# e

For the mathematician, some numbers have an almost spiritual importance. They tend to pop up all over the place, sometimes when least expected, and their significance impresses us beyond their mere practical usefulness. One such number, possibly the “holiest of holies,” is $\pi$. Second only to $\pi$ in the numerical pantheon is today’s subject: e.

e is the base of natural logarithms. The logarithm log is a special function with the interesting property that the logarithm of the product of two numbers equals the sum of the logarithms of the numbers:

$\log(xy) = \log(x) + \log(y)$.

Actually there are many such functions, all multiples of each other. If I take the function $f(x) = \log(x)$ and multiply it by a constant $c$ to define a new function $g(x) = c \log(x)$, the new function also exhibits the log-of-product-equals-sum-of-logs property.

The logarithm function is closely related to the exponential function. If I take any number, say, 10, and raise it to the power of $x$, I have the exponent-base-10 of that number, $10^x$. If I multiply $10^x$ times $10^y$, I observe a complement to the logarithmic behavior: the exponent-of-sum-equals-product-of-exponents property:

$10^x 10^y = 10^{(x+y)}$.

This shows that we can define the logarithm function as the inverse of the exponent function: if $x = 10^y$, then $y = \log(x)$. The particular logarithmic function defined by this prescription is the logarithm to base 10.

We can use any number we want as our base to define logarithms. For example, if $x = 2^y$, then $y = \log_2(x)$; this is the logarithm to base 2. Probably the most common choice is base 10, because our number system is base-10. This makes it easier to calculate, because $10^1 = 10$, $10^2 = 100$, $10^3 = 1000$, etc. But there’s one choice of base which emerges naturally from mathematics itself, it doesn’t depend on using a base-10 number system (a fact which originates in having 10 fingers): base e. When we take the logarithm to base e, we call it the natural logarithm and instead of writing it as $\log_e$, we write it as $\ln$.

What makes base e so natural? All the exponential functions, for any base, have the property that the derivative of the function is proportional to the function itself:

${df \over dx} = c f$,

for some constant c. If we require the constant c to be equal to 1, then

${df \over dx} = f$.

This holds true only for one base: base e.

We can combine this property with Taylor’s Theorem to divine a series expansion for the exponent base e:

$e^x = 1 + x + \frac{1}{2}x^2 + \frac{1}{6}x^3 + ... + \frac{1}{n!}x^n + ... = \sum_{n=0}^\infty x^n / n!$.

The same fellow who posted the most terrifying video you’ll ever see on YouTube has also posted an 8-part series which, combined, constitute what he calls the most important video you’ll ever see. It’s about the exponential function, and its importance is outlining the behavior of systems with a constant growth rate. The whole series is a little over an hour, and is definitely worth viewing.

What’s so special about the exponential function in relation to growth? A constant growth rate r means that the rate of change (the growth) over time of a function is r times the value of the function

${df \over dt} = r f$.

This has the solution

$f(t) = A e^{rt}$,

where the constant A depends on the initial value (the value at time zero). Many phenomena in nature, and in human society, show a constant growth rate — so their growth over time follows an exponential function. In fact, our society and our economy is built on the notion that constant growth is the goal. If the economy doesn’t grow, but remains constant, it’s called “stagnant” and economists pronounce doom and gloom.

It seems to me that we’re rapidly approaching the point at which continued growth will be impossible, and sustainability should be the goal. The important, and sometimes frightening, thing about exponential growth is that the function gets very very large, very very quickly. Suppose, for example, that the world is consuming coal at a rate of 6.2 billion tons per year in 2006. Suppose further that world coal reserves amount to 909 billion tons. Then you might expect coal supplies to last about 909/6.2 = 147 years. But that’s only true if consumption is constant, it doesn’t grow.

Now suppose that coal consumption is growing at a rate of 5% per year. In that case, if $f(t)$ is coal consumption, we can say that roughly

${df \over dt} = 0.05 f$.

Then we know that

$f(t) = A e^{0.05 t}$.

If current consumption (we’ll let $t=0$ be the year 2006, so $t$ is years past 2006) is 6.2 billion tonnes/year, we have

$f(t) = 6.2 e^{0.05 t}$.

At this constant growth rate, here’s what coal consumption would look like for the remainder of the century:

Notice that the curve rises. Dramatically. That’s the nature of exponential growth.

