# 'e' is not a rational number

## an online sermon about population growth

### by Sister Teresa

While visiting the Museum of Science recently, I was dismayed to see the numbers on the world population clock. It read over 5.9 billion with the numbers in the ones and tens column changing every few seconds.. It seems like only yesterday I remember reading about the five billionth human being born on this small planet.

Being a child of the sixties, I remember the world's population presented as being three billion in grade school. That means that the world population has nearly doubled since I was born. If present trends continue, by the time I reach the end of my probable lifespan the population will have nearly doubled again. Barring accident or catastrophic illness most of you here will be alive then too. Over eleven billion humans, all competing for the same limited space, and dwindling resources. Think of it!

Which brings us to the topic of my sermon today: The humble little number known as 'e', more than two, less than three. Like Pi and the square root of two, it is an irrational number, which means it can't be expressed as a ratio between two integers. There are many interesting and fun ways to calculate 'e' and "pi":

and like Pi, you can calculate 'e' out to as many decimal places as you like. But for our purposes we will represent it as approximatly 2.718282 Now 'e' is a very special number. It is the base of a handy little function known as the natural exponential. It governs many processes: from natural ones such chemical metabolism, radioactive decay and population growth, to human processes like savings accounts and loans. To put things in sinple terms: Any quantity which increases by a constant fraction during a fixed amount of time can be said to grow exponentially.

Examples:
Fidelity Contrafund             ~20% per year
Student loans                    ~5-8% per year
World Population                ~1.5% per year

"But the world population growth rate is small", you might think. But is it? When I wrote this, the world population was 5.983,986,796 (it's a lot more now, see for yourself here ). Now, let's multiply that by 1.5% or 0.015 to see what we get... HOLY MOLY!! 89,759,802 nearly ninety million people per year. The population of a country the size of Mexico added to the world every year. Birth of a nation indeed!!

But next year we'll add more, because growth will be 1.5% of what has already increased by 1.5%. and the next year, even more than that. It's a phenomenon bankers refer to as compound interest. But unlike savings accounts where interest may be compounded quarterly, monthly or even daily, exponential growth is interest compounded  continously .

The equation for this type of growth is:

It allows us to relate the growth rate (represented by the greek letter 'lambda' here) to another important quantity known as the doubling time (represented by Tau sub-D)

The doubling time is -you guessed it- the amount of time it takes for a quantity to double in size. Without getting too deeply into the math (too late!!) we can say the doubling time of a quantity is equal to natural log (the inverse of the exponential, or ln) of 2, which is nearly 0.693, divided by the growth rate. This means that that with a 1.5% growth rate, the world population can be expected to double every 46 years.

Let's consider the implications of the doubling time aspect of exponential growth for just a moment. For a given growth rate it is constant. This means that during every doubling time period, a quantity grows as much as it did in all the previous time before it. This is worth repeating: During every doubling time period, a quantity grows as much as it did in all the previous time before it. Say you are a biologist with a jar of bacteria, cells multiplying with a doubling time of two days. Say after week or even a month the culture has grown to fill the specimen jar. No problem, you say, I'll just give them another jar. Now given free access between the two jars does anyone know how long it will take the culture to fill a second jar of equal size?? That's right, two days. And the kicker is, it doesnt matter how big the jar is or how long it takes for the culture to fill it, it will always take only two days for it to fill a second of equal size, and even less to fill a third..

In the last 46 years, our human race has added as many people as we did in all the years previously. In other words. as long as humans have been identifiably human. In another 46 years we will do it again, and after that again, or will we? As we turn more and more acres of wilderness into strip malls and Wal-Marts, as cities grow like malignant tumors on the face of the continents, what happens to what's left? Since this is a finite world, one thing cannot grow without taking from something else, so what is left, becomes less with each passing year. In recent years hundreds of animal species have been wiped out by humanity. In the next 100 years, one-half of the remaining species of animals are likely to be gone too, all sacrificed on the altar of human procreation and "economic growth".
Gone.
Forever.
End of story.

But rest assured, my friend, all is not lost. It is not the fate of the Earth to become one seething, teeming mass of humanity. No, nature will not allow that to happen. The Earth's ecosystem is robust, hardy, full of checks and balances to keep itself alive. (the conservatives are actually right about this one) As humanity grows and inevitably wipes out its fellow creatures, it destroys its natural benefactors as well as its natural enemies. The natural balance upset, resources exhausted: disease, famine, and perhaps human extinction itself will finally bring this unbridled growth to a halt. After many milennia and perhaps eons, new species and ecosystems will arise to replace those that we have destroyed. Yes, Earth will eventually recover and be restored to its natural beauty. Too bad none of us will be alive to see it.

So you see, we have two choices. We can take drastic steps now to curb our population growth, or we can wait and let nature do it for us. By forgoing reproduction, I've done my part. The rest is up to you.

Remember, if we don't stabilize population....

None of what you do
None of what you learn in school
None of your good intentions
Nothing, Will Make One bit of Difference at All