All My Exes

It seems obvious that human beings can do many things bacteria can’t. But have you ever thought about what bacteria can do that human beings can’t? With “binary fission,” they make perfect copies of themselves. Human beings obviously don’t have this skill, although the tiny mitochondria in each of our cells do. But that’s a topic for another time . . . right now we’re discussing my exes. (You do realize I’m talking about exponents, right?)

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 30th 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?

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