The total usage from time zero (year 2006 in our example) to time $t$ is

$U = (A / r) [ e^{rt} - 1]$,

so if we have a total reserve $R$, that reserve will be exhausted in an amount of time

$t = {1 \over r} \ln ({rR \over A} + 1)$.

We can also graph the total usage since 2006, and the reserves remaining, for this example:

We can see (and compute as well), that with 5% growth per year in coal consumption, present reserves don’t last 147 years — they only last 42 years. That means if this case is followed, many of you reading this (who can expect to live more than another 42 years) will see then end of coal reserves on planet earth. Perhaps worse yet, all that coal, if burned, will add well over a trillion tonnes of CO2 to the atmosphere, making global warming a heckuva lot worse.

I didn’t pull these numbers out of thin air. Coal consumption in 2006 was 6.2 billion tonnes, and was rising at a rate of about 5% per year. Worldwide coal reserves are estimated at 909 billion tonnes. The U.S. coal industry is lobbying heavily to dispute global warming, so as not to inhibit coal consumption, and China is planning to expand their use of coal for energy generation greatly, in the near future. Much of the world looks to coal use as a hope for plentiful energy supply far into the future.

But the prospect, in my opinion, is a dreadful one. Not only will coal not be a plentiful supply far into the future — it’s not even likely to last a lifetime — the carbon emissions from coal power generation are sure to make greenhouse-gas pollution so bad as to drive us near to, or over, one of those “tipping points” in global climate that represents not only a terrible environmental consequence, but will also make it nearly impossible to halt global climate change.

All of this emphasizes the urgency of a sane, sustainable energy policy that doesn’t rely on fossil fuels but relies on renewable energy. It also illustrates the folly of the “growth imperative”; we must learn to abandon our attachment to growth and seek sustainability. If you live and work in a coal-rich state or nation, and depend on it for your livelihood, I sympathize. Truly I do. But I’ll let you know up front that I plan to do whatever I can to discourage the use of coal for power generation. If I, and others like me, succeed, it may well make your economic situation more difficult in the next decade. But rest assured that if we don’t succeed, it’ll make your children’s future vastly more difficult — maybe even impossible. The choice is ours: will we make the next generation pay an enormous price, so that we can enjoy a few years of prosperity?

Categories: Global Warming · climate change · mathematics

### 38 responses so far ↓

• John Mashey

If you haven’t read it, and you didn’t grow up in/near coal country, Jeff Goodell’s “Big Coal” is a good read, describing an industry that makes the oil industry look exceptionally enlightened (well, some of them actually are, sometimes). There are lots of people in Appalachian coal country for whom Big Coal is not necessarily so beneficial, even in short term.

Also, I continue to recommend David Strahan’s “The Last Oil Shock” as the other bookend, www.lastoilshock.com.

• Dear Tamino,

let’s go to the “core” of the problem. I think you know something about the R Dawkins book “The Selfish gene”.

Do You think, that people will do “something” (whatever it is) in order to reduce “own” comfort and favour the comfort of their children? Did this happen in the past? It there any reason “why” should people do this?

I give an example. Let’s have a young family. They want to have 3 childrens. You tell them, that this is not “sustainable” and it will probably cause global catastrophe in the future (50-100 years). Will they agree not to have 3, but only 1 children?

Why there is such furry in denying “undeniable”? I see the core of the problem in the short-benefit choises of most people (not only polititians).

To be more pessimistic, even if people are showing some will to solve the GW, when you ask then not to travel by plane, they will hardly do it… another example is carbon off-setting - they will rather pay money for carbon emitting than “not to emitt” CO2. Though I think carbon off-setting is not a solution - it’s even part of the problem - there is no way how to realy off-set fossil carbon. Our only hoe (I think) is to hope for rapid development (in effectivity) of alternative energy sources and massive subsidies into nuclear energy. National gouvernements can make some difference, but who will tell them??

• Alexander,

I’m not familiar with the book “the Selfish Gene.” But I am familiar with the tendencies, first, to place one’s one self-interest above the interest of others, and second, to remain blind to the future while considering the comfort of the present.

I’m also familiar with enough history to know that it *is* possible for people to show — en masse — a different side of human behavior. They can make incredible sacrifices, not only for others, but for future generations as well. And the rapid development and deployment of alternative (renewable) energy sources requires that kind of commitment. I am not at all convinced that nuclear energy is suitable.

