proposed a number of paradoxes describing supposed difficulties with

the division of time and space into infinitely small increments.

Here's an example you've probably heard in many forms before: Suppose

you want to cross a road. Before you get across the road, you must

get halfway across the road. But before you cross the second half, you

must get halfway across that. And so on: there are an infinite

number of halfway points as you get closer and closer to the other

side, so therefore you can never reach it. The total distance you

need to walk is 1/2 + 1/4 + 1/8 + ..., with an infinite total number

of terms in the series.

Some philosophers still talk about this paradox, but in a

mathematical sense, it's pretty well resolved. One concept that Zeno

may not have been aware of, though you probably were taught at some

point in high school algebra, is that an infinite series of

geometrically decreasing elements can indeed add up to an easily

calculated finite sum. As a quick refresher, let S equal the sum

discussed above, 1/2 + 1/4 + 1/8 +... If you multiply S by 1/2, you

will see that you have the same series, except with the first term,

1/2, missing. So S = 1/2 + 1/2S, and solving for S, we get S = 1.

Of course, the fact that we can indeed cross roads also proves to

refute Zeno's paradox, and in fact is not something that we find very

exciting. But this ability to sum up an infinite series can have

interesting consequences when we look at another area: population

sizes. As you have probably heard, many modern countries have

fertility rates below replacement: that means that the number of

people is declining every generation. Let's take Russia as an

example. Their estimated fertility rate in 2007 is 1.39 children per

couple, which means that for every person in the current generation,

there will be about 0.7 people in the next generation. That means

that the ratio of the total number of children that will ever be born

in Russia, compared to the current population, is given by the series

1 + .7 + .7 squared + .7 cubed ..., which by the same method we used

before, sums up to about 3.33.

This is pretty scary if you think about it-- it means that even

after an infinite number of years, at the current fertility rates, the

total number of people that will *ever* have existed in Russia is a

little over 3 times the current population. It's actually a bit less

than that, of course, since after the terms become smaller than whole

numbers, you can't really keep summing up fractional children. Now we

can understand those wacky stories about Russian baby-producing

contests that have been in the media recently. On the other hand,

there's nothing unique about Russia: this issue of declining

population is striking large portions of the Western world. Perhaps

soon other Western countries will emulate Russia's survival strategy,

and baby-producing will be the top event at the 2012 Olympics.

And this has been your math mutation for today.

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