December 16, 2002

One Hundred Interesting Mathematical Calculations, Number 5
#### One Hundred Interesting Mathematical Calculations, Number 5: Exponential Growth and Human Populations

We know from the experience of the colonization of North America that a reasonably well-fed human population with abundant land to hunt, fish, herd, and farm will double every quarter-century or so, even under pre-industrial disease and public health conditions, as long as it has not yet begun to go through the "demographic transition" that reduces birth rates to close to two children per potential mother. Suppose that we start with an initial population size of 100, and watch it double every generation. How long before the total mass of all the humans amounts to as much as the mass of the earth?

Recall that the mass of the earth is 6 x 10^{24} kilograms. If an average person weighs 60 kilograms, then simple division tells us that the earth weighs as much as 10^{23} people. That means that enough time has to elapse for the population to grow by a factor of 10^{21} before the mass of all humans amounts to as much as the mass of the earth. How many doublings are necessary for a population to grow by a factor of 10^{21}? The answer to this question is the value of x in the equation:

2

^{x}= 10^{21}

Let's use an approximation and a shortcut to solve this problem. We know that 10^{21} = (10^{3})^{7}. And we know that 10^{3} = 1000 is approximately equal to 1024 = 2^{10}. So let's replace 10^{3} with 2^{10}, and get:

2

^{x}= (10^{3})^{7 }= (2^{10})^{7 }= 2^{70}

Obviously, the answer is x = 70. It takes 70 doublings. And if each doubling takes 25 years, that's 70 x 25 = 1750 years for human population to grow from 100 to enough people to have the mass of the earth if they are well-fed, and have access to abundant land.

Since it has been not 1750 years but more like 40000 years since the evolution of Cro-Magnon humanity, *homo sapiens sapiens*, the obvious conclusion is that a lot of people have been (a) hungry and malnourished, and (b) short of land over the past tens of thousands of years.

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An interesting (to me) aside. At what point does the expanding of the surface of the earth due to the accumulation of bodies exceed light speed (assuming we allow for the obviously impossible rate to continue)?

Lazy Google search pointed to:

http://www.visi.com/~contra_m//antithesis/v1n4/ant_v1n4_growth.html

This next level takes exponential growth to an even more absurd state, and both examples demonstrate one of the founding pillars of Darwinism.

Posted by: theCoach on December 16, 2002 01:10 PMyou don't even need hungry and malnourished.

Population doubling requires a LOT. You need women having an average of 4 surviving children each. Which means that a woman must

1. survive to reach her childbearing years

2. find a spouse/mate/whatever you call him

3. survive pregnancy and childbirth FOUR times without benefit of modern medicine

4. bear enough extra children to make up for other women who didn't get through steps 1, 2, and 3, above (and step 1 is the big one. Infant mortality rates were traditionally very high).

you don't even need hungry and malnourished.

Population doubling requires a LOT. You need women having an average of 4 surviving children each. Which means that a woman must

1. survive to reach her childbearing years

2. find a spouse/mate/whatever you call him

3. survive pregnancy and childbirth FOUR times without benefit of modern medicine

4. bear enough extra children to make up for other women who didn't get through steps 1, 2, and 3, above (and step 1 is the big one. Infant mortality rates were traditionally very high).

Problem with this analysis is that agriculture wasn't invented until less than 15,000 years ago, probably something more like 10,000.

Up to that point, it was all hunting and gathering...

Best,

Posted by: Stephen J Fromm on December 16, 2002 03:15 PMThis is the most interesting and competent expression of parental concern and constructive confrontation of frustration since Mary Boies (attorney and wife of litigator David Boies) invented Slam Dunk Typing (wonderful software-- used baskeball animations to tutor typing skills). I think this was about a decade ago, but some site still have the software on Mac or Win98.

Professor DeLong, you could make a seriously FUN computer game out of this and other questions -- have you considered assigning this to one of the Berkeley programmers as a collaborative effort?