I suspect that in terms of human behavior, you are something of a pessimist. I think I’m between a realist and an optimist. I think what’s really needed in order to polarize selfless behavior by masses of people is inspiring leadership. Unfortunately, in the United States that seems to be at an all-time low.

I recognize the tendency of humanity toward self-interest, but I also realize that selflessness is overwhelmingly more powerful than selfishness. I might suggest that you counterbalance a reading of “the Selfish Gene” with a viewing of the film, “Gandhi.”

• I recognize the tendency of humanity toward self-interest, but I also realize that selflessness is overwhelmingly more powerful than selfishness

Psychs and sociologists tend to group folk toward those with ’self-regarding’ and ‘other-regarding’ preferences. I tend to use the 80:20 rule here, with ’self-regarding’ being the 80.

Nonetheless, I agree with the assertion that selflessness is more powerful - “Never doubt that a small group of thoughtful, committed citizens can change the world. Indeed, it’s the only thing that ever has. ” — Margaret Mead

Best,

D

• george

Americans really need a leader who will tell them that it is simply not OK to keep wasting energy and resources the way that we have been for so long — someone who will actually make them feel shame for doing so.

It’s a national disgrace. Many developed countries (eg, Germany and Japan) have standards of living that are as high or higher than that in the US with significantly less (in many cases only half) per capita energy use and much higher rates of recycling.

That tells me that we could fairly easily be getting along on half the energy that we currently use, with half the emissions — with little or no drop in our standard of living.

That’s just a start, but significantly reducing our energy use is an absolute requirement if we are ever going to make the transition to alternative energy sources — which would allow us to make much deeper reductions in emissions.

A lot of this stuff would have a “snow-ball” effect, but you first have to get the snow-ball rolling — and so far, we have had nothing but foot-dragging from our leaders in that regard.

One thing is certain: business as usual ain’t going to cut it. Not even close.

• I would really like to know where the “penny saved is a penny earned” attitude went in the U.S. It seems that the idea of conserving, especially with regards to energy, is regarded as unnecessary, or worse, totally rejected/resisted as if consuming energy is our “prize” for being affluent. Thinking or talking about conserving energy is somehow equated to ‘freezing in the dark’. Is it simply because those who actually lived through the Great Depression are all but gone now?

I’m really hoping my generation can help shift the attitude back to being efficient and into conserving for the future. The goal of making energy as efficient as technologically possible could (and should) be a national and global ambition…which would drive advancements in science, engineering, and technology.

The fact that the populous can’t do this on their own, leads to people talking about gov’t having to regulate it, which leads to the idealogical political bickering.

The best way to not run out of oil is to use less of it.

• It may be possible, that I am kind of pessimist redarding the human bahaviour.

There also probably is true, that selflessness is more powerfull than selfishness.
OTOH, there is much more selfish people than selfless people - now indeed there is a question, if this is a matter of leadership - one might guess, what would happen, is Gore was elected instead of Bush.

Maybe another problem is that a lot of people is simply confused regarding the AGW a they simply don’t believe that people could have such an impact on climate, and is so, if this impact is realy bad.

At the moment, I am not confident in predicting the carbon emission in the future, though there is almost certainty that “globally” they will not decrease at least for the next 5-10 years… just hope we have enough time to taking action…

• nanny_govt_sucks

Humans always act in their own self-interest. That’s human nature. If you have the choice of doing what’s best for you and your family, or doing what’s best for some strangers, what are you going to choose? You’ll choose you and your family, of course. There’s nothing evil about this, it’s the way the world works. The problem is when we put people in positions where their self-interested choices harm us. Here’s a very good article at LewRockwell from yesterday on this very subject:
http://www.lewrockwell.com/dilorenzo/dilorenzo128.html

So what’s the solution? Let’s design a system where people’s self-interested choices help us instead of harm us. But we already have that in Capitalism. For more:
http://www.celebratecapitalism.org/bernsteindeclaration/english/index.html

• Humans always act in their own self-interest. That’s human nature.

No they don’t.

Folks often do, but not always, and not all folks.

Certain ideologies appeal to certain personality types, and my link above is to beaucoup studies showing certain ideologies’ base assumptions are wrong, if those who think about altruism haven’t said so already.

