Bacteria duplicate everything in their diminutive bodies and then split in two. When they’ve grown a bit, each of these two divides again. In a short time – sometimes as short as 20 minutes – there are four bacteria where one used to be. In another 20 minutes, there are 8, then 16, then 32, then 64, and so on. If we followed one of these fast-splitting bacteria for one full day, we would have over 4,722,366,482,870,000,000,000 bacteria where one used to be. No wonder there’s so much Lysol on supermarket shelves!

This kind of growth has a lot to do with exponents, which is why, not surprisingly, it is called “exponential growth”. It happens when each critter duplicates itself independent of others. At the end of our bacterium’s first split (Generation 1), there are 2 bacteria. At the end of the second split (Generation 2), there are 4. In mathematics, this can be shown as follows:

2=2

2·2=4

2·2·2=8

2·2·2·2=16

2·2·2·2·2=32

If we keep going like this, by the end of the day, we would need a lot of 2’s to express how many bacteria we have (we would need 72, to be exact). But there is an easier way – exponents:

2¹=2

2²=4

2³=8

2⁴=16

2⁵=32

Now isn’t that better? The raised number tells us how many times the normal number (in this case, 2) is multiplied by itself.  The normal number is called the “base” and the raised number is called the “exponent.” It is interesting to consider that if bacteria split into three instead of two – they don’t ever do this on Earth, but what if they did triplicate on some distant alien moon orbiting, say, Spica – then we could use exponents in the same way to calculate how fast the bizarre Spican bacteria population grows. We’d just need to use a base of 3 instead of 2.

There is a well-known riddle that goes something like this: you rescue a wealthy hedge fund manager from certain death. In return, he gives you a choice of rewards. You may take one million dollars now, or you may take one penny. However, every day for the next month, he will double the value of the penny and give you that amount the next day. He hopes you take the million dollars, because he is a hedge fund manager, after all, and he is careful with his money. But you are acquainted with my exes, so you opt for the penny, which you know to be a penny of exponential growth.

After the first week, you begin to question your wisdom. You have a grand total of $1.27. and tomorrow, you’ll be looking forward to adding $1.28 to that sum. Here’s how your week went:

Total

Day 0 = $ 0.01                        $ 0.01

Day 1 = $ 0.02                        $ 0.03

Day 2 = $ 0.04                        $ 0.07

Day 3 = $ 0.08                        $ 0.15

Day 4 = $ 0.16                        $ 0.31

Day 5 = $ 0.32                        $ 0.63

Day 6 = $ 0.64                        $ 1.27

The hedge fund manager brings you a double latte with sprinkles, and he gives you the option to change your mind. After all, the coffee drink costs a lot more than what you received on Day 6. He points out with a smile that in two more days, you can get a double latte for yourself, and in three days, you might even be able to afford buying him one, too.

“No thank you,” you answer, noting the twinkle in his eye. You’re thinking that you trust math more than hedge fund managers, and you are probably wise to do so. This is how the second week goes:

Total

Day 7 =   $ 1.28            $ 2.55

Day 8 =   $ 2.56             $ 5.11

Day 9 =   $ 5.12            $ 10.23

Day 10 = $ 10.24            $ 20.47

Day 11 = $ 20.48            $ 40.95

Day 12 = $ 40.96            $ 163.83

Day 13 = $ 163.84            $ 327.67

So you are halfway through the month, and you only have a little over three hundred dollars in your pocket. This is still a long way from the million. Should you accept the hedge fund manager’s offer on Day 13, when he drops by your office with a double cherry cheesecake and another chance to change your mind?

You decide against it. You look down hopefully at your list of earnings and notice a few things. First, you notice that your daily total is always one penny less than the amount the hedge fund manager will give you the next day. You also notice that if you use the number of the day as an exponent of 2, the result will tell you how much you will be paid (in pennies) on that day. For example, for Day 7, 2⁷ = 2 · 2 · 2 · 2 · 2 · 2 · 2 = 128. And 128 pennies is, of course, $1.28.

With a grin, you hunt down your calculator from that drawer in the kitchen – you know which drawer – everyone has one like it. Quickly, you calculate the value 2³⁰ (because it is April, and April has 30 days). You are stunned to find that on the 30^th of April, the hedge fund manager will pay you $10,737,418.24, leaving you with a grand total of $21,474,836.48. Over twenty-one-million dollars! That’s better than Lotto!

Aren’t you glad my exes are on your side?

That moment happens, and as we expect, the rabbit sprints away, flashing its cottony tail. It runs in a line so straight it would make Archimedes proud. It streaks away as fast as it possibly can, grey-flecked sides heaving. But the coyote, of course, is speedy too.

Does the rabbit make it, or does the coyote get some fast food?

If I told you the rabbit made it, would you believe me? It’s reasonable. How, then, would you explain this puzzle? In the process of running between the sagebrush and the rabbit hole, the rabbit passes a point exactly halfway between the two. And then it passes another point exactly half of the remaining distance. Then half again, again, again. Wouldn’t this process of cutting the remaining distance in half go on forever because there is always a distance, no matter how small, to cut in half? Wouldn’t that mean the rabbit never reaches the hole?