Posted by: Faith Witryol on December 16, 2002 03:55 PMYour calculations actually argue for a young earth, or one with a cataclysmic history. And Elizabeth is far wide of the mark as to the doubling rate. Here is a **World Population Data Sheet** (HTML link; the chart is PDF) that shows that Africa -- depite its wars, famines and pestilences -- is projected to have a 120 percent population increase by 2050. Elsewhere around the world, to cite a few examples, Nicarauga is projected for a 117 increase by 2050, Paraguay 149 percent, and Bahrain 328 percent!

With high enough birth rates, even the most extreme hardships are not enough to prevent rapid population growth.

Err, Mr. Bubba, there's a difference between the doubling to which Prof. DeLong is referring (each generation twice the size of the previous; a net reproduction rate of 2), and the overall population doubling time referred to on the data sheet from PRB.

The difference--much of the population growth for overall populations, such as those in the African nations you mention, is due to something called population momentum--populations with young age structures grow rapidly as each cohort ages. For example, the population of China is still growing rapidly, despite the fact that Chinese women now average less than 2 children per woman. Why? Mortality (and fertility) rates used to be much higher in China than they are now, and so there are fewer old people than the current number of births and current life expectancies would suggest. Thus, the population grows as larger groups of young people age, and take the place of smaller groups of older people.

Also please note that you're referring to countries doubling in population in 50 years--about 2 generations.

Elizabeth (who used to work for the demographer who wrote the data sheet you linked to--another great source of such data is the International Data Base at the US Census Bureau, http://www.census.gov/ipc/www/idbnew.html)

Posted by: Elizabeth Weber on December 16, 2002 10:57 PMUm, the people are on the Earth, so they contribute to the mass of the Earth. Every one of them dies and contributes to the mass of subsequent generations.

> ...the obvious conclusion is that a lot of

> people have been (a) hungry and malnourished,

> and (b) short of land over the past tens of

> thousands of years.

Only under the assumptions that these people are a.) immortal, and b.) space aliens whose mass originates outside the Earth.

Backtracking, we could likewise look at the possible number and eventual weight of our ancestors. For example, go back 100 years, 4 generations, and what do you have? Thirty-two forebears. Go back a further hundred years, and how many? Not 64. It increases exponentially. So, 200 years back, you'd have 1012 ancestors. But keep going, say, the twenty generations that lay between Jesus and King David. How many ancestors did any given Judean at the time of Jesus have from twenty generations earlier? Not having my calculator at hand, by longhand I get something in excess of 20 million- meaning virtually the entire people of Judea at this point would have been descended from King David. But a few generations beyond that,and the purported population of ancestors would have to exceed the current population of the Earth. A few more, and their weight would equal Everest. Then Earth itself.

Reductio ad absurdum indeed.

Such a mathematical exerise, however ridiculous, puts the matter of "blood" in perspective. If any person consider their ancestry, then the whole human race, not any one race, must have contributed to their genetic makeup. And that name you carry? It was just one that happened to filter down from the thousands you might otherwise have carried.

On the one hand, the less rapid rate of increase doesn't by itself imply that the human race hit those particular Malthusian limits. All sorts of other limits could have hit first, e.g. tribal warfare.

On the other hand, the reproductive rate and resource problems aren't as bad as some of the above comments fear. For one thing, there's some double counting going on - once you say "average", you ALREADY commented on the mothers who have to "bear enough extra children to make up for other women who didn't get through..." For another, a woman at her reproductive peak was mostly going to be selected for being able to go the long haul by her first child; issues of infection etc. get more significant as the Malthusian limits approach. Oh, and again, hunter/gatherer life styles didn't place constraints before Malthusian limits hit either; what they mostly affected was the ability to store reserves. The result was that people were mostly likely not so much chronically near starvation but rather in "feast or famine" (with the latter occasionally leading to balancing deaths).

Posted by: P.M.Lawrence on December 18, 2002 04:51 PMPost a comment