In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. This does not mean merely that for any exponentially growing quantity, the larger the quantity gets, the faster it grows. It implies that the relationship between the size of the dependent variable and its rate of growth is governed by a strict law, of the simplest kind: direct proportion. It is proved in calculus that this law requires that the quantity is given by the exponential function, if we use the correct time scale. This explains the name. ==Intuition== 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 which 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/subexp 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: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Option 1: 1c, 2c, 4c, 8c, 16c, 32c, 64c, $1.28, $2.56, $5.12, $10.24, $20.48, $40.96, $81.92, $163.84, $327.68 Option 2:$1, $2, $3, $4, $5, $6, $7, $8, $9, $10, $11, $12, $13, $14, $15, $16 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. 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 green 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 2x 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 . 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 don't change, the quantity can work as the value k in the formula kn given earlier. ==Technical details== Let ''x'' be a quantity growing exponentially with respect to time ''t''. By definition, the rate of change ''dx/dt'' obeys the differential equation: : where ''k'' > 0 is the constant of proportionality (the average number of offspring per individual in the case of the population). (See logistic function for a simple correction of this growth model where ''k'' is not constant). The solution to this equation is the exponential function -- hence the name ''exponential growth''. The constant is determined by the initial size of the population. In the long run, exponential growth of any kind will overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α: : There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). Growth rates may also be faster than exponential. The linear and exponential models are merely simple candidates but are those of greatest occurrence in nature. In the above differential equation, if ''k'' < 0, then the quantity experiences exponential decay. == Examples of exponential growth == * Biology. ** Microorganisms in a microbiological culture dish will grow exponentially, at first, after the first microorganism appears (but then logistic function until the available food is exhausted, when growth stops). ** A virus (SARS, West Nile virus, smallpox) of sufficient infectivity (''k'' > 0) will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people. ** World population, if the number of births and deaths per person per year remains constant. ** Many responses of living beings to Stimulus (physiology), including human perception, are logarithm responses, which are the inverse of exponential responses; the loudness and frequency of sound are perceived logarithmically, even with very faint stimulus, within the limits of perception. This is the reason that exponentially increasing the brightness of Visual perception is perceived by humans as a smooth (linear) increase, rather than an exponential increase. This has survival value. Generally it is important for the organisms to respond to stimuli in a wide range of levels, from very low levels, to very high levels, while the Accuracy and precision of the estimation of differences at high levels of stimulus is much less important for survival. * Electroengineering ** Electric charge and discharge of capacitors and changes in Current (electricity) in inductors are also exponential growth and Exponential decay phenomena. Engineers use a rule of five RC time constant to estimate when a Steady state (disambiguation) has been reached. *Computer ** Clock rate of computers. See also Moore's law. ** Internet growth. *Investment. The effect of interest over many years has a substantial effect on savings and a person's ability to retire. See also rule of 72 *Physics ** Nuclear chain reaction (the concept behind nuclear weapons). Each uranium atomic nucleus that undergoes Nuclear fission produces multiple neutrons, each of which can be absorption 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) 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. **Newton's law of cooling where ''T'' is temperature, ''t'' is time, and, ''A'' and ''k'' > 0 are constants, is an example of exponential decay. ==See also== *exponential decay *exponential function, *bacterial growth,arthrobacter *logistic curve, *exponential algorithm *asymptotic notation *EXPSPACE *EXPTIME *Rule of 72/Rule of 70 *list of exponential topics Ordinary differential equations Exponentials
It looks like SARS didn't exactly grow exponentially as expected. Maybe a different example would be better? ;) User:Revolver The nuclear reactor example doesn't really work. The damping of the control rods (hopefully!) prevents exponential growth in decay rate. Needs work.--User:Rwinkel 15:56, 8 Aug 2004 (UTC) == 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... User:Kieff — User:Kieff | User talk:Kieff 09:11, Oct 29, 2004 (UTC) == 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? User: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. User:Michael Hardy 02:22, 14 Apr 2005 (UTC) ==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. --User: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! --User: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 [http://web.mit.edu/dikaiser/www/ColdWarReq.pdf 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. --User:Fastfission 17:25, 16 Apr 2005 (UTC) == 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! --User: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: :::World population, if the number of births and deaths per person per year remains constant. ::User: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"? --User:Fastfission 17:31, 5 Jun 2005 (UTC)
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Words begining with Exponential_growth:
