been done if the term redirects here. Fresheneesz 22:52, 26 October 2006 (UTC)

Are you sure they're always the same? I have always thought so too, but lately I've run into a few (seemingly intelligent) articles that refer to them as different phenomena. Always without explaining the difference, of course. An unscientific Googling shows a bunch of articles saying things like "exponential (geometric) growth", and a bunch of others that include "the difference between geometric and exponential". Anybody know math? I don't. JayLevitt 14:58, 15 November 2007 (UTC)

Dominus is right: geometric progression means a progression like 1, 2, 4, 8, 16 when a ratio between neighbours is the same in contrast with arithmetic progression: 2, 4, 6, 8, 10, 12, .. when difference between neighbours is the same. So geometric growth is y=2^x - exponential function; and ariphmetic is y=2x - linear function. Guest 04:45, 18 January 2007 (UTC)

I'd be inclined to use the term "geometric growth" only with discrete time, and "exponential growth" with either discrete or continuous time. Michael Hardy 20:04, 15 November 2007 (UTC)

So is are discrete and continuous time different theories of time, or just different implementations of measurement? In my world, I hear both terms applied to, say, the volume of growth a new server has seen. "Last month, we used to process one hundred messages a minute on that box; now it's two hundred a minute. That's exponential growth!" If [Randall Mundroe] swoops in with a cape and says "No, it's geometric, not exponential!" is he right or wrong, or just being unnecessarily deus-ex-machina? —Preceding unsigned comment added by JayLevitt (talk • contribs) 08:09, 20 November 2007 (UTC)

The above comment makes the very mistake that this article is supposed to be warning people against. "Exponential growth" is NOT synonymous with "fast growth" and if you think it is, you ignored this article. Michael Hardy (talk) 21:32, 10 March 2008 (UTC)

Isn't the key to this whether y=ka^n (geometric) is always capable of alternative expression y=b*exp(x) (exponential) where k,a and b are constants and y,n and x are variables? A hazy memory of mine is that the latter is a specific form of the former - i.e. 'exponential' is sufficient but not necessary for 'geometric'. Some formulae on this in the article would be helpful. Nmcmurdo (talk) 23:11, 13 December 2007 (UTC)

Here is an example that I have always seen to show that 'exponential' is a subset of 'geometric': y=x^a is geometric but not exponential while y=a^x is both geometric and exponential. I really think that Geometric should have its own page. Kmwalke (talk) 19:17, 10 March 2008 (UTC)

The former is not geometric growth; it is polynomial growth. Michael Hardy (talk) 21:30, 10 March 2008 (UTC)

[edit] New graph

Given the previous posts...

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[edit] What about a graph

I think we could use a graphic as an example for this article. There might be a good exponential one already on Wikipedia, so if anyone has the time to check out... — Kieff | Talk 09:11, Oct 29, 2004 (UTC)

(wow, that's OLD!)

[edit] graph

apologies to the person who made it, but this graph is terrible. it looks like it was drawn with technology made obsolite 10years ago. the notiation is rather odd, and can you say pixilated? it could add clarity to the article, but instead left me temperarily dumbfounded. i drew better one, but alas i am too retarded to upload it. can so one better suited to the job fix this? mastodon 20:04, 1 December 2005 (UTC)

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Well then, I decided to make a prettier graph. It'll be at Commons in no time, I'm just waiting for the databases to unlock. Cheers! — Kieff 13:15, 19 January 2007 (UTC)

[edit] What's the opposite?

What do you call the growth rate that slows?

What if I had a million marbles, and a million plus the square root of a million marbles the next day, and that amount plus the square root of that amount the following day. What's that?

Y'know, like rather than the graph curving up like one side of the Eiffel Tower, it would start steep and soon be only slowly increasing, like Uluru. I'm talking about an ever increasing line where the rate of increase constantly slows rather than accelerating. Any help? Thanks. Gronky 18:47, 5 July 2007 (UTC)

Exponential decay--Cronholm¹⁴⁴ 19:44, 5 July 2007 (UTC)

Thanks for the response, but that doesn't seem to be what I'm looking for. I'm looking for something that continually increases, just at an ever slowing rate. Exponential decay seems to be in continual decrease. Gronky 19:47, 5 July 2007 (UTC)

Logarithm, Logistic function, or any function with a positive derivative and a negative second derivative. --Cronholm¹⁴⁴ 20:09, 5 July 2007 (UTC)

Also, if your flip the exponential decay function over the x-axis, then you have the desired function type (y=-e_-x) oops, thanks Michael y = −e^−x--Cronholm¹⁴⁴ 21:49, 8 July 2007 (UTC)

You must have meant −e^−x. Michael Hardy 15:12, 7 July 2007 (UTC)

Asking for a name for something that continues growing, but at a progressively slower rate, is not the same as asking for the "opposite" of exponential growth. Nor is the latter question well-defined enough to admit any answer. Michael Hardy 15:11, 7 July 2007 (UTC)

The differential equation has solutions . The difference relation has solutions of O(k²), so the growth is called quadratic. --Petter 19:43, 30 October 2007 (UTC)

[edit] little more maths

Do people think that some mathematical analysis is apropriate here, i.e. 'exp[x] is the limit as n->oo of ...'

This can be applied to real world problems - continuous interest can be thought of as the limit as the time periods tend to zero of normal compound interest, for example.

Derek dreery 10:09, 3 August 2007 (UTC)

[edit] Wikipedia?

Would Wikipedia be a valid example of Exponential growth, such as the number of wikipedia articles over time?--64.75.187.196 (talk) 09:50, 24 January 2008 (UTC)

[edit] to be recast

a number doesn't grow, according to the definition of number. Also other errors in same section:-

But "the number of people living is Slobovia" or the like does grow. Michael Hardy (talk) 15:30, 21 May 2008 (UTC)

Correct. As drafted in the article it doesn't say number of (members of a set), so it can be easily confused with numeral. BTW number has some ten or more meanings... Quantity would perhaps be more approapriate in the article. Jclerman (talk) 22:38, 21 May 2008 (UTC)

[edit] Intuitition

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.

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	16	17	18
Option 1	$0.01	$0.02	$0.04	$0.08	$0.16	$0.32	$0.64	$1.28	$2.56	$5.12	$10.24	$20.48	$40.96	$81.92	$163.84	$327.68	$655.36	$1310.72	$2621.44
Option 2	$1	$2	$3	$4	$5	$6	$7	$8	$9	$10	$11	$12	$13	$14	$15	$16	$17	$18	$19

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 = 32768¢ = $327.68. All formulas of the form kⁿ, where k is an unchanging number greater than 1 (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:

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 blue 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 (in green) 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 do not change, the quantity can work as the value k in the formula kⁿ given earlier.