Popular Science Feature - Dry Number Line

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Feature - Keeping the Number Line Dry (A paradox of infinity) - Brian Clegg

One of the great delights of infinity is its ability to throw up mind-bending paradoxes. This is about the best I've ever come across. One warning - the wording assumes you know some basic facts about infinity - it would help to read A Brief History of Infinity or a similar book first.

We start by thinking of the number line - let's say for simplicity, the numbers from 0 upwards. So we've got a line, rather like the edge of a ruler, starting from zero and heading off to infinity, featuring all the numbers and fractions along its length.

Now we know the rational fractions (n/m where n and m are whole numbers) have the same cardinality as the integers, thanks to Cantor's proof. And we're going to use another set of fractions alongside them - the sequence 1/2, 1/4, 1/8, 1/16... It is simple enough that these also have the same cardinality. And to show that the sum of the whole series is just 1. With these 'given's the fun begins.

Imagine we wanted to protect the whole number line from getting wet. What we are going to do is issue each rational fraction along the line an umbrella. The umbrella will be a simple T shape. The first umbrella we give out is 1/2 a unit of the number line across the T. The second umbrella is 1/4 of a unit of the number line across and so on. Once every rational fraction has an umbrella, it seems that the whole number line is covered. The umbrella extends half its width in either direction - so, for instance, the first umbrella will cover all numbers for 1/4 of a unit to its left and 1/4 of a unit to its right. Note that this is a rational fraction - and adding it to or subtracting it from the starting point (itself a rational fraction) will reach another rational fraction.

Okay so far? Each umbrella spans from its starting point to a rational fraction on either side of it. Now bearing in mind we've issued an umbrella to each rational fraction, the whole number line is covered, because there's at least a meeting of umbrellas and in most cases an overlap.

We've covered the whole line from 0 to infinity with our umbrellas. But, remember how wide the umbrellas were. Their widths form the infinite series 1/2+1/4+1/8... so with no overlaps, the maximum amount of the number line those umbrellas can cover is 1 unit - and with overlaps they will cover even less. A set of items with a width of just 1 covers a line that goes all the way to infinity.

Spooky!

Many thanks to Neil Sheldon of the Manchester Grammar School for making me aware of this paradox - and for exploring further complications I haven't covered above.

Brian Clegg is a popular science writer whose books include A Brief History of Infinity and Light Years.

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