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Exponential_growth
Exponential growth
In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. This does not mean merely that for any exponentially growing quantity, the larger the quantity gets, the faster it grows. It implies that the relationship between the size of the dependent variable and its rate of growth is governed by a strict law, of the simplest kind: direct proportion. It is proved in calculus that this law requires that the quantity is given by the exponential function, if we use the correct time scale. This explains the name. ==Intuition== 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 which 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/subexp 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: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Option 1: 1c, 2c, 4c, 8c, 16c, 32c, 64c, $1.28, $2.56, $5.12, $10.24, $20.48, $40.96, $81.92, $163.84, $327.68 Option 2:$1, $2, $3, $4, $5, $6, $7, $8, $9, $10, $11, $12, $13, $14, $15, $16 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. 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 green 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 2x 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 . 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 don't change, the quantity can work as the value k in the formula kn given earlier. ==Technical details== Let ''x'' be a quantity growing exponentially with respect to time ''t''. By definition, the rate of change ''dx/dt'' obeys the differential equation: : where ''k'' > 0 is the constant of proportionality (the average number of offspring per individual in the case of the population). (See logistic function for a simple correction of this growth model where ''k'' is not constant). The solution to this equation is the exponential function -- hence the name ''exponential growth''. The constant is determined by the initial size of the population. In the long run, exponential growth of any kind will overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α: : There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). Growth rates may also be faster than exponential. The linear and exponential models are merely simple candidates but are those of greatest occurrence in nature. In the above differential equation, if ''k'' < 0, then the quantity experiences exponential decay. == Examples of exponential growth == * Biology. ** Microorganisms in a microbiological culture dish will grow exponentially, at first, after the first microorganism appears (but then logistic function until the available food is exhausted, when growth stops). ** A virus (SARS, West Nile virus, smallpox) of sufficient infectivity (''k'' > 0) will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people. ** World population, if the number of births and deaths per person per year remains constant. ** Many responses of living beings to Stimulus (physiology), including human perception, are logarithm responses, which are the inverse of exponential responses; the loudness and frequency of sound are perceived logarithmically, even with very faint stimulus, within the limits of perception. This is the reason that exponentially increasing the brightness of Visual perception is perceived by humans as a smooth (linear) increase, rather than an exponential increase. This has survival value. Generally it is important for the organisms to respond to stimuli in a wide range of levels, from very low levels, to very high levels, while the Accuracy and precision of the estimation of differences at high levels of stimulus is much less important for survival. * Electroengineering ** Electric charge and discharge of capacitors and changes in Current (electricity) in inductors are also exponential growth and Exponential decay phenomena. Engineers use a rule of five RC time constant to estimate when a Steady state (disambiguation) has been reached. *Computer ** Clock rate of computers. See also Moore's law. ** Internet growth. *Investment. The effect of interest over many years has a substantial effect on savings and a person's ability to retire. See also rule of 72 *Physics ** Nuclear chain reaction (the concept behind nuclear weapons). Each uranium atomic nucleus that undergoes Nuclear fission produces multiple neutrons, each of which can be absorption 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) 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. **Newton's law of cooling where ''T'' is temperature, ''t'' is time, and, ''A'' and ''k'' > 0 are constants, is an example of exponential decay. ==See also== *exponential decay *exponential function, *bacterial growth,arthrobacter *logistic curve, *exponential algorithm *asymptotic notation *EXPSPACE *EXPTIME *Rule of 72/Rule of 70 *list of exponential topics Ordinary differential equations Exponentials
Exponential growth
It looks like SARS didn't exactly grow exponentially as expected. Maybe a different example would be better? ;) User:Revolver The nuclear reactor example doesn't really work. The damping of the control rods (hopefully!) prevents exponential growth in decay rate. Needs work.--User:Rwinkel 15:56, 8 Aug 2004 (UTC) == 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... User:Kieff — User:Kieff | User talk:Kieff 09:11, Oct 29, 2004 (UTC) == 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? User: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. User:Michael Hardy 02:22, 14 Apr 2005 (UTC) ==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. --User: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! --User: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 [http://web.mit.edu/dikaiser/www/ColdWarReq.pdf 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. --User:Fastfission 17:25, 16 Apr 2005 (UTC) == 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! --User: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: :::World population, if the number of births and deaths per person per year remains constant. ::User: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"? --User:Fastfission 17:31, 5 Jun 2005 (UTC)
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