Best,

D

• Tamino,
Couldn’t one also interpret your trends as suggesting that the need for economic geologists will increase with time, as demand goes up and supply goes down?
Looks to me like yet another good reason to leave academia…

[Response: Only temporarily. We *are* going to run out of coal — and oil. The nature of exponential growth is such that in fact, even if the *entire planet* were made of coal, with consumption rising 5%/year it wouldn’t last 500 years.]

• san quintin

Hi Everyone.
On a (sort of) related topic…..what is the consensus view about the article from David Bellamy and Jack Barrett in the ICE? They claim that climate sensitivity is much lower than IPCC estimate, but seem to have misunderstood feedbacks and some of the details of the greenhouse effect. Any views?

• nanny_govt_sucks

Maybe we’ll be able to switch from coal to salt water:

http://www.wkyc.com/video/player.aspx?aid=42939

I’m constantly amazed at what’s left to discover and what power there is and what world changing events can occur when a guy is left alone to fiddle around in his garage or machine shop.

• I’ve actually wondered how long it will be until there is nowhere left to explore. Being in metals, I can sort of afford to watch petroleum, and extrapolate.

But if the world is committed to addicting itself to finite resources, what better position is there to be in than that of the dealer?

• BikerTrash

Can anyone list a single case for which such a simplistic analysis has been validated. Additionally, why not start the analysis at a past date and check its validity against actual real-world data. The cases for which failure of such simple approaches are well known and have been part of history for centuries. Finally, consider the extent of the efforts that would be necesasary in order that the consumption in the final years of exponential growth could be actually achieved; that would be tons o power plants. Or let me ask the question; how many power plants each producing 1000MWe (about 3300Mwt) would be needed to consume the projected concumption?

• guthrie

NGS- if you remember your school days, you might recall the laws of thermodynamics. The video you point us to is a pile of rubbish. Anyone can split off Hyrdogen from water by putting in lots of energy, you can do it at home. The tricky bit is finding a source of energy to do it in the first place. “Burning” salt water will not replace coal, because all he is doing is burning H2 obtained by breaking apart water, however this will yield less energy than that taken to split the H2 off from the water in the first place.

The days of amateurs in garages making important breakthroughs are long gone.

• Can anyone list a single case for which such a simplistic analysis has been validated. Additionally, why not start the analysis at a past date and check its validity against actual real-world data.

Oil Drum.

Best,

D

• oops. Close tag.

D

• nanny_govt_sucks

Guthrie, LabLemming,

I’m reminded of:

“Everything that can be invented - has already been invented” - AttributedCharles Duell, Commissioner of the United States Patent Office, 1899.

The saltwater burning thing is a discovery. Some say the most important discovery about water in the last 100 years. No one knew that H2 could be liberated in this way, and I think it’s premature to say that nothing will come of this new knowledge.

• Mike B.

“The saltwater burning thing is a discovery. Some say the most important discovery about water in the last 100 years. No one knew that H2 could be liberated in this way, and I think it’s premature to say that nothing will come of this new knowledge.”

Yes it’s a discovery, but it’s not a discovery of a new energy source. It’s a discovery of a new way of doing electrolysis on water. As explained above the energy required to split water into O2 and H2 will exceed the amount of energy produced by burning the hydrogen afterwards.

• Chris O'Neill

“Maybe we’ll be able to switch from coal to salt water”

or more precisely, maybe we’ll be able to switch from coal to salt water and an RF generator powered by a coal-burning power station.

I’m constantly amazed at how many credulous people there are.

• The saltwater burning thing is a discovery. Some say the most important discovery about water in the last 100 years. No one knew that H2 could be liberated in this way…/i> [mphasis added]

I’ve discovered that cat intestines are gooshy. Yay! Some Kindergartners and pre-schoolers say this is the most important discovery in their lifetimes.

Not only does The Google not have a “Wisdom” button, it doesn’t have a “Teach me basic physics” button.

IOW: na_gs, be careful - your cellphone is going to start boiling the blood in your ears any day now.

Best,

D

• What the heck is up with me and closing tags lately? Apologies.

The saltwater burning thing is a discovery. Some say the most important discovery about water in the last 100 years. No one knew that H2 could be liberated in this way… [emphasis added]

I’ve discovered that cat intestines are gooshy. Yay! Some Kindergartners and pre-schoolers say this is the most important discovery in their lifetimes.