So the coyote must catch the rabbit. It’s reasonable. Unfortunately, for the same reason, the coyote can’t reach the rabbit either: there is always a distance between the coyote and the rabbit to divide in half. So how can anything ever go anywhere in this world and actually arrive there???

What if the rabbit was heading for a point twice the distance to its rabbit hole? In this case it would drop down the hole at the first halfway point, and it would get away for sure! All anything would have to do is aim for twice the distance it actually means to cover. But this isn’t reasonable. No rabbit really does this. Besides, no matter how you double or triple things, there is still that distance between the sagebrush and the rabbit hole that can be divided in half infinitely.

In this puzzle, the cutting in half is an infinite process, and infinity has no end. But what if there are some kinds of infinity that actually do “end”? What if the coyote and rabbit puzzle is this kind of infinity?

Take a look at this classic math proof: as you may know, one way to express nine tenths is 0.9; ninety-nine one-hundredths is 0.99. The more nines you add, the closer the value gets to 1. If the 9’s were to continue forever, we would call it “.9 repeating”, which, we may suppose, is still almost, but not quite, equal to 1.

But we would suppose wrong. It’s not almost equal to 1; it’s exactly equal to 1.

If we started with ten chocolate kisses and took away one chocolate kiss, we would be left with nine chocolate kisses, right? In the same way, if we had ten “.9 repeatings” and took away one , we would be left with nine “.9 repeatings”. If we could show that those nine “.9 repeatings” (all taken together) were exactly equal to 9, wouldn’t each “.9 repeating” have to be exactly equal to 1?

If you like to play with numbers, here is the proof:

10  x  ”.9 repeating” = 9.9 …

- 1  x   “.9 repeating”= 0.9 …

9   x   “.9 repeating”  =  9

Therefore, “.9 repeating” = 1

So, why do some infinite series of numbers have exact values like this one? It’s because the digits repeat. If the digits didn’t repeat in a regular pattern infinitely, they wouldn’t have an exact value. Oddly enough, these infinitely repeating numbers are called “rational,” even though they seem to defy reason. (Infinite numbers that don’t repeat are called irrational; it’s ironic, because you would expect an infinite number not to have an exact value, wouldn’t you?)

If you don’t like to play with numbers, think about this:  ”.9 repeating” is exactly equal to 1 precisely because the 9’s are infinite, precisely because the nines don’t end at any point short of one. And no matter how much we divide the distance, the rabbit actually doesn’t stop at any point short of its rabbit hole either.

So, when is there an end to infinity? It turns out to be pretty often!

Set that pie you brought on the table and go find a knife. Cut your pie in half, starting at one edge. Cut right through the middle. This is an exact distance, exactly one diameter.

OK, put your knife down and trace your finger around the outside of the pie. Don’t ask why. This will soon make sense. Trace a full circle, stopping where you began. Like the knife, your finger traced an exact distance: three diameters plus a little more. That three and a little more is pi. Exactly pi. Exactly.

But mathematics has never seen pi calculated to its exact value, never. Supercomputers keep trying, but can’t get there. 3.1415926535 plus a million digits is just an approximation. The digits ramble on into infinity. Contemplate that as you lift a slice of your pie onto a party plate, preferably at 1:59 and 26 seconds, the most significant moment of the day.

What kind of pie did you bring? Coconut cream? Blueberry? Peach or pecan? Hold on. Don’t take a bite just yet. There will be plenty of time for dessert when the time comes. For now, use your fork, or the knife if it’s easier, to trim off the curved part of the crust into a straight line, making your pie piece into a neat little triangle. Why do this? Because now you can measure the trimmed side exactly. If you cut your pie into eight equal pieces, trim all the crusts, measure the lines where you sliced off the crusts and add the measurements up, you would come pretty close to pi. If you cut sixteen pieces instead, you’d come even closer because you’d be trimming off less total crust. Over 1700 years ago, Liu Hui cut his pie into 3072 pieces and measured pi at 3.14159. OK, they didn’t really have pie in ancient China. But he did imagine a circle cut that way and calculated five digits out without a mistake.

Very recently, the known value of pi was extended to 2.7 trillion digits by one Fabrice Bellard of France. Bellard was careful to note that this feat was performed on a relatively inexpensive PC.

How can pi be exact and infinite at the same time? The circle’s built-in enigma, this relationship between its diameter and circumference, contributes to its mystery. Circles are symbols to so many cultures: the circle of seasons, a whirling dervish, the sacred hoop, a wedding band, the horizon from a tall mast at sea. The path around a circle is infinite, but a circle sharply separates the inside from the outside, clear as can be.

Think about these things as you take that well-earned bite of pie. As the sweet filling oozes over your teeth, give a little nod to Mr. William Jones. He is the one who first gave pi a name in 1706. It’s a good thing, because otherwise, every March fourteenth, we’d all have to celebrate “The Quantity Which, When the Diameter is Multiplied by it, Gives the Circumference” Day, and nobody would quite know what to bring to the party.

(copyright 2010 by Linda Sheader)