Not only does The Google not have a “Wisdom” button, it doesn’t have a “Teach me basic physics” button.

IOW: na_gs, be careful - your cellphone is going to start boiling the blood in your ears any day now.

Best,

D

• Say, na_gs, I just had a thought:

I just had a discovery for a perpetual motion machine. No one will look at it, however.

Can you promote it for me on comment boards across The Internets?

Best,

D

• nanny_govt_sucks

It’s a discovery of a new way of doing electrolysis on water.

Well, it may or may not be electrolysis as we understand it. Maybe there are other ways to split the H2-O bond that we didn’t know about. Why such vitriol over a new discovery?

• guthrie

“The saltwater burning thing is a discovery. Some say the most important discovery about water in the last 100 years. No one knew that H2 could be liberated in this way, and I think it’s premature to say that nothing will come of this new knowledge.”

Sure, someone might make some fancy party lighting using it. It’s certainly a good party trick, however consideration of the basic physics and chemistry involved will show that it has fuck all use as a means of power generation. AS I have already said, you can’t get more energy out of such a system than you have put in in splitting the H2 off the O2 in the first place. Secondly, in all the hyping I have seen of it on the net, nobody seems to have bothered doing any efficiency calculations. Alright, not everyone has a physics degree, and my thermodynamics is a bit rusty, but a few calculations would show that it is taking in a huge amount of energy and not putting any out. Look at it this way- the reason it is glowing bright orange is because the electrons in the sodium atoms in the water are bouncing up and down energy levels, giving off light, the same way they do in street lights. This alone shows that large amounts of energy are being dissipated, i.e. lost, quite apart from the energy taken to split off the H2. You can already split off H2 yourself at home, we used to do it at school, but nobody every claimed it was a wonderful new energy source.

If you don’t believe me, I’m sure there must be some right wing college professors who will explain the science to you, or some defense department weapons engineers or someone you trust, who will explain to you why it doesn’t work.

• nanny_govt_sucks

Where did I claim it was a new energy source? You all need to relax and have a beer.

What fascinates me is discovery. When/if a theory comes about to describe this phenomenon there will undoubtably be applications of that theory that we can’t even imagine right now. Just think of where we were before we began to understand coherent light.

• This is making me nostalgic for the days of cold fusion.

• Where did I claim it was a new energy source? You all need to relax and have a beer.

Um…scroll wheel don’t fail me now:

Maybe we’ll be able to switch from coal to ,a href=”http://tamino.wordpress.com/2007/09/11/e/#comment-5103″>salt water

That’s standard na_gs: misdirection type #4a, I believe. Maybe distraction #3c(2)(i). The tap dance pattern is certainly #2.

Best,

D

• Sigh…

Where did I claim it was a new energy source? You all need to relax and have a beer.

Um…scroll wheel don’t fail me now:

Maybe we’ll be able to switch from coal to salt water

That’s standard na_gs: misdirection type #4a, I believe. Maybe distraction #3c(2)(i). The tap dance pattern is certainly #2.

Best,

D

• elspi

NGS giving the AG a run for his money in the early onset Altheimers category.

“Maybe we’ll be able to switch from coal to salt water:”

Followed by

“Where did I claim it was a new energy source?”

• guthrie

Umm, NGS, I think we’re all trying to point out that this is a well understood phenomenon with obvious limits. You put energy into water, hey presto, Hydrogen comes out.

• BikerTrash

So called “models” that have not been validated by comparison with data are always denoted to be useless.

• guthrie

OK, I exagerated somewhat with the “well understood” statement. Nevertheless, this is not exactly a breakthrough on the order of making lasers, and appealing to chance and serendipity for lauding something to the skies is silly. I shall also remind you that you said:

“Maybe we’ll be able to switch from coal to salt water”

which is basically a claim that it may well be a new energy source!

• People have been running RF discharges forever. This is neither new nor particularly interesting.

Here are a couple of recent links
http://www.jstage.jst.go.jp/article/cl/36/7/36_870/_article

• nanny_govt_sucks

Sorry all, but “Maybe … ” anything doesn’t result in a claim. Perhaps it would have helped if I put a smiley face after that sentence. Folks, have a beer and chill for a while.

• guthrie

Yup, smiley’s are good.

• Ah.